Understanding Arithmetic Circuits

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  <page>
    <title>Understanding Arithmetic Circuits</title>
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      <parentid>2688876</parentid>
      <timestamp>2024-11-28T14:54:59Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20241127.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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      <parentid>2689190</parentid>
      <timestamp>2024-11-28T14:58:04Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20241128.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2689188</id>
      <parentid>2688890</parentid>
      <timestamp>2024-11-28T14:53:47Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2689188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5241" sha1="2df1wium96dduwh7tzhsve5rr7mxxve" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241127.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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    </revision>
    <revision>
      <id>2689194</id>
      <parentid>2689188</parentid>
      <timestamp>2024-11-28T14:57:32Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2689194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5241" sha1="nf28g2odawduaxx9ivl7lqxwmmhiw66" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241128.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
      <sha1>nf28g2odawduaxx9ivl7lqxwmmhiw66</sha1>
    </revision>
  </page>
  <page>
    <title>User:ThaniosAkro/sandbox</title>
    <ns>2</ns>
    <id>219126</id>
    <revision>
      <id>2689203</id>
      <parentid>2689183</parentid>
      <timestamp>2024-11-28T15:32:49Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Implementation */</comment>
      <origin>2689203</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71849" sha1="el92en5vsi0aslx2cwcv4o9gdlnu6lf" xml:space="preserve">= Hyperbola =
{{RoundBoxTop|theme=2}}
[[File:0911hyperbola00.png|thumb|400px|'''
Figure 1: Hyperbola at origin with transverse axis horizontal.''' &lt;/br&gt;&lt;/br&gt;

Origin at point &lt;math&gt;O&lt;/math&gt;&lt;math&gt;: (0,0)&lt;/math&gt;.&lt;/br&gt;
Foci are points &lt;math&gt;F_1 (-c,0),\ F_2 (c,0). OF_1 = OF_2 = c.&lt;/math&gt;&lt;/br&gt;
Vertices are points &lt;math&gt;V_1 (-a,0),\ V_2 (a,0). OV_1 = OV_2 = a.&lt;/math&gt;&lt;/br&gt;
Line segment &lt;math&gt;V_1OV_2&lt;/math&gt; is the &lt;math&gt;transverse\ axis.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;PF_1 - PF_2 = 2a.&lt;/math&gt;
]]

In cartesian [[geometry]] in two dimensions hyperbola is locus of a point &lt;math&gt;P&lt;/math&gt; that moves relative to two fixed points called &lt;math&gt;foci&lt;/math&gt;&lt;math&gt;: F_1, F_2.&lt;/math&gt;
The distance &lt;math&gt;F_1 F_2&lt;/math&gt; from one &lt;math&gt;focus\ (F_1)&lt;/math&gt; to the other &lt;math&gt;focus\ (F_2)&lt;/math&gt; is non-zero. The absolute difference of the distances &lt;math&gt;(PF_1, PF_2)&lt;/math&gt; from point to foci is constant.


&lt;math&gt;PF_1 - PF_2 = K.&lt;/math&gt; See figure 1. 


Center of hyperbola is located at the origin &lt;math&gt;O (0,0)&lt;/math&gt; and the foci &lt;math&gt;(F_1, F_2)&lt;/math&gt; are on the &lt;math&gt;X\ axis&lt;/math&gt;
at distance &lt;math&gt;c&lt;/math&gt; from &lt;math&gt;O. &lt;/math&gt;


&lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(-c, 0). F_2&lt;/math&gt; has coordinates &lt;math&gt;(c,0)&lt;/math&gt;. Line segments &lt;math&gt;OF_1 = OF_2 = c.&lt;/math&gt;


Each point &lt;math&gt;(V_1,V_2)&lt;/math&gt; where the curve intersects the transverse axis is called a &lt;math&gt;vertex.\ V_1,V_2&lt;/math&gt; are the vertices of the ellipse.


By definition &lt;math&gt;PF_1 - PF_2 = V_2F_1 - V_2F_2 = V_1F_2 - V_1F_1 = K.&lt;/math&gt;


&lt;math&gt;\therefore V_2F_1 - V_2F_2 = V_2F_1 - V_1F_1 = V_1V_2 = K = 2a,&lt;/math&gt; the length of the
&lt;math&gt;transverse\ axis\ (V_1V_2).&lt;/math&gt;
&lt;math&gt;OV_1 = OV_2 = a.&lt;/math&gt;
{{RoundBoxBottom}}
==Radians, the natural angle==
If you were a mathematician among the ancient Sumerians of the 3rd millennium BC and you were determined to define 
the angle that could be adopted as a standard to be used by all users of trigonometry, you would probably suggest
the angle in an equilateral triangle.
This angle is easily defined, easily constructed, easily understood and easily reproduced. It would be easy to call
this angle the &quot;natural&quot; angle.

The numeral system used by the ancient Sumerians was Sexagesimal, also known as base 60, a numeral system with sixty as its base.
In practice the natural angle could be divided into 60 parts, now called degrees, and each degree could be divided into
60 parts, now called minutes, and so on.

Three equilateral triangles fit neatly into a semi-circle, hence 180 degrees in a semi-circle.

We know that &lt;math&gt;\tan 30^\circ = \frac{\sqrt{3}}{3}.&lt;/math&gt;
Therefore, &lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; should be &lt;math&gt;0.5,&lt;/math&gt; or one half of our concept of the natural angle.

Whatever the natural angle might be, it has existed for billions of years, but it has come to light only in recent times
with invention of the calculus.

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:

&lt;math&gt;\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + \cdots.&lt;/math&gt;

The following python code calculates 
&lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; using Gregory's series:
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

r3 = 3 ** .5
x = r3/3

arctan_x = (
      x        -  x**3/3  +  x**5/5  -  x**7/7  +  x**9/9  - x**11/11 + x**13/13 - x**15/15 + x**17/17 - x**19/19
    + x**21/21 - x**23/23 + x**25/25 - x**27/27 + x**29/29 - x**31/31 + x**33/33 - x**35/35 + x**37/37 - x**39/39
    + x**41/41 - x**43/43 + x**45/45 - x**47/47 + x**49/49 - x**51/51 + x**53/53 - x**55/55 + x**57/57 - x**59/59
    + x**61/61 - x**63/63 + x**65/65 - x**67/67 + x**69/69
)

sx = 'arctan_x' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x = 0.5235987755982988
&lt;/syntaxhighlight&gt;
Our assessment of the natural angle as the angle in an equilateral triangle was a very reasonable guess.

However, the natural angle is the radian, the angle that subtends an arc on the circumference of a circle equal to the radius.

Six times arctan_x &lt;math&gt;= 180^\circ&lt;/math&gt; or the number of radians in a semi-circle:

&lt;syntaxhighlight lang=python&gt;
# python code

sx = 'arctan_x * 6' ; print (sx, '=', eval(sx))
sx = '180/(arctan_x * 6)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x * 6 = 3.141592653589793
180/(arctan_x * 6) = 57.29577951308232
&lt;/syntaxhighlight&gt;
&lt;math&gt;\pi = 3.141592653589793\dots,&lt;/math&gt; number of radians in semi-circle.

One radian &lt;math&gt;= 57.29577951308232^\circ,&lt;/math&gt; slightly less than &lt;math&gt;60^\circ.&lt;/math&gt;


Because the value &lt;math&gt;\frac\sqrt{3}{3}&lt;/math&gt; is fairly large, calculation of &lt;code&gt;arctan_x&lt;/code&gt; above required 34 operations
to produce result accurate to 16 places of decimals. The calculation did not converge quickly.

Python code below uses much smaller values of &lt;math&gt;x&lt;/math&gt; and calculation of &lt;code&gt;arctan_x&lt;/code&gt; for precision of 1001 is quite fast.

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

==tan(A/2)==

{{RoundBoxTop|theme=2}}
[[File:1122tanA_200.png|thumb|400px|'''Graphical calculation of &lt;math&gt;\tan \frac{A}{2}&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;OQ = 1;\ QP = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\tan(A) = \frac{QP}{OQ} = \frac{t}{1} = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;OP = OR = \sqrt{1 + t^2}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Point &lt;math&gt;P&lt;/math&gt; has coordinates &lt;math&gt;(1,t).&lt;/math&gt;

Point &lt;math&gt;R&lt;/math&gt; has coordinates &lt;math&gt;(\sqrt{1 + t^2},0).&lt;/math&gt;

Mid point of &lt;math&gt;PR,\ M&lt;/math&gt; has coordinates &lt;math&gt;(   \frac{ 1 +  \sqrt{1 + t^2}  }{2}, \frac{t}{2}            ).&lt;/math&gt;

&lt;math&gt;\tan \frac{A}{2} = \frac{t}{2} / \frac{ 1 +  \sqrt{1 + t^2}  }{2} = \frac{t}{1 +  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t}{1 +  \sqrt{1 + t^2} } \cdot \frac{1 -  \sqrt{1 + t^2}}{1 -  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{1-(1+t^2)}&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{-t^2}&lt;/math&gt;
&lt;math&gt;= \frac{-1 +  \sqrt{1 + t^2} }{t}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


* &lt;math&gt;\tan \frac{A}{2} = \frac{-1 +  \sqrt{1 + \tan^2 (A)} }{\tan (A)}&lt;/math&gt;

* &lt;math&gt;\tan (2A) = \frac{2\tan (A)}{ 1 - \tan^2 (A) }&lt;/math&gt;
{{RoundBoxBottom}}

==Implementation==
{{RoundBoxTop|theme=2}}
This section calculates five values of &lt;math&gt;\pi&lt;/math&gt; using the following known values of &lt;math&gt;\tan(A):&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;A&lt;/math&gt; || &lt;math&gt;\tan(A)&lt;/math&gt;
|-
| &lt;math&gt;45^\circ&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| &lt;math&gt;36^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{   5 - 2\sqrt{5}  }&lt;/math&gt;
|-
| &lt;math&gt;30^\circ&lt;/math&gt;
| &lt;math&gt;\frac{\sqrt{3}}{3}&lt;/math&gt;
|-
| &lt;math&gt;27^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{    11 - 4\sqrt{5} + (\sqrt{5} - 3)   \sqrt{   10 - 2\sqrt{5}      }   }&lt;/math&gt;
|-
| &lt;math&gt;24^\circ&lt;/math&gt;
| &lt;math&gt;\frac{       (3\sqrt{5} + 7)     \sqrt{5 - 2\sqrt{5}} - (\sqrt{5} + 3)\sqrt{3}     }{2}&lt;/math&gt;
|}

Values of &lt;math&gt;x&lt;/math&gt; in table below are derived from the above values by using identity &lt;math&gt;\tan(\frac{A}{2}) = \frac{-1 + \sqrt{1 + \tan^2(A)}}{\tan(A)}&lt;/math&gt;:

{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;\theta&lt;/math&gt; || &lt;math&gt;x = \tan(\theta)&lt;/math&gt;
|-
| &lt;math&gt;\frac{45^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_91432_37995_4197.....089_03901_63759_3912&lt;/code&gt;
|-
| &lt;math&gt;\frac{36^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_73145_90396_3357.....211_97500_56173_0713&lt;/code&gt;
|-
| &lt;math&gt;\frac{30^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_60954_91996_9464.....024_32806_94580_0689&lt;/code&gt;
|-
| &lt;math&gt;\frac{27^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_54859_42797_2518.....791_30634_03540_9738&lt;/code&gt;
|-
| &lt;math&gt;\frac{24^\circ}{2^{32}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_97527_87195_1143.....736_60376_04724_6778&lt;/code&gt;
|}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

desired_precision = 1001
number_of_leading_zeroes = 10 # See below.

import decimal
dD = decimal.Decimal # decimal object is like float with (almost) infinite precision.
dgt = decimal.getcontext()

Precision = dgt.prec = desired_precision + 3  # Adjust as necessary.
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))      # Adjust as necessary.
adjustment_to_precision = number_of_leading_zeroes * 2 + 3

def tan_halfA(tan_A) :
    dgt.prec += adjustment_to_precision
    top = -1 + (1+tan_A**2).sqrt()
    dgt.prec -= adjustment_to_precision
    tan_A_2 = top/tan_A
    return tan_A_2
    
def tan_2A (tanA) :
    '''
          2 * tanA
tan(2A) = -----------
          1 - tanA**2
    '''
    if tanA in (1,-1) : return '1/0'
    dgt.prec += adjustment_to_precision
    bottom = (1 - tanA**2)
    output = 2*tanA/bottom
    dgt.prec -= adjustment_to_precision
    return output+0

def θ_tanθ_from_A_tanA (angleA, tanA) :
    '''
if input == 45,1
output is:
&quot;dD(45) / (2 ** (33))&quot;, &quot;0.00000_00000_91432_37995_....._63759_3912&quot;
                           ^^^^^^^^^^^
number_of_leading_zeroes refers to these zeroes.
θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA)
    '''
    θ, tanθ = angleA, tanA
    for p in range (1,100) :
        θ /= 2
        tanθ = tan_halfA(tanθ)
        if tanθ &gt;= dD('1e-' + str(number_of_leading_zeroes)) : continue
        str1 = str(tanθ)
        # str1 = &quot;n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11&quot;
        str1a = str1[0] + str1[2:-4]
        list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ]
        str2 = '0.00000_00000_' + ('_'.join(list1))
        dD2 = dD(str2)
        (dD2 == tanθ) or ({}[2])
        ((θ * (2**p)) == angleA ) or ({}[3])
        str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p)
        (θ == eval(str3)) or ({}[4])
        return str3, str2
    ({}[5])


r3 = dD(3).sqrt()
r5 = dD(5).sqrt()
tan36 = (5 - 2*r5).sqrt()
tan45 = dD(1)
tan30 = r3/3

v1 = 3*r5+7
v2 = (5 - 2*r5).sqrt()
v3 = (r5+3)*r3
tan24 = ( v1*v2 - v3 )/2

v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt()
tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt()


values_of_A_tanA = (
    (dD(45), tan45),
    (dD(36), tan36),
    (dD(30), tan30),
    (dD(27), tan27),
    (dD(24), tan24),
)

values_of_θ_tanθ = []
for (A, tanA) in values_of_A_tanA :
    θ, tanθ = θ_tanθ_from_A_tanA (A, tanA)
    print()
    sx = 'θ' ; print (sx, '=', eval(sx))
#    sx = 'tanθ' ; print (sx, '=', eval(sx))
    print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:]))
    values_of_θ_tanθ += [ (θ, tanθ) ]

# Check
for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) :
    A, tanA = v1,v2
    θ = eval(v3)
    tanθ = dD(v4)
    status = 0
    for p in range (1,100) :
        θ *= 2
        tanθ = tan_2A (tanθ) 
        if θ == A :
            dgt.prec = desired_precision
            (+tanθ == +tanA) or ({}[10])
            dgt.prec = Precision            
            status = 1
            break
    status or ({}[11])
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
θ = dD(45) / (2 ** (33))
tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912

θ = dD(36) / (2 ** (33))
tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713

θ = dD(30) / (2 ** (33))
tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689

θ = dD(27) / (2 ** (33))
tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738

θ = dD(24) / (2 ** (32))
tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

def calculate_π (angleθ, tanθ) :
    '''
angleθ may be: &quot;dD(27) / (2 ** (33))&quot;
tanθ may be: &quot;0.00000_00000_54859_42797_ ..... _03540_9738&quot;
π = calculate_π (angleθ, tanθ)
    '''
    thisName = 'calculate_π (angleθ, tanθ) :'

    if isinstance(angleθ, dD) : pass
    elif isinstance(angleθ, str) : angleθ = eval(angleθ)
    else : ({}[21])
    if isinstance(tanθ, dD) : pass
    elif isinstance(tanθ, str) : tanθ = dD(tanθ)
    else : ({}[22])
    
    x = tanθ ; multiplier = -1 ; sum = x ; count = 0; status = 0

    #         x**3   x**5   x**7   x**9
    # y = x - ---- + ---- - ---- + ----
    #           3      5      7      9
    # 
    # Each term in the sequence is roughly the previous term multiplied by x**2
    # Each value of x contains 10 leading zeroes.
    # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes.
    # Each pass through main loop adds about 20 digits to the current value of sum.
    # and θ is calculated to precision of 1004 digits with about 50 passes through main loop.
    # 
    for p in range (3,200,2) :
        # This is main loop.
        count += 1
        addendum = (multiplier * (x**p)) / p
        sum += addendum
        if abs(addendum) &lt; Tolerance :
            status = 1; break
        multiplier = -multiplier
    status or ({}[23])
    print(thisName, 'count =',count)
    π = sum * 180 / angleθ
    dgt.prec = desired_precision
    π += 0 # This forces π to adopt precision of desired_precision.
    dgt.prec = Precision
    return π

# Calculate five values of π:
values_of_π = []
for θ,tanθ in values_of_θ_tanθ :
    π = calculate_π (θ,tanθ)
    values_of_π += [ π ]
&lt;/syntaxhighlight&gt;
Each calculation of π required about 50 passes through main loop:
&lt;syntaxhighlight&gt;
calculate_π (angleθ, tanθ) : count = 50
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 50
&lt;/syntaxhighlight&gt;
Check that all 5 values of π are equal:
&lt;syntaxhighlight lang=python&gt;
# python code

set1 = set(values_of_π)
sx = 'len(values_of_π)' ; print (sx, '=', eval(sx))
sx = 'len(set1)' ; print (sx, '=', eval(sx))
sx = 'set1' ; print (sx, '=', eval(sx))
π, = set1
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(values_of_π) = 5
len(set1) = 1
set1 = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')}
&lt;/syntaxhighlight&gt;
Print value of π as python command formatted:
&lt;syntaxhighlight lang=python&gt;
# python code

newLine = '''
'''[-1:]

def print_π (π) :
    '''
Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989')
This function prints:
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     .....
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
    '''
    πstr = str(π)
    (len(πstr) == (desired_precision + 1)) or ({}[31])
    (πstr[:2] == '3.') or ({}[32])

    ten_rows = []
    for p in range (2, len(πstr), 100) :
        str1a = πstr[p:p+100]
        list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ]
        str1b = '_'.join(list1a)
        ten_rows += [str1b]
    ten_rows[0] = '3.' + ten_rows[0]
    joiner = '&quot;{}     +  &quot;'.format(newLine)
    str3 = '( &quot;{}&quot; )'.format(joiner.join(ten_rows))
    
    str4 = eval(str3)
    (dD(str4) == π) or ({}[33])
    print ('π =', str3)
    return str3

π1 = print_π (π)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     +  &quot;44288_10975_66593_34461_28475_64823_37867_83165_27120_19091_45648_56692_34603_48610_45432_66482_13393_60726_02491_41273&quot;
     +  &quot;72458_70066_06315_58817_48815_20920_96282_92540_91715_36436_78925_90360_01133_05305_48820_46652_13841_46951_94151_16094&quot;
     +  &quot;33057_27036_57595_91953_09218_61173_81932_61179_31051_18548_07446_23799_62749_56735_18857_52724_89122_79381_83011_94912&quot;
     +  &quot;98336_73362_44065_66430_86021_39494_63952_24737_19070_21798_60943_70277_05392_17176_29317_67523_84674_81846_76694_05132&quot;
     +  &quot;00056_81271_45263_56082_77857_71342_75778_96091_73637_17872_14684_40901_22495_34301_46549_58537_10507_92279_68925_89235&quot;
     +  &quot;42019_95611_21290_21960_86403_44181_59813_62977_47713_09960_51870_72113_49999_99837_29780_49951_05973_17328_16096_31859&quot;
     +  &quot;50244_59455_34690_83026_42522_30825_33446_85035_26193_11881_71010_00313_78387_52886_58753_32083_81420_61717_76691_47303&quot;
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Asymptotes of hyperbola==

===Line and hyperbola===
This section describes possibilities that arise when we consider intersection of line and hyperbola.
====With two common points====

{{RoundBoxTop|theme=2}}
[[File:01hyperbola01.png|thumb|400px|'''Diagram of hyperbola and line.'''
&lt;/br&gt;
Line and hyperbola have two common points.
&lt;/br&gt;
When line and hyperbola have two common points, line cannot be parallel to asymptote.
&lt;/br&gt;
]]
Line 1:
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 704, -1404, 1344, -11040, -41220, -161775

abc = a,b,c = .96, .28, .2
result = hyperbola_and_line (ABCDEF, abc) 
sx = 'result' ; print (sx, eval(sx))
&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
result [ (1.425,-5.6), (4.575,-16.4) ]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Length of latus rectum==
-----------------------

&lt;math&gt;b^2x^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2c^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2(a^2 - b^2) + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2a^2 - b^4 + a^2y^2 - a^2b^2 =0&lt;/math&gt;

&lt;math&gt;a^2y^2 = b^4&lt;/math&gt;

&lt;math&gt;y^2 = \frac{b^4}{a^2}&lt;/math&gt;

&lt;math&gt;y = \frac{b^2}{a}&lt;/math&gt;

Length of latus rectum &lt;math&gt;= L_1R_1 = L_2R_2 = \frac{2b^2}{a}.&lt;/math&gt;

=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Parabola====


&lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(18)x + (-24)y + (104) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-24)x + (32)y + (28) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-30)x + (40)y + (160) = 0&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never &lt;math&gt;0.75&lt;/math&gt; because axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.
]]
Consider conic section: &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;

This curve is a parabola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 9,16,-24,104,28,-144 # parabola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0
For slope vertical: (-24)x + (32)y + (28) = 0
For slope 2: (-30)x + (40)y + (160) = 0
&lt;/syntaxhighlight&gt;
Because all 3 lines are parallel to axis, all 3 lines have slope &lt;math&gt;\frac{3}{4}.&lt;/math&gt;


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.75:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

three_slopes (ABCDEF, 0.75, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0 # Same as above.
For slope vertical: (-24)x + (32)y + (28) = 0    # Same as above.
For slope 0.75: (0.0)x + (0.0)y + (125.0) = 0    # Impossible.
&lt;/syntaxhighlight&gt;
Axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.

&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Ellipse====

&lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(3542)x + (1944)y + (-44860) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(1944)x + (2408)y + (-18520) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(1598)x + (-464)y + (-26340) = 0&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;-1.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;

This curve is an ellipse.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;-1:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse
three_slopes (ABCDEF, -1, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
For slope horizontal: (3542)x + (1944)y + (-44860) = 0
For slope vertical: (1944)x + (2408)y + (-18520) = 0
For slope -1: (1598)x + (-464)y + (-26340) = 0
&lt;/syntaxhighlight&gt;
Because curve is closed loop, slope of curve may be any value including &lt;math&gt;\frac{1}{0}.&lt;/math&gt;

If slope of curve is given as &lt;math&gt;\frac{1}{0},&lt;/math&gt; it means that curve is vertical at that
point and tangent to curve has equation &lt;math&gt;x = k.&lt;/math&gt;

For any given slope there are always 2 points on opposite sides of curve where tangent to curve
at those points has the given slope.
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Hyperbola====


&lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(-702)x + (-336)y + (4182) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-336)x + (352)y + (-3824) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-1374)x + (368)y + (-3466) = 0&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0505hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
&lt;math&gt;\text{Line 2}&lt;/math&gt; and curve do not intersect. Slope is never vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;

This curve is a hyperbola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -351, 176, -336, 4182, -3824, -16231 # hyperbola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0
For slope horizontal: (-702)x + (-336)y + (4182) = 0
For slope vertical: (-336)x + (352)y + (-3824) = 0
For slope 2: (-1374)x + (368)y + (-3466) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
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===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
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All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
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[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
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[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Second definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
{{RoundBoxBottom}}
===Proof===
{{RoundBoxTop|theme=2}}
[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Heading==
===Properties of ellipse===
{{RoundBoxTop|theme=2}}
[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

* Equation: &lt;math&gt;\frac {x^2} {a^2}  + \frac {y^2} {b^2} = 1&lt;/math&gt;

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

&lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>el92en5vsi0aslx2cwcv4o9gdlnu6lf</sha1>
    </revision>
    <revision>
      <id>2689281</id>
      <parentid>2689203</parentid>
      <timestamp>2024-11-29T11:51:58Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Implementation */</comment>
      <origin>2689281</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71865" sha1="dz937yr4d84dwr31yns45rugxttjftd" xml:space="preserve">= Hyperbola =
{{RoundBoxTop|theme=2}}
[[File:0911hyperbola00.png|thumb|400px|'''
Figure 1: Hyperbola at origin with transverse axis horizontal.''' &lt;/br&gt;&lt;/br&gt;

Origin at point &lt;math&gt;O&lt;/math&gt;&lt;math&gt;: (0,0)&lt;/math&gt;.&lt;/br&gt;
Foci are points &lt;math&gt;F_1 (-c,0),\ F_2 (c,0). OF_1 = OF_2 = c.&lt;/math&gt;&lt;/br&gt;
Vertices are points &lt;math&gt;V_1 (-a,0),\ V_2 (a,0). OV_1 = OV_2 = a.&lt;/math&gt;&lt;/br&gt;
Line segment &lt;math&gt;V_1OV_2&lt;/math&gt; is the &lt;math&gt;transverse\ axis.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;PF_1 - PF_2 = 2a.&lt;/math&gt;
]]

In cartesian [[geometry]] in two dimensions hyperbola is locus of a point &lt;math&gt;P&lt;/math&gt; that moves relative to two fixed points called &lt;math&gt;foci&lt;/math&gt;&lt;math&gt;: F_1, F_2.&lt;/math&gt;
The distance &lt;math&gt;F_1 F_2&lt;/math&gt; from one &lt;math&gt;focus\ (F_1)&lt;/math&gt; to the other &lt;math&gt;focus\ (F_2)&lt;/math&gt; is non-zero. The absolute difference of the distances &lt;math&gt;(PF_1, PF_2)&lt;/math&gt; from point to foci is constant.


&lt;math&gt;PF_1 - PF_2 = K.&lt;/math&gt; See figure 1. 


Center of hyperbola is located at the origin &lt;math&gt;O (0,0)&lt;/math&gt; and the foci &lt;math&gt;(F_1, F_2)&lt;/math&gt; are on the &lt;math&gt;X\ axis&lt;/math&gt;
at distance &lt;math&gt;c&lt;/math&gt; from &lt;math&gt;O. &lt;/math&gt;


&lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(-c, 0). F_2&lt;/math&gt; has coordinates &lt;math&gt;(c,0)&lt;/math&gt;. Line segments &lt;math&gt;OF_1 = OF_2 = c.&lt;/math&gt;


Each point &lt;math&gt;(V_1,V_2)&lt;/math&gt; where the curve intersects the transverse axis is called a &lt;math&gt;vertex.\ V_1,V_2&lt;/math&gt; are the vertices of the ellipse.


By definition &lt;math&gt;PF_1 - PF_2 = V_2F_1 - V_2F_2 = V_1F_2 - V_1F_1 = K.&lt;/math&gt;


&lt;math&gt;\therefore V_2F_1 - V_2F_2 = V_2F_1 - V_1F_1 = V_1V_2 = K = 2a,&lt;/math&gt; the length of the
&lt;math&gt;transverse\ axis\ (V_1V_2).&lt;/math&gt;
&lt;math&gt;OV_1 = OV_2 = a.&lt;/math&gt;
{{RoundBoxBottom}}
==Radians, the natural angle==
If you were a mathematician among the ancient Sumerians of the 3rd millennium BC and you were determined to define 
the angle that could be adopted as a standard to be used by all users of trigonometry, you would probably suggest
the angle in an equilateral triangle.
This angle is easily defined, easily constructed, easily understood and easily reproduced. It would be easy to call
this angle the &quot;natural&quot; angle.

The numeral system used by the ancient Sumerians was Sexagesimal, also known as base 60, a numeral system with sixty as its base.
In practice the natural angle could be divided into 60 parts, now called degrees, and each degree could be divided into
60 parts, now called minutes, and so on.

Three equilateral triangles fit neatly into a semi-circle, hence 180 degrees in a semi-circle.

We know that &lt;math&gt;\tan 30^\circ = \frac{\sqrt{3}}{3}.&lt;/math&gt;
Therefore, &lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; should be &lt;math&gt;0.5,&lt;/math&gt; or one half of our concept of the natural angle.

Whatever the natural angle might be, it has existed for billions of years, but it has come to light only in recent times
with invention of the calculus.

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:

&lt;math&gt;\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + \cdots.&lt;/math&gt;

The following python code calculates 
&lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; using Gregory's series:
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

r3 = 3 ** .5
x = r3/3

arctan_x = (
      x        -  x**3/3  +  x**5/5  -  x**7/7  +  x**9/9  - x**11/11 + x**13/13 - x**15/15 + x**17/17 - x**19/19
    + x**21/21 - x**23/23 + x**25/25 - x**27/27 + x**29/29 - x**31/31 + x**33/33 - x**35/35 + x**37/37 - x**39/39
    + x**41/41 - x**43/43 + x**45/45 - x**47/47 + x**49/49 - x**51/51 + x**53/53 - x**55/55 + x**57/57 - x**59/59
    + x**61/61 - x**63/63 + x**65/65 - x**67/67 + x**69/69
)

sx = 'arctan_x' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x = 0.5235987755982988
&lt;/syntaxhighlight&gt;
Our assessment of the natural angle as the angle in an equilateral triangle was a very reasonable guess.

However, the natural angle is the radian, the angle that subtends an arc on the circumference of a circle equal to the radius.

Six times arctan_x &lt;math&gt;= 180^\circ&lt;/math&gt; or the number of radians in a semi-circle:

&lt;syntaxhighlight lang=python&gt;
# python code

sx = 'arctan_x * 6' ; print (sx, '=', eval(sx))
sx = '180/(arctan_x * 6)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x * 6 = 3.141592653589793
180/(arctan_x * 6) = 57.29577951308232
&lt;/syntaxhighlight&gt;
&lt;math&gt;\pi = 3.141592653589793\dots,&lt;/math&gt; number of radians in semi-circle.

One radian &lt;math&gt;= 57.29577951308232^\circ,&lt;/math&gt; slightly less than &lt;math&gt;60^\circ.&lt;/math&gt;


Because the value &lt;math&gt;\frac\sqrt{3}{3}&lt;/math&gt; is fairly large, calculation of &lt;code&gt;arctan_x&lt;/code&gt; above required 34 operations
to produce result accurate to 16 places of decimals. The calculation did not converge quickly.

Python code below uses much smaller values of &lt;math&gt;x&lt;/math&gt; and calculation of &lt;code&gt;arctan_x&lt;/code&gt; for precision of 1001 is quite fast.

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

==tan(A/2)==

{{RoundBoxTop|theme=2}}
[[File:1122tanA_200.png|thumb|400px|'''Graphical calculation of &lt;math&gt;\tan \frac{A}{2}&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;OQ = 1;\ QP = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\tan(A) = \frac{QP}{OQ} = \frac{t}{1} = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;OP = OR = \sqrt{1 + t^2}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Point &lt;math&gt;P&lt;/math&gt; has coordinates &lt;math&gt;(1,t).&lt;/math&gt;

Point &lt;math&gt;R&lt;/math&gt; has coordinates &lt;math&gt;(\sqrt{1 + t^2},0).&lt;/math&gt;

Mid point of &lt;math&gt;PR,\ M&lt;/math&gt; has coordinates &lt;math&gt;(   \frac{ 1 +  \sqrt{1 + t^2}  }{2}, \frac{t}{2}            ).&lt;/math&gt;

&lt;math&gt;\tan \frac{A}{2} = \frac{t}{2} / \frac{ 1 +  \sqrt{1 + t^2}  }{2} = \frac{t}{1 +  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t}{1 +  \sqrt{1 + t^2} } \cdot \frac{1 -  \sqrt{1 + t^2}}{1 -  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{1-(1+t^2)}&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{-t^2}&lt;/math&gt;
&lt;math&gt;= \frac{-1 +  \sqrt{1 + t^2} }{t}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


* &lt;math&gt;\tan \frac{A}{2} = \frac{-1 +  \sqrt{1 + \tan^2 (A)} }{\tan (A)}&lt;/math&gt;

* &lt;math&gt;\tan (2A) = \frac{2\tan (A)}{ 1 - \tan^2 (A) }&lt;/math&gt;
{{RoundBoxBottom}}

==Implementation==
{{RoundBoxTop|theme=2}}
This section calculates five values of &lt;math&gt;\pi&lt;/math&gt; using the following known values of &lt;math&gt;\tan(A):&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;A&lt;/math&gt; || &lt;math&gt;\tan(A)&lt;/math&gt;
|-
| &lt;math&gt;45^\circ&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| &lt;math&gt;36^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{   5 - 2\sqrt{5}  }&lt;/math&gt;
|-
| &lt;math&gt;30^\circ&lt;/math&gt;
| &lt;math&gt;\frac{\sqrt{3}}{3}&lt;/math&gt;
|-
| &lt;math&gt;27^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{    11 - 4\sqrt{5} + (\sqrt{5} - 3)   \sqrt{   10 - 2\sqrt{5}      }   }&lt;/math&gt;
|-
| &lt;math&gt;24^\circ&lt;/math&gt;
| &lt;math&gt;\frac{       (3\sqrt{5} + 7)     \sqrt{5 - 2\sqrt{5}} - (\sqrt{5} + 3)\sqrt{3}     }{2}&lt;/math&gt;
|}

Values of &lt;math&gt;x&lt;/math&gt; in table below are derived from the above values by using identity &lt;math&gt;\tan(\frac{A}{2}) = \frac{-1 + \sqrt{1 + \tan^2(A)}}{\tan(A)}&lt;/math&gt;:

{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;\theta&lt;/math&gt; || &lt;math&gt;x = \tan(\theta)&lt;/math&gt;
|-
| &lt;math&gt;\frac{45^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_91432_37995_4197.....089_03901_63759_3912&lt;/code&gt;
|-
| &lt;math&gt;\frac{36^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_73145_90396_3357.....211_97500_56173_0713&lt;/code&gt;
|-
| &lt;math&gt;\frac{30^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_60954_91996_9464.....024_32806_94580_0689&lt;/code&gt;
|-
| &lt;math&gt;\frac{27^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_54859_42797_2518.....791_30634_03540_9738&lt;/code&gt;
|-
| &lt;math&gt;\frac{24^\circ}{2^{32}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_97527_87195_1143.....736_60376_04724_6778&lt;/code&gt;
|}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

desired_precision = 1001
number_of_leading_zeroes = 10 # See below.

import decimal
dD = decimal.Decimal # decimal object is like float with (almost) infinite precision.
dgt = decimal.getcontext()

Precision = dgt.prec = desired_precision + 3  # Adjust as necessary.
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))      # Adjust as necessary.
adjustment_to_precision = number_of_leading_zeroes * 2 + 3

def tan_halfA(tan_A) :
    dgt.prec += adjustment_to_precision
    top = -1 + (1+tan_A**2).sqrt()
    dgt.prec -= adjustment_to_precision
    tan_A_2 = top/tan_A
    return tan_A_2
    
def tan_2A (tanA) :
    '''
          2 * tanA
tan(2A) = -----------
          1 - tanA**2
    '''
    if tanA in (1,-1) : return '1/0'
    dgt.prec += adjustment_to_precision
    bottom = (1 - tanA**2)
    output = 2*tanA/bottom
    dgt.prec -= adjustment_to_precision
    return output+0

def θ_tanθ_from_A_tanA (angleA, tanA) :
    '''
if input == 45,1
output is:
&quot;dD(45) / (2 ** (33))&quot;, &quot;0.00000_00000_91432_37995_....._63759_3912&quot;
                           ^^^^^^^^^^^
number_of_leading_zeroes refers to these zeroes.
θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA)
    '''
    θ, tanθ = angleA, tanA
    for p in range (1,100) :
        θ /= 2
        tanθ = tan_halfA(tanθ)
        if tanθ &gt;= dD('1e-' + str(number_of_leading_zeroes)) : continue
        str1 = str(tanθ)
        # str1 = &quot;n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11&quot;
        str1a = str1[0] + str1[2:-4]
        list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ]
        str2 = '0.00000_00000_' + ('_'.join(list1))
        dD2 = dD(str2)
        (dD2 == tanθ) or ({}[2])
        ((θ * (2**p)) == angleA ) or ({}[3])
        str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p)
        (θ == eval(str3)) or ({}[4])
        return str3, str2
    ({}[5])


r3 = dD(3).sqrt()
r5 = dD(5).sqrt()
tan36 = (5 - 2*r5).sqrt()
tan45 = dD(1)
tan30 = r3/3

v1 = 3*r5+7
v2 = (5 - 2*r5).sqrt()
v3 = (r5+3)*r3
tan24 = ( v1*v2 - v3 )/2

v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt()
tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt()


values_of_A_tanA = (
    (dD(45), tan45),
    (dD(36), tan36),
    (dD(30), tan30),
    (dD(27), tan27),
    (dD(24), tan24),
)

values_of_θ_tanθ = []
for (A, tanA) in values_of_A_tanA :
    θ, tanθ = θ_tanθ_from_A_tanA (A, tanA)
    print()
    sx = 'θ' ; print (sx, '=', eval(sx))
#    sx = 'tanθ' ; print (sx, '=', eval(sx))
    print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:]))
    values_of_θ_tanθ += [ (θ, tanθ) ]

# Check
for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) :
    A, tanA = v1,v2
    θ = eval(v3)
    tanθ = dD(v4)
    status = 0
    for p in range (1,100) :
        θ *= 2
        tanθ = tan_2A (tanθ) 
        if θ == A :
            dgt.prec = desired_precision
            (+tanθ == +tanA) or ({}[10])
            dgt.prec = Precision            
            status = 1
            break
    status or ({}[11])
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
θ = dD(45) / (2 ** (33))
tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912

θ = dD(36) / (2 ** (33))
tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713

θ = dD(30) / (2 ** (33))
tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689

θ = dD(27) / (2 ** (33))
tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738

θ = dD(24) / (2 ** (32))
tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

def calculate_π (angleθ, tanθ) :
    '''
angleθ may be: &quot;dD(27) / (2 ** (33))&quot;
tanθ may be: &quot;0.00000_00000_54859_42797_ ..... _03540_9738&quot;
π = calculate_π (angleθ, tanθ)
    '''
    thisName = 'calculate_π (angleθ, tanθ) :'

    if isinstance(angleθ, dD) : pass
    elif isinstance(angleθ, str) : angleθ = eval(angleθ)
    else : ({}[21])
    if isinstance(tanθ, dD) : pass
    elif isinstance(tanθ, str) : tanθ = dD(tanθ)
    else : ({}[22])
    
    x = tanθ ; multiplier = -1 ; sum = x ; count = 0; status = 0

    #         x**3   x**5   x**7   x**9
    # y = x - ---- + ---- - ---- + ----
    #           3      5      7      9
    # 
    # Each term in the sequence is roughly the previous term multiplied by x**2.
    # Each value of x contains 10 leading zeroes after decimal point.
    # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes.
    # Each pass through main loop adds about 20 digits to current value of sum
    # and θ is calculated to precision of 1004 digits with about 50 passes through main loop.
    # 
    for p in range (3,200,2) :
        # This is main loop.
        count += 1
        addendum = (multiplier * (x**p)) / p
        sum += addendum
        if abs(addendum) &lt; Tolerance :
            status = 1; break
        multiplier = -multiplier
    status or ({}[23])
    print(thisName, 'count =',count)
    π = sum * 180 / angleθ
    dgt.prec = desired_precision
    π += 0 # This forces π to adopt precision of desired_precision.
    dgt.prec = Precision
    return π

# Calculate five values of π:
values_of_π = []
for θ,tanθ in values_of_θ_tanθ :
    π = calculate_π (θ,tanθ)
    values_of_π += [ π ]
&lt;/syntaxhighlight&gt;
Each calculation of π required about 50 passes through main loop:
&lt;syntaxhighlight&gt;
calculate_π (angleθ, tanθ) : count = 50
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 50
&lt;/syntaxhighlight&gt;
Check that all 5 values of π are equal:
&lt;syntaxhighlight lang=python&gt;
# python code

set1 = set(values_of_π)
sx = 'len(values_of_π)' ; print (sx, '=', eval(sx))
sx = 'len(set1)' ; print (sx, '=', eval(sx))
sx = 'set1' ; print (sx, '=', eval(sx))
π, = set1
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(values_of_π) = 5
len(set1) = 1
set1 = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')}
&lt;/syntaxhighlight&gt;
Print value of π as python command formatted:
&lt;syntaxhighlight lang=python&gt;
# python code

newLine = '''
'''[-1:]

def print_π (π) :
    '''
Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989')
This function prints:
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     .....
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
    '''
    πstr = str(π)
    (len(πstr) == (desired_precision + 1)) or ({}[31])
    (πstr[:2] == '3.') or ({}[32])

    ten_rows = []
    for p in range (2, len(πstr), 100) :
        str1a = πstr[p:p+100]
        list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ]
        str1b = '_'.join(list1a)
        ten_rows += [str1b]
    ten_rows[0] = '3.' + ten_rows[0]
    joiner = '&quot;{}     +  &quot;'.format(newLine)
    str3 = '( &quot;{}&quot; )'.format(joiner.join(ten_rows))
    
    str4 = eval(str3)
    (dD(str4) == π) or ({}[33])
    print ('π =', str3)
    return str3

π1 = print_π (π)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     +  &quot;44288_10975_66593_34461_28475_64823_37867_83165_27120_19091_45648_56692_34603_48610_45432_66482_13393_60726_02491_41273&quot;
     +  &quot;72458_70066_06315_58817_48815_20920_96282_92540_91715_36436_78925_90360_01133_05305_48820_46652_13841_46951_94151_16094&quot;
     +  &quot;33057_27036_57595_91953_09218_61173_81932_61179_31051_18548_07446_23799_62749_56735_18857_52724_89122_79381_83011_94912&quot;
     +  &quot;98336_73362_44065_66430_86021_39494_63952_24737_19070_21798_60943_70277_05392_17176_29317_67523_84674_81846_76694_05132&quot;
     +  &quot;00056_81271_45263_56082_77857_71342_75778_96091_73637_17872_14684_40901_22495_34301_46549_58537_10507_92279_68925_89235&quot;
     +  &quot;42019_95611_21290_21960_86403_44181_59813_62977_47713_09960_51870_72113_49999_99837_29780_49951_05973_17328_16096_31859&quot;
     +  &quot;50244_59455_34690_83026_42522_30825_33446_85035_26193_11881_71010_00313_78387_52886_58753_32083_81420_61717_76691_47303&quot;
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Asymptotes of hyperbola==

===Line and hyperbola===
This section describes possibilities that arise when we consider intersection of line and hyperbola.
====With two common points====

{{RoundBoxTop|theme=2}}
[[File:01hyperbola01.png|thumb|400px|'''Diagram of hyperbola and line.'''
&lt;/br&gt;
Line and hyperbola have two common points.
&lt;/br&gt;
When line and hyperbola have two common points, line cannot be parallel to asymptote.
&lt;/br&gt;
]]
Line 1:
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 704, -1404, 1344, -11040, -41220, -161775

abc = a,b,c = .96, .28, .2
result = hyperbola_and_line (ABCDEF, abc) 
sx = 'result' ; print (sx, eval(sx))
&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
result [ (1.425,-5.6), (4.575,-16.4) ]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Length of latus rectum==
-----------------------

&lt;math&gt;b^2x^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2c^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2(a^2 - b^2) + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2a^2 - b^4 + a^2y^2 - a^2b^2 =0&lt;/math&gt;

&lt;math&gt;a^2y^2 = b^4&lt;/math&gt;

&lt;math&gt;y^2 = \frac{b^4}{a^2}&lt;/math&gt;

&lt;math&gt;y = \frac{b^2}{a}&lt;/math&gt;

Length of latus rectum &lt;math&gt;= L_1R_1 = L_2R_2 = \frac{2b^2}{a}.&lt;/math&gt;

=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Parabola====


&lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(18)x + (-24)y + (104) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-24)x + (32)y + (28) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-30)x + (40)y + (160) = 0&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never &lt;math&gt;0.75&lt;/math&gt; because axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.
]]
Consider conic section: &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;

This curve is a parabola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 9,16,-24,104,28,-144 # parabola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0
For slope vertical: (-24)x + (32)y + (28) = 0
For slope 2: (-30)x + (40)y + (160) = 0
&lt;/syntaxhighlight&gt;
Because all 3 lines are parallel to axis, all 3 lines have slope &lt;math&gt;\frac{3}{4}.&lt;/math&gt;


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.75:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

three_slopes (ABCDEF, 0.75, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0 # Same as above.
For slope vertical: (-24)x + (32)y + (28) = 0    # Same as above.
For slope 0.75: (0.0)x + (0.0)y + (125.0) = 0    # Impossible.
&lt;/syntaxhighlight&gt;
Axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.

&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Ellipse====

&lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(3542)x + (1944)y + (-44860) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(1944)x + (2408)y + (-18520) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(1598)x + (-464)y + (-26340) = 0&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;-1.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;

This curve is an ellipse.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;-1:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse
three_slopes (ABCDEF, -1, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
For slope horizontal: (3542)x + (1944)y + (-44860) = 0
For slope vertical: (1944)x + (2408)y + (-18520) = 0
For slope -1: (1598)x + (-464)y + (-26340) = 0
&lt;/syntaxhighlight&gt;
Because curve is closed loop, slope of curve may be any value including &lt;math&gt;\frac{1}{0}.&lt;/math&gt;

If slope of curve is given as &lt;math&gt;\frac{1}{0},&lt;/math&gt; it means that curve is vertical at that
point and tangent to curve has equation &lt;math&gt;x = k.&lt;/math&gt;

For any given slope there are always 2 points on opposite sides of curve where tangent to curve
at those points has the given slope.
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Hyperbola====


&lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(-702)x + (-336)y + (4182) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-336)x + (352)y + (-3824) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-1374)x + (368)y + (-3466) = 0&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0505hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
&lt;math&gt;\text{Line 2}&lt;/math&gt; and curve do not intersect. Slope is never vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;

This curve is a hyperbola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -351, 176, -336, 4182, -3824, -16231 # hyperbola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0
For slope horizontal: (-702)x + (-336)y + (4182) = 0
For slope vertical: (-336)x + (352)y + (-3824) = 0
For slope 2: (-1374)x + (368)y + (-3466) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
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===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
{{RoundBoxTop|theme=2}}
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
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[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Second definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
{{RoundBoxBottom}}
===Proof===
{{RoundBoxTop|theme=2}}
[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Heading==
===Properties of ellipse===
{{RoundBoxTop|theme=2}}
[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

* Equation: &lt;math&gt;\frac {x^2} {a^2}  + \frac {y^2} {b^2} = 1&lt;/math&gt;

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

&lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>dz937yr4d84dwr31yns45rugxttjftd</sha1>
    </revision>
    <revision>
      <id>2689282</id>
      <parentid>2689281</parentid>
      <timestamp>2024-11-29T11:54:34Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Implementation */</comment>
      <origin>2689282</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71934" sha1="hutzo4lenw86n07f8m0rll9qkztwis4" xml:space="preserve">= Hyperbola =
{{RoundBoxTop|theme=2}}
[[File:0911hyperbola00.png|thumb|400px|'''
Figure 1: Hyperbola at origin with transverse axis horizontal.''' &lt;/br&gt;&lt;/br&gt;

Origin at point &lt;math&gt;O&lt;/math&gt;&lt;math&gt;: (0,0)&lt;/math&gt;.&lt;/br&gt;
Foci are points &lt;math&gt;F_1 (-c,0),\ F_2 (c,0). OF_1 = OF_2 = c.&lt;/math&gt;&lt;/br&gt;
Vertices are points &lt;math&gt;V_1 (-a,0),\ V_2 (a,0). OV_1 = OV_2 = a.&lt;/math&gt;&lt;/br&gt;
Line segment &lt;math&gt;V_1OV_2&lt;/math&gt; is the &lt;math&gt;transverse\ axis.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;PF_1 - PF_2 = 2a.&lt;/math&gt;
]]

In cartesian [[geometry]] in two dimensions hyperbola is locus of a point &lt;math&gt;P&lt;/math&gt; that moves relative to two fixed points called &lt;math&gt;foci&lt;/math&gt;&lt;math&gt;: F_1, F_2.&lt;/math&gt;
The distance &lt;math&gt;F_1 F_2&lt;/math&gt; from one &lt;math&gt;focus\ (F_1)&lt;/math&gt; to the other &lt;math&gt;focus\ (F_2)&lt;/math&gt; is non-zero. The absolute difference of the distances &lt;math&gt;(PF_1, PF_2)&lt;/math&gt; from point to foci is constant.


&lt;math&gt;PF_1 - PF_2 = K.&lt;/math&gt; See figure 1. 


Center of hyperbola is located at the origin &lt;math&gt;O (0,0)&lt;/math&gt; and the foci &lt;math&gt;(F_1, F_2)&lt;/math&gt; are on the &lt;math&gt;X\ axis&lt;/math&gt;
at distance &lt;math&gt;c&lt;/math&gt; from &lt;math&gt;O. &lt;/math&gt;


&lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(-c, 0). F_2&lt;/math&gt; has coordinates &lt;math&gt;(c,0)&lt;/math&gt;. Line segments &lt;math&gt;OF_1 = OF_2 = c.&lt;/math&gt;


Each point &lt;math&gt;(V_1,V_2)&lt;/math&gt; where the curve intersects the transverse axis is called a &lt;math&gt;vertex.\ V_1,V_2&lt;/math&gt; are the vertices of the ellipse.


By definition &lt;math&gt;PF_1 - PF_2 = V_2F_1 - V_2F_2 = V_1F_2 - V_1F_1 = K.&lt;/math&gt;


&lt;math&gt;\therefore V_2F_1 - V_2F_2 = V_2F_1 - V_1F_1 = V_1V_2 = K = 2a,&lt;/math&gt; the length of the
&lt;math&gt;transverse\ axis\ (V_1V_2).&lt;/math&gt;
&lt;math&gt;OV_1 = OV_2 = a.&lt;/math&gt;
{{RoundBoxBottom}}
==Radians, the natural angle==
If you were a mathematician among the ancient Sumerians of the 3rd millennium BC and you were determined to define 
the angle that could be adopted as a standard to be used by all users of trigonometry, you would probably suggest
the angle in an equilateral triangle.
This angle is easily defined, easily constructed, easily understood and easily reproduced. It would be easy to call
this angle the &quot;natural&quot; angle.

The numeral system used by the ancient Sumerians was Sexagesimal, also known as base 60, a numeral system with sixty as its base.
In practice the natural angle could be divided into 60 parts, now called degrees, and each degree could be divided into
60 parts, now called minutes, and so on.

Three equilateral triangles fit neatly into a semi-circle, hence 180 degrees in a semi-circle.

We know that &lt;math&gt;\tan 30^\circ = \frac{\sqrt{3}}{3}.&lt;/math&gt;
Therefore, &lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; should be &lt;math&gt;0.5,&lt;/math&gt; or one half of our concept of the natural angle.

Whatever the natural angle might be, it has existed for billions of years, but it has come to light only in recent times
with invention of the calculus.

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:

&lt;math&gt;\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + \cdots.&lt;/math&gt;

The following python code calculates 
&lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; using Gregory's series:
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

r3 = 3 ** .5
x = r3/3

arctan_x = (
      x        -  x**3/3  +  x**5/5  -  x**7/7  +  x**9/9  - x**11/11 + x**13/13 - x**15/15 + x**17/17 - x**19/19
    + x**21/21 - x**23/23 + x**25/25 - x**27/27 + x**29/29 - x**31/31 + x**33/33 - x**35/35 + x**37/37 - x**39/39
    + x**41/41 - x**43/43 + x**45/45 - x**47/47 + x**49/49 - x**51/51 + x**53/53 - x**55/55 + x**57/57 - x**59/59
    + x**61/61 - x**63/63 + x**65/65 - x**67/67 + x**69/69
)

sx = 'arctan_x' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x = 0.5235987755982988
&lt;/syntaxhighlight&gt;
Our assessment of the natural angle as the angle in an equilateral triangle was a very reasonable guess.

However, the natural angle is the radian, the angle that subtends an arc on the circumference of a circle equal to the radius.

Six times arctan_x &lt;math&gt;= 180^\circ&lt;/math&gt; or the number of radians in a semi-circle:

&lt;syntaxhighlight lang=python&gt;
# python code

sx = 'arctan_x * 6' ; print (sx, '=', eval(sx))
sx = '180/(arctan_x * 6)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x * 6 = 3.141592653589793
180/(arctan_x * 6) = 57.29577951308232
&lt;/syntaxhighlight&gt;
&lt;math&gt;\pi = 3.141592653589793\dots,&lt;/math&gt; number of radians in semi-circle.

One radian &lt;math&gt;= 57.29577951308232^\circ,&lt;/math&gt; slightly less than &lt;math&gt;60^\circ.&lt;/math&gt;


Because the value &lt;math&gt;\frac\sqrt{3}{3}&lt;/math&gt; is fairly large, calculation of &lt;code&gt;arctan_x&lt;/code&gt; above required 34 operations
to produce result accurate to 16 places of decimals. The calculation did not converge quickly.

Python code below uses much smaller values of &lt;math&gt;x&lt;/math&gt; and calculation of &lt;code&gt;arctan_x&lt;/code&gt; for precision of 1001 is quite fast.

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

==tan(A/2)==

{{RoundBoxTop|theme=2}}
[[File:1122tanA_200.png|thumb|400px|'''Graphical calculation of &lt;math&gt;\tan \frac{A}{2}&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;OQ = 1;\ QP = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\tan(A) = \frac{QP}{OQ} = \frac{t}{1} = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;OP = OR = \sqrt{1 + t^2}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Point &lt;math&gt;P&lt;/math&gt; has coordinates &lt;math&gt;(1,t).&lt;/math&gt;

Point &lt;math&gt;R&lt;/math&gt; has coordinates &lt;math&gt;(\sqrt{1 + t^2},0).&lt;/math&gt;

Mid point of &lt;math&gt;PR,\ M&lt;/math&gt; has coordinates &lt;math&gt;(   \frac{ 1 +  \sqrt{1 + t^2}  }{2}, \frac{t}{2}            ).&lt;/math&gt;

&lt;math&gt;\tan \frac{A}{2} = \frac{t}{2} / \frac{ 1 +  \sqrt{1 + t^2}  }{2} = \frac{t}{1 +  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t}{1 +  \sqrt{1 + t^2} } \cdot \frac{1 -  \sqrt{1 + t^2}}{1 -  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{1-(1+t^2)}&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{-t^2}&lt;/math&gt;
&lt;math&gt;= \frac{-1 +  \sqrt{1 + t^2} }{t}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


* &lt;math&gt;\tan \frac{A}{2} = \frac{-1 +  \sqrt{1 + \tan^2 (A)} }{\tan (A)}&lt;/math&gt;

* &lt;math&gt;\tan (2A) = \frac{2\tan (A)}{ 1 - \tan^2 (A) }&lt;/math&gt;
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==Implementation==
{{RoundBoxTop|theme=2}}
This section calculates five values of &lt;math&gt;\pi&lt;/math&gt; using the following known values of &lt;math&gt;\tan(A):&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;A&lt;/math&gt; || &lt;math&gt;\tan(A)&lt;/math&gt;
|-
| &lt;math&gt;45^\circ&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| &lt;math&gt;36^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{   5 - 2\sqrt{5}  }&lt;/math&gt;
|-
| &lt;math&gt;30^\circ&lt;/math&gt;
| &lt;math&gt;\frac{\sqrt{3}}{3}&lt;/math&gt;
|-
| &lt;math&gt;27^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{    11 - 4\sqrt{5} + (\sqrt{5} - 3)   \sqrt{   10 - 2\sqrt{5}      }   }&lt;/math&gt;
|-
| &lt;math&gt;24^\circ&lt;/math&gt;
| &lt;math&gt;\frac{       (3\sqrt{5} + 7)     \sqrt{5 - 2\sqrt{5}} - (\sqrt{5} + 3)\sqrt{3}     }{2}&lt;/math&gt;
|}

Values of &lt;math&gt;x&lt;/math&gt; in table below are derived from the above values by using identity &lt;math&gt;\tan(\frac{A}{2}) = \frac{-1 + \sqrt{1 + \tan^2(A)}}{\tan(A)}&lt;/math&gt;:

{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;\theta&lt;/math&gt; || &lt;math&gt;x = \tan(\theta)&lt;/math&gt;
|-
| &lt;math&gt;\frac{45^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_91432_37995_4197.....089_03901_63759_3912&lt;/code&gt;
|-
| &lt;math&gt;\frac{36^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_73145_90396_3357.....211_97500_56173_0713&lt;/code&gt;
|-
| &lt;math&gt;\frac{30^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_60954_91996_9464.....024_32806_94580_0689&lt;/code&gt;
|-
| &lt;math&gt;\frac{27^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_54859_42797_2518.....791_30634_03540_9738&lt;/code&gt;
|-
| &lt;math&gt;\frac{24^\circ}{2^{32}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_97527_87195_1143.....736_60376_04724_6778&lt;/code&gt;
|}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

desired_precision = 1001
number_of_leading_zeroes = 10 # See below.

import decimal
dD = decimal.Decimal # decimal object is like float with (almost) infinite precision.
dgt = decimal.getcontext()

Precision = dgt.prec = desired_precision + 3  # Adjust as necessary.
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))      # Adjust as necessary.
adjustment_to_precision = number_of_leading_zeroes * 2 + 3

def tan_halfA(tan_A) :
    dgt.prec += adjustment_to_precision
    top = -1 + (1+tan_A**2).sqrt()
    dgt.prec -= adjustment_to_precision
    tan_A_2 = top/tan_A
    return tan_A_2
    
def tan_2A (tanA) :
    '''
          2 * tanA
tan(2A) = -----------
          1 - tanA**2
    '''
    if tanA in (1,-1) : return '1/0'
    dgt.prec += adjustment_to_precision
    bottom = (1 - tanA**2)
    output = 2*tanA/bottom
    dgt.prec -= adjustment_to_precision
    return output+0

def θ_tanθ_from_A_tanA (angleA, tanA) :
    '''
if input == 45,1
output is:
&quot;dD(45) / (2 ** (33))&quot;, &quot;0.00000_00000_91432_37995_....._63759_3912&quot;
                           ^^^^^^^^^^^
number_of_leading_zeroes refers to these zeroes.
θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA)
    '''
    θ, tanθ = angleA, tanA
    for p in range (1,100) :
        θ /= 2
        tanθ = tan_halfA(tanθ)
        if tanθ &gt;= dD('1e-' + str(number_of_leading_zeroes)) : continue
        str1 = str(tanθ)
        # str1 = &quot;n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11&quot;
        str1a = str1[0] + str1[2:-4]
        list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ]
        str2 = '0.00000_00000_' + ('_'.join(list1))
        dD2 = dD(str2)
        (dD2 == tanθ) or ({}[2])
        ((θ * (2**p)) == angleA ) or ({}[3])
        str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p)
        (θ == eval(str3)) or ({}[4])
        return str3, str2
    ({}[5])


r3 = dD(3).sqrt()
r5 = dD(5).sqrt()
tan36 = (5 - 2*r5).sqrt()
tan45 = dD(1)
tan30 = r3/3

v1 = 3*r5+7
v2 = (5 - 2*r5).sqrt()
v3 = (r5+3)*r3
tan24 = ( v1*v2 - v3 )/2

v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt()
tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt()


values_of_A_tanA = (
    (dD(45), tan45),
    (dD(36), tan36),
    (dD(30), tan30),
    (dD(27), tan27),
    (dD(24), tan24),
)

values_of_θ_tanθ = []
for (A, tanA) in values_of_A_tanA :
    θ, tanθ = θ_tanθ_from_A_tanA (A, tanA)
    print()
    sx = 'θ' ; print (sx, '=', eval(sx))
#    sx = 'tanθ' ; print (sx, '=', eval(sx))
    print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:]))
    values_of_θ_tanθ += [ (θ, tanθ) ]

# Check
for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) :
    A, tanA = v1,v2
    θ = eval(v3)
    tanθ = dD(v4)
    status = 0
    for p in range (1,100) :
        θ *= 2
        tanθ = tan_2A (tanθ) 
        if θ == A :
            dgt.prec = desired_precision
            (+tanθ == +tanA) or ({}[10])
            dgt.prec = Precision            
            status = 1
            break
    status or ({}[11])
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
θ = dD(45) / (2 ** (33))
tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912

θ = dD(36) / (2 ** (33))
tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713

θ = dD(30) / (2 ** (33))
tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689

θ = dD(27) / (2 ** (33))
tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738

θ = dD(24) / (2 ** (32))
tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

def calculate_π (angleθ, tanθ) :
    '''
angleθ may be: &quot;dD(27) / (2 ** (33))&quot;
tanθ may be: &quot;0.00000_00000_54859_42797_ ..... _03540_9738&quot;
π = calculate_π (angleθ, tanθ)
    '''
    thisName = 'calculate_π (angleθ, tanθ) :'

    if isinstance(angleθ, dD) : pass
    elif isinstance(angleθ, str) : angleθ = eval(angleθ)
    else : ({}[21])
    if isinstance(tanθ, dD) : pass
    elif isinstance(tanθ, str) : tanθ = dD(tanθ)
    else : ({}[22])
    
    x = tanθ ; multiplier = -1 ; sum = x ; count = 0; status = 0

    #         x**3   x**5   x**7   x**9
    # y = x - ---- + ---- - ---- + ----
    #           3      5      7      9
    # 
    # Each term in the sequence is roughly the previous term multiplied by x**2.
    # Each value of x contains 10 leading zeroes after decimal point.
    # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes.
    # Each pass through main loop adds about 20 digits to current value of sum
    # and θ is calculated to precision of 1004 digits with about 50 passes through main loop.
    # 
    for p in range (3,200,2) :
        # This is main loop.
        count += 1
        addendum = (multiplier * (x**p)) / p
        sum += addendum
        if abs(addendum) &lt; Tolerance :
            status = 1; break
        multiplier = -multiplier
    status or ({}[23])
    print(thisName, 'count =',count)
    π = sum * 180 / angleθ
    dgt.prec = desired_precision
    π += 0 # This forces π to adopt precision of desired_precision.
    dgt.prec = Precision
    return π

# Calculate five values of π:
values_of_π = []
for θ,tanθ in values_of_θ_tanθ :
    π = calculate_π (θ,tanθ)
    values_of_π += [ π ]
&lt;/syntaxhighlight&gt;
Each calculation of π required about 50 passes through main loop:
&lt;syntaxhighlight&gt;
calculate_π (angleθ, tanθ) : count = 50
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 50
&lt;/syntaxhighlight&gt;
Check that all 5 values of π are equal:
&lt;syntaxhighlight lang=python&gt;
# python code

set1 = set(values_of_π)
sx = 'len(values_of_π)' ; print (sx, '=', eval(sx))
sx = 'len(set1)' ; print (sx, '=', eval(sx))
sx = 'set1' ; print (sx, '=', eval(sx))
π, = set1 # Note the syntax. If length of set1 is not 1, this statement fails.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(values_of_π) = 5
len(set1) = 1
set1 = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')}
&lt;/syntaxhighlight&gt;
Print value of π as python command formatted:
&lt;syntaxhighlight lang=python&gt;
# python code

newLine = '''
'''[-1:]

def print_π (π) :
    '''
Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989')
This function prints:
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     .....
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
    '''
    πstr = str(π)
    (len(πstr) == (desired_precision + 1)) or ({}[31])
    (πstr[:2] == '3.') or ({}[32])

    ten_rows = []
    for p in range (2, len(πstr), 100) :
        str1a = πstr[p:p+100]
        list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ]
        str1b = '_'.join(list1a)
        ten_rows += [str1b]
    ten_rows[0] = '3.' + ten_rows[0]
    joiner = '&quot;{}     +  &quot;'.format(newLine)
    str3 = '( &quot;{}&quot; )'.format(joiner.join(ten_rows))
    
    str4 = eval(str3)
    (dD(str4) == π) or ({}[33])
    print ('π =', str3)
    return str3

π1 = print_π (π)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     +  &quot;44288_10975_66593_34461_28475_64823_37867_83165_27120_19091_45648_56692_34603_48610_45432_66482_13393_60726_02491_41273&quot;
     +  &quot;72458_70066_06315_58817_48815_20920_96282_92540_91715_36436_78925_90360_01133_05305_48820_46652_13841_46951_94151_16094&quot;
     +  &quot;33057_27036_57595_91953_09218_61173_81932_61179_31051_18548_07446_23799_62749_56735_18857_52724_89122_79381_83011_94912&quot;
     +  &quot;98336_73362_44065_66430_86021_39494_63952_24737_19070_21798_60943_70277_05392_17176_29317_67523_84674_81846_76694_05132&quot;
     +  &quot;00056_81271_45263_56082_77857_71342_75778_96091_73637_17872_14684_40901_22495_34301_46549_58537_10507_92279_68925_89235&quot;
     +  &quot;42019_95611_21290_21960_86403_44181_59813_62977_47713_09960_51870_72113_49999_99837_29780_49951_05973_17328_16096_31859&quot;
     +  &quot;50244_59455_34690_83026_42522_30825_33446_85035_26193_11881_71010_00313_78387_52886_58753_32083_81420_61717_76691_47303&quot;
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Asymptotes of hyperbola==

===Line and hyperbola===
This section describes possibilities that arise when we consider intersection of line and hyperbola.
====With two common points====

{{RoundBoxTop|theme=2}}
[[File:01hyperbola01.png|thumb|400px|'''Diagram of hyperbola and line.'''
&lt;/br&gt;
Line and hyperbola have two common points.
&lt;/br&gt;
When line and hyperbola have two common points, line cannot be parallel to asymptote.
&lt;/br&gt;
]]
Line 1:
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 704, -1404, 1344, -11040, -41220, -161775

abc = a,b,c = .96, .28, .2
result = hyperbola_and_line (ABCDEF, abc) 
sx = 'result' ; print (sx, eval(sx))
&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
result [ (1.425,-5.6), (4.575,-16.4) ]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Length of latus rectum==
-----------------------

&lt;math&gt;b^2x^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2c^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2(a^2 - b^2) + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2a^2 - b^4 + a^2y^2 - a^2b^2 =0&lt;/math&gt;

&lt;math&gt;a^2y^2 = b^4&lt;/math&gt;

&lt;math&gt;y^2 = \frac{b^4}{a^2}&lt;/math&gt;

&lt;math&gt;y = \frac{b^2}{a}&lt;/math&gt;

Length of latus rectum &lt;math&gt;= L_1R_1 = L_2R_2 = \frac{2b^2}{a}.&lt;/math&gt;

=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Parabola====


&lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(18)x + (-24)y + (104) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-24)x + (32)y + (28) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-30)x + (40)y + (160) = 0&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never &lt;math&gt;0.75&lt;/math&gt; because axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.
]]
Consider conic section: &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;

This curve is a parabola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 9,16,-24,104,28,-144 # parabola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0
For slope vertical: (-24)x + (32)y + (28) = 0
For slope 2: (-30)x + (40)y + (160) = 0
&lt;/syntaxhighlight&gt;
Because all 3 lines are parallel to axis, all 3 lines have slope &lt;math&gt;\frac{3}{4}.&lt;/math&gt;


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.75:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

three_slopes (ABCDEF, 0.75, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0 # Same as above.
For slope vertical: (-24)x + (32)y + (28) = 0    # Same as above.
For slope 0.75: (0.0)x + (0.0)y + (125.0) = 0    # Impossible.
&lt;/syntaxhighlight&gt;
Axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.

&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Ellipse====

&lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(3542)x + (1944)y + (-44860) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(1944)x + (2408)y + (-18520) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(1598)x + (-464)y + (-26340) = 0&lt;/math&gt;


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[[File:0504ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;-1.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;

This curve is an ellipse.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;-1:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse
three_slopes (ABCDEF, -1, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
For slope horizontal: (3542)x + (1944)y + (-44860) = 0
For slope vertical: (1944)x + (2408)y + (-18520) = 0
For slope -1: (1598)x + (-464)y + (-26340) = 0
&lt;/syntaxhighlight&gt;
Because curve is closed loop, slope of curve may be any value including &lt;math&gt;\frac{1}{0}.&lt;/math&gt;

If slope of curve is given as &lt;math&gt;\frac{1}{0},&lt;/math&gt; it means that curve is vertical at that
point and tangent to curve has equation &lt;math&gt;x = k.&lt;/math&gt;

For any given slope there are always 2 points on opposite sides of curve where tangent to curve
at those points has the given slope.
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Hyperbola====


&lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(-702)x + (-336)y + (4182) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-336)x + (352)y + (-3824) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-1374)x + (368)y + (-3466) = 0&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

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[[File:0505hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
&lt;math&gt;\text{Line 2}&lt;/math&gt; and curve do not intersect. Slope is never vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;

This curve is a hyperbola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -351, 176, -336, 4182, -3824, -16231 # hyperbola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0
For slope horizontal: (-702)x + (-336)y + (4182) = 0
For slope vertical: (-336)x + (352)y + (-3824) = 0
For slope 2: (-1374)x + (368)y + (-3466) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
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[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
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===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
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Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
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All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
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Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
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[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


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==Ellipse at origin==
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[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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==Second definition of ellipse==
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[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
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===Proof===
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[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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==Heading==
===Properties of ellipse===
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[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

* Equation: &lt;math&gt;\frac {x^2} {a^2}  + \frac {y^2} {b^2} = 1&lt;/math&gt;

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
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====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
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===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


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[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

&lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>hutzo4lenw86n07f8m0rll9qkztwis4</sha1>
    </revision>
    <revision>
      <id>2689283</id>
      <parentid>2689282</parentid>
      <timestamp>2024-11-29T11:58:47Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* tan(A/2) */</comment>
      <origin>2689283</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71977" sha1="rxj0tc3dfnp7yb5xadq0plhpkz0ogqv" xml:space="preserve">= Hyperbola =
{{RoundBoxTop|theme=2}}
[[File:0911hyperbola00.png|thumb|400px|'''
Figure 1: Hyperbola at origin with transverse axis horizontal.''' &lt;/br&gt;&lt;/br&gt;

Origin at point &lt;math&gt;O&lt;/math&gt;&lt;math&gt;: (0,0)&lt;/math&gt;.&lt;/br&gt;
Foci are points &lt;math&gt;F_1 (-c,0),\ F_2 (c,0). OF_1 = OF_2 = c.&lt;/math&gt;&lt;/br&gt;
Vertices are points &lt;math&gt;V_1 (-a,0),\ V_2 (a,0). OV_1 = OV_2 = a.&lt;/math&gt;&lt;/br&gt;
Line segment &lt;math&gt;V_1OV_2&lt;/math&gt; is the &lt;math&gt;transverse\ axis.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;PF_1 - PF_2 = 2a.&lt;/math&gt;
]]

In cartesian [[geometry]] in two dimensions hyperbola is locus of a point &lt;math&gt;P&lt;/math&gt; that moves relative to two fixed points called &lt;math&gt;foci&lt;/math&gt;&lt;math&gt;: F_1, F_2.&lt;/math&gt;
The distance &lt;math&gt;F_1 F_2&lt;/math&gt; from one &lt;math&gt;focus\ (F_1)&lt;/math&gt; to the other &lt;math&gt;focus\ (F_2)&lt;/math&gt; is non-zero. The absolute difference of the distances &lt;math&gt;(PF_1, PF_2)&lt;/math&gt; from point to foci is constant.


&lt;math&gt;PF_1 - PF_2 = K.&lt;/math&gt; See figure 1. 


Center of hyperbola is located at the origin &lt;math&gt;O (0,0)&lt;/math&gt; and the foci &lt;math&gt;(F_1, F_2)&lt;/math&gt; are on the &lt;math&gt;X\ axis&lt;/math&gt;
at distance &lt;math&gt;c&lt;/math&gt; from &lt;math&gt;O. &lt;/math&gt;


&lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(-c, 0). F_2&lt;/math&gt; has coordinates &lt;math&gt;(c,0)&lt;/math&gt;. Line segments &lt;math&gt;OF_1 = OF_2 = c.&lt;/math&gt;


Each point &lt;math&gt;(V_1,V_2)&lt;/math&gt; where the curve intersects the transverse axis is called a &lt;math&gt;vertex.\ V_1,V_2&lt;/math&gt; are the vertices of the ellipse.


By definition &lt;math&gt;PF_1 - PF_2 = V_2F_1 - V_2F_2 = V_1F_2 - V_1F_1 = K.&lt;/math&gt;


&lt;math&gt;\therefore V_2F_1 - V_2F_2 = V_2F_1 - V_1F_1 = V_1V_2 = K = 2a,&lt;/math&gt; the length of the
&lt;math&gt;transverse\ axis\ (V_1V_2).&lt;/math&gt;
&lt;math&gt;OV_1 = OV_2 = a.&lt;/math&gt;
{{RoundBoxBottom}}
==Radians, the natural angle==
If you were a mathematician among the ancient Sumerians of the 3rd millennium BC and you were determined to define 
the angle that could be adopted as a standard to be used by all users of trigonometry, you would probably suggest
the angle in an equilateral triangle.
This angle is easily defined, easily constructed, easily understood and easily reproduced. It would be easy to call
this angle the &quot;natural&quot; angle.

The numeral system used by the ancient Sumerians was Sexagesimal, also known as base 60, a numeral system with sixty as its base.
In practice the natural angle could be divided into 60 parts, now called degrees, and each degree could be divided into
60 parts, now called minutes, and so on.

Three equilateral triangles fit neatly into a semi-circle, hence 180 degrees in a semi-circle.

We know that &lt;math&gt;\tan 30^\circ = \frac{\sqrt{3}}{3}.&lt;/math&gt;
Therefore, &lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; should be &lt;math&gt;0.5,&lt;/math&gt; or one half of our concept of the natural angle.

Whatever the natural angle might be, it has existed for billions of years, but it has come to light only in recent times
with invention of the calculus.

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:

&lt;math&gt;\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} + \cdots.&lt;/math&gt;

The following python code calculates 
&lt;math&gt;\arctan (\frac{\sqrt{3}}{3})&lt;/math&gt; using Gregory's series:
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

r3 = 3 ** .5
x = r3/3

arctan_x = (
      x        -  x**3/3  +  x**5/5  -  x**7/7  +  x**9/9  - x**11/11 + x**13/13 - x**15/15 + x**17/17 - x**19/19
    + x**21/21 - x**23/23 + x**25/25 - x**27/27 + x**29/29 - x**31/31 + x**33/33 - x**35/35 + x**37/37 - x**39/39
    + x**41/41 - x**43/43 + x**45/45 - x**47/47 + x**49/49 - x**51/51 + x**53/53 - x**55/55 + x**57/57 - x**59/59
    + x**61/61 - x**63/63 + x**65/65 - x**67/67 + x**69/69
)

sx = 'arctan_x' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x = 0.5235987755982988
&lt;/syntaxhighlight&gt;
Our assessment of the natural angle as the angle in an equilateral triangle was a very reasonable guess.

However, the natural angle is the radian, the angle that subtends an arc on the circumference of a circle equal to the radius.

Six times arctan_x &lt;math&gt;= 180^\circ&lt;/math&gt; or the number of radians in a semi-circle:

&lt;syntaxhighlight lang=python&gt;
# python code

sx = 'arctan_x * 6' ; print (sx, '=', eval(sx))
sx = '180/(arctan_x * 6)' ; print (sx, '=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
arctan_x * 6 = 3.141592653589793
180/(arctan_x * 6) = 57.29577951308232
&lt;/syntaxhighlight&gt;
&lt;math&gt;\pi = 3.141592653589793\dots,&lt;/math&gt; number of radians in semi-circle.

One radian &lt;math&gt;= 57.29577951308232^\circ,&lt;/math&gt; slightly less than &lt;math&gt;60^\circ.&lt;/math&gt;


Because the value &lt;math&gt;\frac\sqrt{3}{3}&lt;/math&gt; is fairly large, calculation of &lt;code&gt;arctan_x&lt;/code&gt; above required 34 operations
to produce result accurate to 16 places of decimals. The calculation did not converge quickly.

Python code below uses much smaller values of &lt;math&gt;x&lt;/math&gt; and calculation of &lt;code&gt;arctan_x&lt;/code&gt; for precision of 1001 is quite fast.

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

==tan(A/2)==

{{RoundBoxTop|theme=2}}
[[File:1122tanA_200.png|thumb|400px|'''Graphical calculation of &lt;math&gt;\tan \frac{A}{2}&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;OQ = 1;\ QP = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\tan(A) = \frac{QP}{OQ} = \frac{t}{1} = t.&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;OP = OR = \sqrt{1 + t^2}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Point &lt;math&gt;P&lt;/math&gt; has coordinates &lt;math&gt;(1,t).&lt;/math&gt;

Point &lt;math&gt;R&lt;/math&gt; has coordinates &lt;math&gt;(\sqrt{1 + t^2},0).&lt;/math&gt;

Mid point of &lt;math&gt;PR,\ M&lt;/math&gt; has coordinates &lt;math&gt;(   \frac{ 1 +  \sqrt{1 + t^2}  }{2}, \frac{t}{2}            ).&lt;/math&gt;

&lt;math&gt;\tan \frac{A}{2} = \frac{t}{2} / \frac{ 1 +  \sqrt{1 + t^2}  }{2} = \frac{t}{1 +  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t}{1 +  \sqrt{1 + t^2} } \cdot \frac{1 -  \sqrt{1 + t^2}}{1 -  \sqrt{1 + t^2} }&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{1-(1+t^2)}&lt;/math&gt;
&lt;math&gt;= \frac{t(   1 -  \sqrt{1 + t^2}    )}{-t^2}&lt;/math&gt;
&lt;math&gt;= \frac{-1 +  \sqrt{1 + t^2} }{t}&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


* &lt;math&gt;\tan \frac{A}{2} = \frac{\tan(A)}{1 + \sqrt{1 + \tan^2(A)}} = \frac{-1 +  \sqrt{1 + \tan^2 (A)} }{\tan (A)}&lt;/math&gt;

* &lt;math&gt;\tan (2A) = \frac{2\tan (A)}{ 1 - \tan^2 (A) }&lt;/math&gt;
{{RoundBoxBottom}}

==Implementation==
{{RoundBoxTop|theme=2}}
This section calculates five values of &lt;math&gt;\pi&lt;/math&gt; using the following known values of &lt;math&gt;\tan(A):&lt;/math&gt;
{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;A&lt;/math&gt; || &lt;math&gt;\tan(A)&lt;/math&gt;
|-
| &lt;math&gt;45^\circ&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|-
| &lt;math&gt;36^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{   5 - 2\sqrt{5}  }&lt;/math&gt;
|-
| &lt;math&gt;30^\circ&lt;/math&gt;
| &lt;math&gt;\frac{\sqrt{3}}{3}&lt;/math&gt;
|-
| &lt;math&gt;27^\circ&lt;/math&gt;
| &lt;math&gt;\sqrt{    11 - 4\sqrt{5} + (\sqrt{5} - 3)   \sqrt{   10 - 2\sqrt{5}      }   }&lt;/math&gt;
|-
| &lt;math&gt;24^\circ&lt;/math&gt;
| &lt;math&gt;\frac{       (3\sqrt{5} + 7)     \sqrt{5 - 2\sqrt{5}} - (\sqrt{5} + 3)\sqrt{3}     }{2}&lt;/math&gt;
|}

Values of &lt;math&gt;x&lt;/math&gt; in table below are derived from the above values by using identity &lt;math&gt;\tan(\frac{A}{2}) = \frac{-1 + \sqrt{1 + \tan^2(A)}}{\tan(A)}&lt;/math&gt;:

{| class=&quot;wikitable&quot;
|- 
! Angle &lt;math&gt;\theta&lt;/math&gt; || &lt;math&gt;x = \tan(\theta)&lt;/math&gt;
|-
| &lt;math&gt;\frac{45^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_91432_37995_4197.....089_03901_63759_3912&lt;/code&gt;
|-
| &lt;math&gt;\frac{36^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_73145_90396_3357.....211_97500_56173_0713&lt;/code&gt;
|-
| &lt;math&gt;\frac{30^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_60954_91996_9464.....024_32806_94580_0689&lt;/code&gt;
|-
| &lt;math&gt;\frac{27^\circ}{2^{33}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_54859_42797_2518.....791_30634_03540_9738&lt;/code&gt;
|-
| &lt;math&gt;\frac{24^\circ}{2^{32}}&lt;/math&gt;
| &lt;code&gt;0.00000_00000_97527_87195_1143.....736_60376_04724_6778&lt;/code&gt;
|}


&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

desired_precision = 1001
number_of_leading_zeroes = 10 # See below.

import decimal
dD = decimal.Decimal # decimal object is like float with (almost) infinite precision.
dgt = decimal.getcontext()

Precision = dgt.prec = desired_precision + 3  # Adjust as necessary.
Tolerance = dD(&quot;1e-&quot; + str(Precision-2))      # Adjust as necessary.
adjustment_to_precision = number_of_leading_zeroes * 2 + 3

def tan_halfA(tan_A) :
    dgt.prec += adjustment_to_precision
    top = -1 + (1+tan_A**2).sqrt()
    dgt.prec -= adjustment_to_precision
    tan_A_2 = top/tan_A
    return tan_A_2
    
def tan_2A (tanA) :
    '''
          2 * tanA
tan(2A) = -----------
          1 - tanA**2
    '''
    if tanA in (1,-1) : return '1/0'
    dgt.prec += adjustment_to_precision
    bottom = (1 - tanA**2)
    output = 2*tanA/bottom
    dgt.prec -= adjustment_to_precision
    return output+0

def θ_tanθ_from_A_tanA (angleA, tanA) :
    '''
if input == 45,1
output is:
&quot;dD(45) / (2 ** (33))&quot;, &quot;0.00000_00000_91432_37995_....._63759_3912&quot;
                           ^^^^^^^^^^^
number_of_leading_zeroes refers to these zeroes.
θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA)
    '''
    θ, tanθ = angleA, tanA
    for p in range (1,100) :
        θ /= 2
        tanθ = tan_halfA(tanθ)
        if tanθ &gt;= dD('1e-' + str(number_of_leading_zeroes)) : continue
        str1 = str(tanθ)
        # str1 = &quot;n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11&quot;
        str1a = str1[0] + str1[2:-4]
        list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ]
        str2 = '0.00000_00000_' + ('_'.join(list1))
        dD2 = dD(str2)
        (dD2 == tanθ) or ({}[2])
        ((θ * (2**p)) == angleA ) or ({}[3])
        str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p)
        (θ == eval(str3)) or ({}[4])
        return str3, str2
    ({}[5])


r3 = dD(3).sqrt()
r5 = dD(5).sqrt()
tan36 = (5 - 2*r5).sqrt()
tan45 = dD(1)
tan30 = r3/3

v1 = 3*r5+7
v2 = (5 - 2*r5).sqrt()
v3 = (r5+3)*r3
tan24 = ( v1*v2 - v3 )/2

v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt()
tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt()


values_of_A_tanA = (
    (dD(45), tan45),
    (dD(36), tan36),
    (dD(30), tan30),
    (dD(27), tan27),
    (dD(24), tan24),
)

values_of_θ_tanθ = []
for (A, tanA) in values_of_A_tanA :
    θ, tanθ = θ_tanθ_from_A_tanA (A, tanA)
    print()
    sx = 'θ' ; print (sx, '=', eval(sx))
#    sx = 'tanθ' ; print (sx, '=', eval(sx))
    print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:]))
    values_of_θ_tanθ += [ (θ, tanθ) ]

# Check
for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) :
    A, tanA = v1,v2
    θ = eval(v3)
    tanθ = dD(v4)
    status = 0
    for p in range (1,100) :
        θ *= 2
        tanθ = tan_2A (tanθ) 
        if θ == A :
            dgt.prec = desired_precision
            (+tanθ == +tanA) or ({}[10])
            dgt.prec = Precision            
            status = 1
            break
    status or ({}[11])
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
θ = dD(45) / (2 ** (33))
tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912

θ = dD(36) / (2 ** (33))
tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713

θ = dD(30) / (2 ** (33))
tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689

θ = dD(27) / (2 ** (33))
tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738

θ = dD(24) / (2 ** (32))
tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
# python code

def calculate_π (angleθ, tanθ) :
    '''
angleθ may be: &quot;dD(27) / (2 ** (33))&quot;
tanθ may be: &quot;0.00000_00000_54859_42797_ ..... _03540_9738&quot;
π = calculate_π (angleθ, tanθ)
    '''
    thisName = 'calculate_π (angleθ, tanθ) :'

    if isinstance(angleθ, dD) : pass
    elif isinstance(angleθ, str) : angleθ = eval(angleθ)
    else : ({}[21])
    if isinstance(tanθ, dD) : pass
    elif isinstance(tanθ, str) : tanθ = dD(tanθ)
    else : ({}[22])
    
    x = tanθ ; multiplier = -1 ; sum = x ; count = 0; status = 0

    #         x**3   x**5   x**7   x**9
    # y = x - ---- + ---- - ---- + ----
    #           3      5      7      9
    # 
    # Each term in the sequence is roughly the previous term multiplied by x**2.
    # Each value of x contains 10 leading zeroes after decimal point.
    # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes.
    # Each pass through main loop adds about 20 digits to current value of sum
    # and θ is calculated to precision of 1004 digits with about 50 passes through main loop.
    # 
    for p in range (3,200,2) :
        # This is main loop.
        count += 1
        addendum = (multiplier * (x**p)) / p
        sum += addendum
        if abs(addendum) &lt; Tolerance :
            status = 1; break
        multiplier = -multiplier
    status or ({}[23])
    print(thisName, 'count =',count)
    π = sum * 180 / angleθ
    dgt.prec = desired_precision
    π += 0 # This forces π to adopt precision of desired_precision.
    dgt.prec = Precision
    return π

# Calculate five values of π:
values_of_π = []
for θ,tanθ in values_of_θ_tanθ :
    π = calculate_π (θ,tanθ)
    values_of_π += [ π ]
&lt;/syntaxhighlight&gt;
Each calculation of π required about 50 passes through main loop:
&lt;syntaxhighlight&gt;
calculate_π (angleθ, tanθ) : count = 50
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 50
&lt;/syntaxhighlight&gt;
Check that all 5 values of π are equal:
&lt;syntaxhighlight lang=python&gt;
# python code

set1 = set(values_of_π)
sx = 'len(values_of_π)' ; print (sx, '=', eval(sx))
sx = 'len(set1)' ; print (sx, '=', eval(sx))
sx = 'set1' ; print (sx, '=', eval(sx))
π, = set1 # Note the syntax. If length of set1 is not 1, this statement fails.
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
len(values_of_π) = 5
len(set1) = 1
set1 = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')}
&lt;/syntaxhighlight&gt;
Print value of π as python command formatted:
&lt;syntaxhighlight lang=python&gt;
# python code

newLine = '''
'''[-1:]

def print_π (π) :
    '''
Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989')
This function prints:
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     .....
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
    '''
    πstr = str(π)
    (len(πstr) == (desired_precision + 1)) or ({}[31])
    (πstr[:2] == '3.') or ({}[32])

    ten_rows = []
    for p in range (2, len(πstr), 100) :
        str1a = πstr[p:p+100]
        list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ]
        str1b = '_'.join(list1a)
        ten_rows += [str1b]
    ten_rows[0] = '3.' + ten_rows[0]
    joiner = '&quot;{}     +  &quot;'.format(newLine)
    str3 = '( &quot;{}&quot; )'.format(joiner.join(ten_rows))
    
    str4 = eval(str3)
    (dD(str4) == π) or ({}[33])
    print ('π =', str3)
    return str3

π1 = print_π (π)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
π = ( &quot;3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510_58209_74944_59230_78164_06286_20899_86280_34825_34211_70679&quot;
     +  &quot;82148_08651_32823_06647_09384_46095_50582_23172_53594_08128_48111_74502_84102_70193_85211_05559_64462_29489_54930_38196&quot;
     +  &quot;44288_10975_66593_34461_28475_64823_37867_83165_27120_19091_45648_56692_34603_48610_45432_66482_13393_60726_02491_41273&quot;
     +  &quot;72458_70066_06315_58817_48815_20920_96282_92540_91715_36436_78925_90360_01133_05305_48820_46652_13841_46951_94151_16094&quot;
     +  &quot;33057_27036_57595_91953_09218_61173_81932_61179_31051_18548_07446_23799_62749_56735_18857_52724_89122_79381_83011_94912&quot;
     +  &quot;98336_73362_44065_66430_86021_39494_63952_24737_19070_21798_60943_70277_05392_17176_29317_67523_84674_81846_76694_05132&quot;
     +  &quot;00056_81271_45263_56082_77857_71342_75778_96091_73637_17872_14684_40901_22495_34301_46549_58537_10507_92279_68925_89235&quot;
     +  &quot;42019_95611_21290_21960_86403_44181_59813_62977_47713_09960_51870_72113_49999_99837_29780_49951_05973_17328_16096_31859&quot;
     +  &quot;50244_59455_34690_83026_42522_30825_33446_85035_26193_11881_71010_00313_78387_52886_58753_32083_81420_61717_76691_47303&quot;
     +  &quot;59825_34904_28755_46873_11595_62863_88235_37875_93751_95778_18577_80532_17122_68066_13001_92787_66111_95909_21642_01989&quot; )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Asymptotes of hyperbola==

===Line and hyperbola===
This section describes possibilities that arise when we consider intersection of line and hyperbola.
====With two common points====

{{RoundBoxTop|theme=2}}
[[File:01hyperbola01.png|thumb|400px|'''Diagram of hyperbola and line.'''
&lt;/br&gt;
Line and hyperbola have two common points.
&lt;/br&gt;
When line and hyperbola have two common points, line cannot be parallel to asymptote.
&lt;/br&gt;
]]
Line 1:
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 704, -1404, 1344, -11040, -41220, -161775

abc = a,b,c = .96, .28, .2
result = hyperbola_and_line (ABCDEF, abc) 
sx = 'result' ; print (sx, eval(sx))
&lt;/syntaxhighlight&gt;
Code returns list containing two points:
&lt;syntaxhighlight&gt;
result [ (1.425,-5.6), (4.575,-16.4) ]
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==Length of latus rectum==
-----------------------

&lt;math&gt;b^2x^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2c^2 + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2(a^2 - b^2) + a^2y^2 - a^2b^2 = 0&lt;/math&gt;

&lt;math&gt;b^2a^2 - b^4 + a^2y^2 - a^2b^2 =0&lt;/math&gt;

&lt;math&gt;a^2y^2 = b^4&lt;/math&gt;

&lt;math&gt;y^2 = \frac{b^4}{a^2}&lt;/math&gt;

&lt;math&gt;y = \frac{b^2}{a}&lt;/math&gt;

Length of latus rectum &lt;math&gt;= L_1R_1 = L_2R_2 = \frac{2b^2}{a}.&lt;/math&gt;

=Conic sections generally=
Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere
and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Slope of curve==

Given equation of conic section: &lt;math&gt;Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,&lt;/math&gt;

differentiate both sides with respect to &lt;math&gt;x.&lt;/math&gt;


&lt;math&gt;2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0&lt;/math&gt;

&lt;math&gt;2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0&lt;/math&gt;

&lt;math&gt;y'(2By + Cx + E) = -(2Ax + Cy + D)&lt;/math&gt;


&lt;math&gt;y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;


For slope horizontal: &lt;math&gt;2Ax + Cy + D = 0.&lt;/math&gt;

For slope vertical: &lt;math&gt;Cx + 2By + E = 0.&lt;/math&gt;

For given slope &lt;math&gt;m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}&lt;/math&gt;

&lt;math&gt;m(Cx + 2By + E) = -2Ax - Cy - D&lt;/math&gt;

&lt;math&gt;mCx + 2Ax + m2By + Cy + mE + D = 0&lt;/math&gt;

&lt;math&gt;(mC + 2A)x + (m2B + C)y + (mE + D) = 0.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
===Implementation===
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
# python code

def three_slopes (ABCDEF, slope, flag = 0) :
    '''
equation1, equation2, equation3 = three_slopes (ABCDEF, slope[, flag]) 
equation1 is equation for slope horizontal.
equation2 is equation for slope vertical.
equation3 is equation for slope supplied.
All equations are in format (a,b,c) where ax + by + c = 0.
    '''
    A,B,C,D,E,F = ABCDEF
    output = []

    abc = 2*A, C, D ; output += [ abc ]
    abc = C, 2*B, E ; output += [ abc ]

    m = slope
    # m(Cx + 2By + E) = -2Ax - Cy - D
    # mCx + m2By + mE = -2Ax - Cy - D
    # mCx + 2Ax + m2By + Cy + mE + D = 0
    abc = m*C + 2*A, m*2*B + C, m*E + D ; output += [ abc ]

    if flag :
        str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format (A,B,C,D,E,F)
        print (str1)
        a,b,c = output[0]
        str1 = 'For slope horizontal: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[1]
        str1 = 'For slope vertical: ({})x + ({})y + ({}) = 0'.format (a,b,c)
        print (str1)
        a,b,c = output[2]
        str1 = 'For slope {}: ({})x + ({})y + ({}) = 0'.format (slope, a,b,c)
        print (str1)
        
    return output
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
===Examples===
====Quadratic function====

&lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

&lt;math&gt;\text{line 1:}\ x = 7&lt;/math&gt;

&lt;math&gt;\text{line 2:}\ x = 17&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
=====y = f(x)=====
{{RoundBoxTop|theme=2}}
[[File:0502quadratic01.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope = &lt;math&gt;0&lt;/math&gt;.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never vertical.
]]
Consider conic section: &lt;math&gt;(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;y = \frac{x^2 - 14x - 39}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;m = y' = \frac{2x - 14}{4}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -1,0,0,14,4,39 # quadratic 
three_slopes (ABCDEF, 5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0
For slope horizontal: (-2)x + (0)y + (14) = 0  # x = 7
For slope vertical: (0)x + (0)y + (4) = 0      # This does not make sense. 
                                               # Slope is never vertical. 
For slope 5: (-2)x + (0)y + (34) = 0           # x = 17. 
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for x in (7,17) :
    m = (2*x - 14)/4
    s1 = 'x,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
x,m (7, 0.0)   # When x = 7, slope = 0. 
x,m (17, 5.0)  # When x =17, slope = 5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=====x = f(y)=====

&lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;


&lt;math&gt;\text{line 1:}\ y = -7&lt;/math&gt;


&lt;math&gt;\text{line 2:}\ y = -11&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0502quadratic02.png|thumb|400px|'''Graph of quadratic function &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 1}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{line 2}&lt;/math&gt; and curve, slope = &lt;math&gt;0.5&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never horizontal.
]]
Consider conic section: &lt;math&gt;(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.&lt;/math&gt;

This is quadratic function: &lt;math&gt;x = \frac{-(y^2 + 14y + 5)}{4}&lt;/math&gt;

Slope of this curve: &lt;math&gt;\frac{dx}{dy} = \frac{-2y - 14}{4}&lt;/math&gt;

&lt;math&gt;m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}&lt;/math&gt;

Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.5:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 0,-1,0,-4,-14,-5 # quadratic x = f(y)
three_slopes (ABCDEF, 0.5, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0
For slope horizontal: (0)x + (0)y + (-4) = 0    # This does not make sense.
                                                # Slope is never horizontal.
For slope vertical: (0)x + (-2)y + (-14) = 0    # y = -7
For slope 0.5: (0.0)x + (-1.0)y + (-11.0) = 0   # y = -11
&lt;/syntaxhighlight&gt;
Check results:

&lt;syntaxhighlight lang=python&gt;
# python code

for y in (-7,-11) :
    top = -4 ; bottom = 2*y + 14
    if bottom == 0 :
        print ('y,m',y,'{}/{}'.format(top,bottom))
        continue
    m = top/bottom
    s1 = 'y,m' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
y,m -7 -4/0     # When y = -7, slope is vertical.
y,m (-11, 0.5)  # When y = -11, slope is 0.5.
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Parabola====


&lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(18)x + (-24)y + (104) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-24)x + (32)y + (28) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-30)x + (40)y + (160) = 0&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504parabola01.png|thumb|400px|'''Graph of parabola &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At interscetion of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2&lt;/math&gt;.&lt;/br&gt;
Slope of curve is never &lt;math&gt;0.75&lt;/math&gt; because axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.
]]
Consider conic section: &lt;math&gt;(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.&lt;/math&gt;

This curve is a parabola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 9,16,-24,104,28,-144 # parabola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0
For slope vertical: (-24)x + (32)y + (28) = 0
For slope 2: (-30)x + (40)y + (160) = 0
&lt;/syntaxhighlight&gt;
Because all 3 lines are parallel to axis, all 3 lines have slope &lt;math&gt;\frac{3}{4}.&lt;/math&gt;


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;0.75:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

three_slopes (ABCDEF, 0.75, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0
For slope horizontal: (18)x + (-24)y + (104) = 0 # Same as above.
For slope vertical: (-24)x + (32)y + (28) = 0    # Same as above.
For slope 0.75: (0.0)x + (0.0)y + (125.0) = 0    # Impossible.
&lt;/syntaxhighlight&gt;
Axis has slope &lt;math&gt;0.75&lt;/math&gt; and curve is never parallel to axis.

&lt;syntaxhighlight lang=python&gt;
# python code

&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Ellipse====

&lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(3542)x + (1944)y + (-44860) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(1944)x + (2408)y + (-18520) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(1598)x + (-464)y + (-26340) = 0&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0504ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 2}&lt;/math&gt; and curve, slope is vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;-1.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.&lt;/math&gt;

This curve is an ellipse.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;-1:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse
three_slopes (ABCDEF, -1, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
For slope horizontal: (3542)x + (1944)y + (-44860) = 0
For slope vertical: (1944)x + (2408)y + (-18520) = 0
For slope -1: (1598)x + (-464)y + (-26340) = 0
&lt;/syntaxhighlight&gt;
Because curve is closed loop, slope of curve may be any value including &lt;math&gt;\frac{1}{0}.&lt;/math&gt;

If slope of curve is given as &lt;math&gt;\frac{1}{0},&lt;/math&gt; it means that curve is vertical at that
point and tangent to curve has equation &lt;math&gt;x = k.&lt;/math&gt;

For any given slope there are always 2 points on opposite sides of curve where tangent to curve
at those points has the given slope.
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

====Hyperbola====


&lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 1:}&lt;/math&gt;

&lt;math&gt;(-702)x + (-336)y + (4182) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 2:}&lt;/math&gt;

&lt;math&gt;(-336)x + (352)y + (-3824) = 0&lt;/math&gt;


&lt;math&gt;\text{Line 3:}&lt;/math&gt;

&lt;math&gt;(-1374)x + (368)y + (-3466) = 0&lt;/math&gt;


&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

{{RoundBoxTop|theme=2}}
[[File:0505hyperbola01.png|thumb|400px|'''Graph of hyperbola &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;'''
&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 1}&lt;/math&gt; and curve, slope is horizontal.&lt;/br&gt;
&lt;math&gt;\text{Line 2}&lt;/math&gt; and curve do not intersect. Slope is never vertical.&lt;/br&gt;
At intersection of &lt;math&gt;\text{Line 3}&lt;/math&gt; and curve, slope = &lt;math&gt;2.&lt;/math&gt;
]]
Consider conic section: &lt;math&gt;(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.&lt;/math&gt;

This curve is a hyperbola.


Produce values for slope horizontal, slope vertical and slope &lt;math&gt;2:&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = -351, 176, -336, 4182, -3824, -16231 # hyperbola
three_slopes (ABCDEF, 2, 1)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0
For slope horizontal: (-702)x + (-336)y + (4182) = 0
For slope vertical: (-336)x + (352)y + (-3824) = 0
For slope 2: (-1374)x + (368)y + (-3466) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
{{RoundBoxBottom}}

===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
{{RoundBoxTop|theme=2}}
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Second definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
{{RoundBoxBottom}}
===Proof===
{{RoundBoxTop|theme=2}}
[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Heading==
===Properties of ellipse===
{{RoundBoxTop|theme=2}}
[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

* Equation: &lt;math&gt;\frac {x^2} {a^2}  + \frac {y^2} {b^2} = 1&lt;/math&gt;

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

&lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
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&lt;syntaxhighlight lang=python&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>rxj0tc3dfnp7yb5xadq0plhpkz0ogqv</sha1>
    </revision>
  </page>
  <page>
    <title>Algebra II/Functions</title>
    <ns>0</ns>
    <id>248581</id>
    <revision>
      <id>2689255</id>
      <parentid>2005577</parentid>
      <timestamp>2024-11-29T02:28:50Z</timestamp>
      <contributor>
        <username>Patricielaa</username>
        <id>2993063</id>
      </contributor>
      <minor />
      <origin>2689255</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1015" sha1="f1flk04xnzqwq2k7f1me5843fpu2h9a" xml:space="preserve">:''Intro to Functions: [[Speak Math Now!/Week 4: Functions]]''

A '''vertical line test''' serves to check if a relation is a function or not. If more than two dots touch the &quot;vertical line&quot;, it is ''not'' a function. Therefore, a function cannot be vertical (but, on the contrast, it can be horizontal). 
[[Category:Algebra]]

Functions are graphical representation of algebra. Thus there will always be an algebraic equation to solve before representing it on the Cartesian plane.

A Cartesian plane is a cross that consist of two axis, the x and y axis. On this cross the horizontal line of it is the 'x axis' and the vertical line is the 'y axis'. The point of intersection of these two axis is called the origin. Numbers above the origin and on the y axis are positive and numbers below the origin are negative numbers of y axis. Numbers on the left side of x axis are negative and numbers on the right side are positive.Between these numbers is the point of intersection and the numerical value there is zero.</text>
      <sha1>f1flk04xnzqwq2k7f1me5843fpu2h9a</sha1>
    </revision>
  </page>
  <page>
    <title>Quantum gravity (Planck)</title>
    <ns>0</ns>
    <id>250057</id>
    <revision>
      <id>2689254</id>
      <parentid>2688712</parentid>
      <timestamp>2024-11-29T01:31:59Z</timestamp>
      <contributor>
        <username>Platos Cave (physics)</username>
        <id>2562653</id>
      </contributor>
      <minor />
      <origin>2689254</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49020" sha1="55weyta2se83vxa3ej8g7zefqwyuhju" xml:space="preserve">'''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale'''

An orbital simulation program is described that emulates gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations at the [[w:Planck_units |Planck scale]]. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the Planck units for comparison to real world orbits.

[[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]]
For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''&lt;sub&gt;P&lt;/sub&gt; (for example, a 1kg satellite would be divided into 1kg/''m''&lt;sub&gt;P&lt;/sub&gt; = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=3. Gravitational orbits emerge from Planck scale n-body rotating orbital pairs |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.11496.93445/17}}&lt;/ref&gt;. 

Each orbital pair rotates 1 unit of length ([[w:Planck length |Planck length]] ''l''&lt;sub&gt;p&lt;/sub&gt;)  per unit of time ([[w:Planck time |Planck time]] ''t''&lt;sub&gt;p&lt;/sub&gt;) which in Planck units equates to velocity ''c'' (''c'' = ''l''&lt;sub&gt;p&lt;/sub&gt;/''t''&lt;sub&gt;p&lt;/sub&gt;) in [[v:Relativity (Planck) |hypersphere space]] co-ordinates, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... 

The basic simulation uses only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs, information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). 

For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together and a 'photon' is added in a series of steps. As the electron continues to orbit the nucleus during the transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although the spiral path is semi-classical, it exhibits the quantum states, and suggests that quantization could have geometrical origins.  


=== Theory ===
The simulation itself is a dimensionless geometrical model, but it can be measured in Planck units. In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model.

If the electron '''is''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have''' 1 unit of Planck mass at every unit of Planck time, the object will require 10&lt;sup&gt;23&lt;/sup&gt; electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (our measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. 

Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the photon-orbital model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;, in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). 

It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels (at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, we may ask if quantization of the atom has a geometrical origin.

Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition periods correlate closely with the observed experimental data. We also note a 'transition signature' where the duration of period between different energy levels (''n'' shells) is not consistent, and this signature appears in both the experimental and the simulated results. This deviation from the expected results is most pronounced in the ''n'' = 1 to ''n'' = 2 transition (where the transition period is comparatively longer).
 
In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; mass to mass or wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). 

The gravitational orbits that we observe are the time averaging sum of the underlying gravitational orbitals. Gravity is therefore not weaker than the electric force, rather it is stronger at the Planck scale (for point-point orbitals rotate faster than wave-wave), its apparent weakness is simply because point-point rotations seldom occur relative to wave-wave orbitals (the probability of occurrence as the inverse of the gravitational coupling constant which for each electron would be the ratio &lt;math&gt;1:10^{23}&lt;/math&gt;). 


=== N-body orbitals ===
[[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]]

The simulation universe is a 4-axis hypersphere expanding in increments &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}&lt;/ref&gt; with 3-axis (the [[v:Black-hole_(Planck)  |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'.

The simulation itself is dimensionless, simply rotating geometries. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |mass m&lt;sub&gt;P&lt;/sub&gt;]], [[w:Planck length |length l&lt;sub&gt;p&lt;/sub&gt;]], [[w:Planck time |time t&lt;sub&gt;p&lt;/sub&gt;]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''&lt;sub&gt;p&lt;/sub&gt;/''t''&lt;sub&gt;p&lt;/sub&gt;). These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. 

Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). 

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt; 

As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation.

Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt; is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence.


[[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]]
==== 2 body orbits ('''x, y''' plane) ====
For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. 

The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units

:&lt;math&gt;r_s = \frac{2 l_p M}{m_P}&lt;/math&gt;

As the simulation itself is dimensionless, we can remove the dimensioned length component &lt;math&gt;2 l_p&lt;/math&gt;, and as each point is analogous to 1 unit of Planck mass &lt;math&gt;m_P&lt;/math&gt;, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius &lt;math&gt;r_s&lt;/math&gt; = number of central points = total points - 1 (the orbiting point).


After every orbital has rotated 1 (Planck) length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. 

Key:

1. &lt;math&gt;r_s&lt;/math&gt; = '''i'''; number of center points in the orbit (the center mass). 

2. '''j''' = total number of points (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 = 82).

3. '''j&lt;sub&gt;max&lt;/sub&gt;''' = radius to mass co-efficient.

4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 

5. '''r&lt;sub&gt;α&lt;/sub&gt;''' = a radius constant, here r&lt;sub&gt;α&lt;/sub&gt; = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals.

:&lt;math&gt;r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} &lt;/math&gt;


==== Orbital formulas (2-D plane)====

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2&lt;/math&gt;, orbital radius

:&lt;math&gt;r_{barycenter} = \frac{r_{outer}}{j}&lt;/math&gt;, barycenter

:&lt;math&gt;v_{outer} = \frac{i}{j_{max} r_{\alpha}} &lt;/math&gt;, orbiting point velocity  

:&lt;math&gt;v_{inner} = \frac{1}{j_{max} r_{\alpha}}&lt;/math&gt;, orbited point(s) velocity  

:&lt;math&gt;t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 &lt;/math&gt;, orbiting point period

:&lt;math&gt;l_{outer} = 2 \pi (r_{outer} - r_{barycenter})&lt;/math&gt;, distance travelled 


Simulation data:

:period &lt;math&gt;t_{sim}&lt;/math&gt;

:length &lt;math&gt;l_{sim} = t_{sim} \frac{i}{j_{max}r_{\alpha}}&lt;/math&gt;

:radius &lt;math&gt;r_{sim} = \frac{l_{sim}}{2 \pi}&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = \frac{l_{sim}}{t_{sim}} = \frac{i}{r_{\alpha} j_{max}}&lt;/math&gt; 

:barycenter &lt;math&gt;b_{sim} = \frac{x_{max} + x_{min}}{2}&lt;/math&gt;


[[File:gravity-orbit-hyperbolic-spiral.jpg|thumb|right|576px|Object leaving a gravitational circular orbit (j&lt;sub&gt;max&lt;/sub&gt; = j) with constant outward motion follows the same [[v:Fine-structure_constant_(spiral) |alpha hyperbolic spiral]] as an ionizing electron]]
For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 

1) ''j''&lt;sub&gt;max&lt;/sub&gt; = i+1 = 8 

:period &lt;math&gt;t = 74465.0516,\; t_{outer} = 74471.6125&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 626.3951&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/18.920137&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 89.5241,\; r_{barycenter} = 89.4929&lt;/math&gt;


2) ''j''&lt;sub&gt;max&lt;/sub&gt; = 32*i+1 = 225 

:period &lt;math&gt;t = 1656793370.3483,\; t_{outer} = 1656793381.3051&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 495531.959&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/532.128856&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 70790.283, \;r_{barycenter} = 70790.280&lt;/math&gt;


3) Moon orbit. 

From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1.

:&lt;math&gt;\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663&lt;/math&gt;

:&lt;math&gt;r_{earth-moon}&lt;/math&gt; = 384400km

:&lt;math&gt;M_{earth}&lt;/math&gt; = 0.597378 10&lt;sup&gt;25&lt;/sup&gt;kg


Solving &lt;math&gt;j_{max}&lt;/math&gt;

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}&lt;/math&gt;

:&lt;math&gt;j_{max} = 1440443&lt;/math&gt;


Gives

:&lt;math&gt;t_{outer} =  4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}&lt;/math&gt;s

:&lt;math&gt;t_{outer} \frac{M_{earth}} {m_P } = 2371844&lt;/math&gt;s (27.452 days)

:&lt;math&gt;v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s&lt;/math&gt;

:&lt;math&gt;v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}}  = 12.57m/s&lt;/math&gt;

:&lt;math&gt;r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km&lt;/math&gt;


==== Gravitational coupling constant ====
In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α&lt;sub&gt;G&lt;/sub&gt;'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

:&lt;math&gt;\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}&lt;/math&gt;

For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α&lt;sub&gt;G&lt;/sub&gt; then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt;). 

:&lt;math&gt;{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}&lt;/math&gt;

As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α&lt;sub&gt;G&lt;/sub&gt; is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10&lt;sup&gt;42&lt;/sup&gt; units of Planck time (&lt;math&gt;t_p = 10^{-42}s&lt;/math&gt;), this occurs about once every 3 minutes.

:&lt;math&gt;\frac{{\alpha_G}^{-1}}{t_p}&lt;/math&gt;

Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak.  

If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f&lt;sub&gt;e&lt;/sub&gt;, then the point will require 10&lt;sup&gt;23&lt;/sup&gt; electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time).

:&lt;math&gt;f_e = \frac{m_P}{m_e} = 10^{23}&lt;/math&gt;

For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have &lt;math&gt;1kg/m_P = 45940509&lt;/math&gt; particles in the point-state. The earth (A) will have &lt;math&gt;5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}&lt;/math&gt; particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite;

:&lt;math&gt;N_{orbitals} =  (\frac{m_A}{m_P})(\frac{m_B}{m_P})  = 0.1261\; x10^{41}&lt;/math&gt;
 

Examples:

:&lt;math&gt;i = \frac{M_{earth}}{m_P} = 0.27444  \;x10^{33}&lt;/math&gt; (earth as the center mass)

:&lt;math&gt;i 2 l_p = 0.00887&lt;/math&gt; (earth Schwarzschild radius)

:&lt;math&gt;s = \frac{1kg}{m_P} = 45940509&lt;/math&gt; (1kg orbiting satellite)

:&lt;math&gt;j = N_{orbitals} = i*s = 0.1261  \;x10^{41}&lt;/math&gt;


1) 1kg satellite at earth surface orbit 

:&lt;math&gt;r_{o} = 6371000 km&lt;/math&gt; (earth surface)

:&lt;math&gt;j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 2289.41&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 7909.7924&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374&lt;/math&gt; s 


2) 1kg satellite at a synchronous orbit radius

:&lt;math&gt;r_o = 42164.17 km&lt;/math&gt; 

:&lt;math&gt;j_{max} =  \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}}  = 0.74256\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 5889.674&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 3074.66&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165&lt;/math&gt; s 


3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). 

:&lt;math&gt;E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J&lt;/math&gt; (energy per orbital)

:&lt;math&gt;N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}&lt;/math&gt; (number of orbitals)

:&lt;math&gt;E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg&lt;/math&gt;


4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.

:&lt;math&gt;N_{sun} = \frac{M_{sun}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{planet} = \frac{M_{planet}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{orbitals} = N_{sun}N_{planet} &lt;/math&gt;

:&lt;math&gt;n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} &lt;/math&gt;

:&lt;math&gt;L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} &lt;/math&gt;

The orbital angular momentum of the planets;

 mercury = .9153 x10&lt;sup&gt;39&lt;/sup&gt;  
 venus    = .1844 x10&lt;sup&gt;41&lt;/sup&gt;  
 earth    = .2662 x10&lt;sup&gt;41&lt;/sup&gt; 
 mars     = .3530 x10&lt;sup&gt;40&lt;/sup&gt; 
 jupiter   = .1929 x10&lt;sup&gt;44&lt;/sup&gt;   
 pluto   = .365 x10&lt;sup&gt;39&lt;/sup&gt;   


Orbital angular momentum combined with orbit velocity cancels ''n&lt;sub&gt;g&lt;/sub&gt;'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.

:&lt;math&gt;L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} &lt;/math&gt;


[[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2&lt;sup&gt;32&lt;/sup&gt; steps they have clumped to form 1 large mass and 2 orbiting masses.]]
==== Freely moving points ====
The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted.   

In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2&lt;sup&gt;32&lt;/sup&gt; steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. 


[[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]]
==== Orbital trajectory (circular vs. straight) ====
Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. 

In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit.

The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). 

The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other.


==== Precession ====
semi-minor axis: &lt;math&gt;b = \alpha l^2 \lambda_A&lt;/math&gt;

semi-major axis: &lt;math&gt;a = \alpha n^2 \lambda_A&lt;/math&gt;

radius of curvature :&lt;math&gt;L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}&lt;/math&gt;

:&lt;math&gt;\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}&lt;/math&gt;


arc secs per 100 years (drift):

:&lt;math&gt;T_{earth}&lt;/math&gt; = 365.25 days 

drift = &lt;math&gt;\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}&lt;/math&gt; 

 Mercury (eccentricity = 0.205630)
 T = 87.9691 days
 a = 57909050 km (''n'' = 378.2734) 
 b = 56671523 km (''l'' = 374.2096)
 drift = 42.98

 Venus (eccentricity = 0.006772) 
 T = 224.701 days
 a = 108208000 km (''n'' = 517.085) 
 b = 108205519 km (''l'' = 517.079)
 drift = 8.6247

 Earth (eccentricity = 0.0167)
 T = 365.25 days
 a = 149598000 km (''n'' = 607.989) 
 b = 149577138 km (''l'' = 607.946)
 drift = 3.8388

 Mars (eccentricity = 0.0934)
 T = 686.980 days
 a = 227939366 km (''n'' = 750.485) 
 b = 226942967 km (''l'' = 748.843)
 drift = 1.351


[[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]]
==== Hyper-sphere orbit ====
{{main|Relativity (Planck)}}

Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt;. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. 

While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B&lt;sup&gt;1&lt;/sup&gt; to B&lt;sup&gt;11&lt;/sup&gt;) around the ''A'' time-line (vertical expansion) axis ('''t&lt;sub&gt;d&lt;/sub&gt;''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. 

:&lt;math&gt;d = r_{\alpha} n_g&lt;/math&gt;

:&lt;math&gt;t_0 = 2 \pi r = 2 \pi \frac{t}{2 \pi d}&lt;/math&gt;

:&lt;math&gt;v_{outer} = \frac{1}{d}&lt;/math&gt;

For object '''B'''

:&lt;math&gt;t_d = \sqrt{t^2 - {t_0}^2} = t \sqrt{1 - v_{outer}^2}&lt;/math&gt; 

For object '''A'''

:&lt;math&gt;t_d = t \sqrt{1 - v_{inner}^2}&lt;/math&gt;


==== Planck force ====

:&lt;math&gt;F_p = \frac{m_P c^2}{l_p}&lt;/math&gt;

:&lt;math&gt;M_a = \frac{m_P \lambda_a}{2 l_p} ,\;m_b = \frac{m_P \lambda_b}{2 l_p}&lt;/math&gt;

:&lt;math&gt;F_g = \frac{M_a m_b G}{R^2} = \frac{\lambda_a \lambda_b F_p}{4 R_g^2}  = \frac{\lambda_a \lambda_b F_p}{4 \alpha^2 n^4 (\lambda_a + \lambda_b)^2} &lt;/math&gt;

a) &lt;math&gt;M_a = m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{F_p}{{(4 \alpha n^2)}^2} &lt;/math&gt;

b) &lt;math&gt;M_a &gt;&gt; m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{\lambda_b F_p}{{(2 \alpha n^2)}^2 \lambda_a} = \frac{m_b c^2}{2 \alpha^2 n^4 \lambda_a} = m_b a_g&lt;/math&gt;


=== Atomic orbitals ===
[[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]]
In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. 

During electron transition between orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components cancel reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r'').

As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, we may ask if quantization of the atom has a geometrical origin. &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;.


==== Simulation ====
The simulation program can be adapted for atomic orbitals by simply including an additional alpha term in the rotation angle '''β'''. 

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}&lt;/math&gt; 


The following example simulates an electron transition, the electron begins at radius &lt;math&gt;r = r_{orbital}&lt;/math&gt; and makes a 360° rotation at orbital radius ''r'' (the orbital phase) and then moves in incremental steps to higher orbitals (transition phase). The period and length are measured at these [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles (360°; 360°; 360+120°; 360+180°, 360+216°, 360+240°) which in a (theoretical) Rydberg atom (of point size, infinite mass and disregarding wavelength) correspond precisely to the [[w:principal quantum number |principal quantum number]] ''n'' = 1, 2, 3, 4, 5, 6...


Key: nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''&lt;sub&gt;sim&lt;/sub&gt; = period and ''l''&lt;sub&gt;sim&lt;/sub&gt; = distance travelled by the electron, the radius coefficient ''r''&lt;sub&gt;n&lt;/sub&gt; = radius divided by &lt;math&gt;r_{orbital}&lt;/math&gt;. 
[[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]]

:&lt;math&gt;j_{atom} = 250&lt;/math&gt; (atomic mass)

:&lt;math&gt;i_{nucleus} = j_{atom} -1 = 249&lt;/math&gt; (relative nucleus mass)

:&lt;math&gt;r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2&lt;/math&gt;

:&lt;math&gt;r_{orbital} = 2 \alpha \;*\; r_{wavelength} &lt;/math&gt; (radius)

:&lt;math&gt;t_{calc} = \frac{t_{sim}}{r_{wavelength} (r_n - r_{1})}&lt;/math&gt; (i.e.: transition radius - orbital radius)

:&lt;math&gt;t_n = \frac{t_{sim}}{2\pi 2 \alpha r_{orbital}}&lt;/math&gt; 

:&lt;math&gt;l_n = \frac{l_{sim}}{2\pi r_{orbital}}&lt;/math&gt; 

:&lt;math&gt;r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}&lt;/math&gt; 

:&lt;math&gt;r_n = \frac{r_b}{r_{orbital}}&lt;/math&gt; 


Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]).

H(1s-2s) =  2466 061 413 187.035 kHz &lt;ref&gt;http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf&lt;/ref&gt;  

H(1s-3s) =  2922 743 278 665.79 kHz &lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/33243883/&lt;/ref&gt;  

H(1s-4s) =  3082 581 563 822.63 kHz &lt;ref&gt;https://codata.org/&lt;/ref&gt;   

H(1s-∞s) =  3288 086 857 127.60 kHz &lt;ref&gt;https://codata.org/ (109678.77174307cm-1)&lt;/ref&gt; (''n'' = ∞)


R = 10973731.568157 &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt; ([[w:Rydberg constant |Rydberg constant]])

α =137.035999177 (inverse fine structure constant &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt;

proton/electron mass ratio pe = 1836.152673426 &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt;
:&lt;math&gt;r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p} = 8\pi c \alpha^2 R \frac{pe}{pe+1}&lt;/math&gt;


Dividing (dimensioned) wavelength (&lt;math&gt;r_{wavelength}&lt;/math&gt;) by (dimensioned) transition returns a dimensionless number (the alpha component of the photon). The &lt;math&gt;(n^2 - 1)&lt;/math&gt; term refers to the number of orbital wavelengths in the transition phase;

:&lt;math&gt;t_{expt} = \frac{(n^2 - 1) \lambda_H }{H(1s-ns)}&lt;/math&gt;


There is no ''n''=1 to ''n''=1 transition, but the ionization energy will be virtually equivalent to the ''n''=1 orbital, and so for this orbital we can use a single wavelength and the ''n''=infinity transition as a close approximation. 

:&lt;math&gt;t_{1s} = \frac{\lambda_H }{H(1s-\infty s)} = 471959.243&lt;/math&gt;


Note: For a Rydberg atom, each segment on the 2-D plane is a multiple of &lt;math&gt;t_{ref}&lt;/math&gt; (the base orbital at ''n'' = 1)

:&lt;math&gt;t_{ref} = 2\pi 2\alpha 2\alpha&lt;/math&gt;


The experimental data will be relativistic and so a similar term is added to the reference data. The simulation data &lt;math&gt;t_{sim}&lt;/math&gt; however is for a 2-D plane and no relativistic term is added.

:&lt;math&gt;t_{rel} = t_{ref} \sqrt{1 - \frac{1}{4\alpha^2}} = 471961.21478&lt;/math&gt;


{| class=&quot;wikitable&quot;
|+Electron transition (mass = 250 ''n''=1 to ''n''=5)
! angle
! ''t''&lt;sub&gt;rel&lt;/sub&gt; 
! ''t''&lt;sub&gt;expt&lt;/sub&gt; 
! ''t''&lt;sub&gt;sim&lt;/sub&gt;
! ''l''&lt;sub&gt;sim&lt;/sub&gt;
! ''l''&lt;sub&gt;n&lt;/sub&gt;
! ''x'', ''y'' (electron) 
! ''x'', ''y'' (nucleus) 
! ''x'', ''y'' (barycenter) 
|-
| 360° n=1
| 471961.215
| 471959.243*
| 471957.072
| 3457.8864
| 0.99999
| 550.3341,  0.0036
| -2.2125,  -0.0075
| -0.0023,  -0.0075
|-
| 720° n=2
| 1887844.859
| 1887839.826
| 1888356.002
| 6917.7127
| 3.00053
| 2202.8558,  0.0001
| -7.9565,  -1.9475
| 0.8868,  -1.9397
|-
| 840° n=3 
| 4247650.933
| 4247634.049
| 4247567.984 
| 13831.460 
| 4.999924
| -2472.8741,  4296.3067 
| 13.5590,  -10.3184 
| 3.6133,  6.9081
|-
| 900° n=4 
| 7551379.436
| 7551347.553 
| 7551165.283
| 20747.092
| 6.99986 
| -8814.9421,  13.3054
| 25.6293,  13.3056
| -9.7330,  13.3056
|-
| 936° n=5 
| 11799030.370
|  
| 11798590.909
| 31118.741  
| 8.99978 
| -11157.39,  -8079.27
| 16.573,  39.086
| -28.123,  6.612 
|}

''*'' at infinity the second photon energy reduces to 0. 


===== Comparative data =====
[[File:H-transitions-simulation_vs_experimental.jpg|thumb|right|800px|H atom transitions spiral data &lt;math&gt;\delta_t&lt;/math&gt; for different mass; orange=64, gray=126, red=250, blue=500 (graph 1) and blue=experimental (graph 2)]]
If we include simulations for mass = 64 and mass = 126 as reference and calculate all data relative to &lt;math&gt;t_{rel}&lt;/math&gt; then we can compare results &lt;math&gt;\delta_t&lt;/math&gt; (see simulation and experimental data graphs). We find a correlation between mass and data peaks as a function of nucleus mass. This is most pronounced in the ''n'' = 1 to ''n'' = 2 transition 'signature' peak and in the simulation it is a function of nucleus mass and shape - as the electron travels further from the center, the nucleus expands (this is using the gravity simulation program without modification and so the points that represent the nucleus are not at a fixed radius - each of the mass points are calculated independently). Conversely a Rydberg atom gives a straight line (multiples of &lt;math&gt;t_{ref}&lt;/math&gt; with no peaks) - i.e.: the barycenter is always (0, 0) and the nucleus is a point (with mass but no size) and so does not expand. This suggests that by selecting nucleus mass and shape it may be possible to approach the experimental results, it also suggests that the proton shape could be influenced by the vicinity of the electron (electron charge)&lt;ref&gt;https://theprogrammergod.com/ The Programmer God, are we living in a simulation? (chapter 9. Atomic orbitals, 2014 edition)&lt;/ref&gt;. 
 

:&lt;math&gt;\delta_t = \frac{t_{ns} - n^2 t_{rel}}{n^2}&lt;/math&gt;

{| class=&quot;wikitable&quot;
|+ &lt;math&gt;\delta_t&lt;/math&gt; values for different mass
! ''n'' 
! ''m''&lt;sub&gt;64&lt;/sub&gt; 
! ''m''&lt;sub&gt;126&lt;/sub&gt; 
! ''m''&lt;sub&gt;250&lt;/sub&gt; 
! ''m''&lt;sub&gt;500&lt;/sub&gt; 
! ''m''&lt;sub&gt;expt&lt;/sub&gt; 
|-
| 1 
| -3.8058
| -4.0272 
| -4.1427
| -4.2015
| -1.9720 
|-
| 2 
| 519.5311 
| 259.4329
| 127.7856 
| 61.2898
| -1.2582 
|-
| 3
| -11.5856
| -9.3487
| -9.2166
| -9.4759
| -1.8760 
|-
| 4
| -14.7931
| -14.0301 
| -13.3846
|  
| -1.9927 
|-
| 5
| -18.4095 
| -18.4353
| -17.5784 
|  
|  
|}


[[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]]
==== Theory ====

===== Bohr model =====
The H atom has 1 proton and 1 electron orbiting the proton, the electron can be found at fixed radius (the [[w:Bohr radius |Bohr radius]]) from the proton (nucleus), these radius represent different energy levels (orbitals) at which the electron may be found orbiting the proton and so are described as quantum levels. Electron transition (to higher energy levels) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to lower energy levels.

The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron is further from the nucleus). Each ''n'' ([[w:electron shell|electron shell]]) can accommodate up to ''n''&lt;sup&gt;2&lt;/sup&gt; electrons (1, 4, 9, 16, 25...), and accounting for two states of spin, 2''n''&lt;sup&gt;2&lt;/sup&gt;. As these orbitals are fixed according to integer ''n'', the atom can be said to be quantized.


The basic (alpha) radius for each ''n'' level uses the fine structure constant alpha (α = 137.036) whereby;

&lt;math&gt;r_{orbital} = 2\alpha n^2&lt;/math&gt;


Such that at ''n'' = 1, the start radius ''r'' = 2α. We can map the electron orbit around the orbital as a series of steps with the duration of each step the frequency of the electron + proton wavelengths (&lt;math&gt;\lambda_p + \lambda_e&lt;/math&gt;). The steps are defined according to angle β;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;


[[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]]
At specific ''n'' levels;

:&lt;math&gt;\beta = \frac{1}{4\alpha^2 n^3}&lt;/math&gt;


This gives a length travelled per step as the inverse of the radius

:&lt;math&gt;l_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 

:&lt;math&gt;v_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 


The number of steps (orbital period) for 1 orbit of the electron then becomes 

:&lt;math&gt;t_{orbital} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha n^3&lt;/math&gt;


A base (reference) orbital (''n''=1)

:&lt;math&gt;t_{ref} = 2\pi 4\alpha^2&lt;/math&gt;


===== Photon orbital model =====
The electron can jump between ''n'' levels via the absorption or emission of a photon. In the [[Quantum_gravity_(Planck)#Atomic_orbitals|photon-orbital]] model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that &lt;math&gt;orbital \;radius + photon = zero&lt;/math&gt; (cancel).

The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius). Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role.


The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

:&lt;math&gt;\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}&lt;/math&gt;

:&lt;math&gt;\lambda_{photon} = (+\lambda_i) - (+\lambda_f)&lt;/math&gt;


The wavelength of the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon corresponds to the wavelength of the orbital radius. The (+&lt;math&gt;\lambda_i&lt;/math&gt;) will then delete the orbital radius as described above (orbital + photon = zero), however the (-&lt;math&gt;\lambda_f&lt;/math&gt;), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+&lt;math&gt;\lambda_i&lt;/math&gt;) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

:&lt;math&gt;r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)&lt;/math&gt;


However, the (&lt;math&gt;\lambda_f&lt;/math&gt;) has the longer wavelength, and so after the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon has been absorbed, and for the remaining duration of this (&lt;math&gt;\lambda_f&lt;/math&gt;) photon wavelength, at each transition step the orbital radius will be extended until the (&lt;math&gt;\lambda_f&lt;/math&gt;) is also absorbed. At each step, as the orbital radius increases, the orbital rotation angle β will conversely decrease, and as the velocity of orbital rotation depends on β, the velocity will adjust accordingly.

For example, the electron is at the ''n'' = 1 orbital. To jump from an initial &lt;math&gt;n_i = 1&lt;/math&gt; orbital to a final &lt;math&gt;n_f = 2&lt;/math&gt; orbital, first the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed (&lt;math&gt;\lambda_i + \lambda_{orbital} = zero&lt;/math&gt; which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (&lt;math&gt;\lambda_f&lt;/math&gt;) photon continues until it too is absorbed (the '''transition phase''').

:&lt;math&gt;\lambda_i = 1t_{ref}&lt;/math&gt;

:&lt;math&gt;\lambda_f = 4t_{ref}&lt;/math&gt; (''n'' = 2)


After &lt;math&gt;t_{ref}&lt;/math&gt; steps, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed, but the (&lt;math&gt;\lambda_f&lt;/math&gt;) photon still has &lt;math&gt;\lambda_f = (4-1)t_{ref}&lt;/math&gt; steps remaining until it too is absorbed.


[[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]]
Instead of a discrete jump between energy levels by the electron, absorption/emission takes place in steps, each step corresponds to a unit of &lt;math&gt;r_{incr}&lt;/math&gt;;

:&lt;math&gt;r_{incr} = -\frac{1}{2 \pi 2\alpha}&lt;/math&gt;


As &lt;math&gt;r_{incr}&lt;/math&gt; has a minus value, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon will shrink the orbital radius accordingly, per step

:&lt;math&gt;r_{orbital} = r_{orbital} + r_{incr}&lt;/math&gt; 


Conversely, because of its minus term, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon will extend the orbital radius accordingly;

:&lt;math&gt;r_{orbital} = r_{orbital} - r_{incr}&lt;/math&gt; 


The transition frequency is a combination of the orbital phase and the transition phase. 

:&lt;math&gt;\frac{{n_f}^2 - {n_i}^2}{t_{orbital} + t_{transition}}&lt;/math&gt;


The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]]

:&lt;math&gt;t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}&lt;/math&gt; 

The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;

:&lt;math&gt;t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}&lt;/math&gt; 


===== Alpha spiral =====
[[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]]
A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a''').

In its simplest form, a [[w:fine structure constant|fine structure constant]] spiral (or alpha spiral) is a specific hyperbolic spiral that appears in [[w:Atomic electron transition|electron transitions]] between [[w:atomic orbital|atomic orbitals]] in a [[w:Hydrogen atom|Hydrogen atom]].


It can be represented in Cartesian coordinates by

:&lt;math&gt;x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 &lt; \varphi &lt; 4\pi&lt;/math&gt;


This spiral has only 2 revolutions approaching 720° as the radius approaches infinity. If we set start radius '''r''' = 1, then at given angles radius '''r''' will have integer values (the angle components cancel).

:&lt;math&gt;\varphi = (2)\pi, \; r = 4&lt;/math&gt; (360°)
:&lt;math&gt;\varphi = (4/3)\pi,\; r = 9&lt;/math&gt; (240°)
:&lt;math&gt;\varphi = (1)\pi, \;  r = 16&lt;/math&gt; (180°)
:&lt;math&gt;\varphi = (4/5)\pi, \;  r = 25&lt;/math&gt; (144°)
:&lt;math&gt;\varphi = (2/3)\pi, \;  r = 36&lt;/math&gt; (120°)


Starting with &lt;math&gt;\varphi = 0, \;r = 2\alpha&lt;/math&gt; (''n''=1), for each step during transition;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;

:&lt;math&gt;\varphi = \varphi + \beta&lt;/math&gt;


As &lt;math&gt;\beta&lt;/math&gt; is proportional to the radius, as the radius increases the value of &lt;math&gt;\beta&lt;/math&gt; will reduce correspondingly (likewise reducing the orbital velocity). 

{{see|Fine-structure_constant_(spiral)}}


Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and the &lt;math&gt;n_f^2&lt;/math&gt; at each step &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;.

:&lt;math&gt;r = 2 \alpha + \frac{t}{2\pi 2\alpha}&lt;/math&gt; (number of increments ''t'' of &lt;math&gt;r_{incr}&lt;/math&gt;)

:&lt;math&gt;n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}&lt;/math&gt; (&lt;math&gt;n_f^2&lt;/math&gt; as a function of ''t'') 

:&lt;math&gt;\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}&lt;/math&gt; (&lt;math&gt;\varphi&lt;/math&gt; at any &lt;math&gt;n_f^2&lt;/math&gt;) 


[[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]]
===== H atom =====
The Bohr radius for an ionizing electron (H atom) follows this hyperbolic spiral. At specific spiral angles, the angle components (for this particular spiral) cancel returning an integer value for the radius (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). 

In the classical [[w:Bohr model|Bohr model]], the electron orbits around the barycenter (center of mass) and for this is used the reduced mass (the CODATA proton-electron mass ratio ''μ'' = 1836.152673426(32)).

:&lt;math&gt;\mu_n = \frac{m_e + m_p}{m_p} = 1 + \frac{1}{\mu}&lt;/math&gt; = 1.000544 617 021


However, the &lt;math&gt;H_{1s-\infty s}&lt;/math&gt; (ionization) vs. Rydberg constant shows slight divergence

:&lt;math&gt;\frac{R}{H(1s-\infty s)}&lt;/math&gt; = 1.000533 776 387


We can then determine the precise value for &lt;math&gt;\mu_n&lt;/math&gt; for each energy level using the literature values as reference:

H(1s-2s) =  2466 061 413 187.035 kHz &lt;ref&gt;http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf&lt;/ref&gt;

H(1s-3s) =  2922 743 278 665.79 kHz &lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/33243883/&lt;/ref&gt;

H(1s-4s) =  3082 581 563 822.63 kHz &lt;ref&gt;https://codata.org/&lt;/ref&gt;

''n'' = 2,    &lt;math&gt;\mu_n&lt;/math&gt; = 1.000539 387 875  

''n'' = 3,    &lt;math&gt;\mu_n&lt;/math&gt; = 1.000536 337 888  

''n'' = 4,   &lt;math&gt;\mu_n&lt;/math&gt; = 1.000535 460 372  


[[File:Hatom-alpha-orbital-spiral-angle-vs-transition-frequency-eV.jpg|thumb|right|480px|H atom transition (n=1 to n=64); alpha orbital spiral angles (pi) vs transition energies (eV)]]
By using ''n'' as a function of the rotation angle, we can plot a continuous value for &lt;math&gt;\mu_n&lt;/math&gt; (see graph right)


== External links ==
* [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]]
* [[v:Physical_constant_(anomaly) | Physical constant anomalies]]
* [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] 
* [[v:electron_(mathematical) | The mathematical electron]]
* [[v:Relativity_(Planck) | Programming relativity at the Planck scale]]
* [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]]
* [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]]
* [[v:God_(programmer) | The Programmer God]]
* [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] 
* [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] 


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
      <sha1>55weyta2se83vxa3ej8g7zefqwyuhju</sha1>
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        <id>1669821</id>
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      <comment>Wasps associated with plant galls</comment>
      <origin>2689243</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="eowiy6guij2t2vzof5ruzt6dj4176vc" xml:space="preserve">&lt;!--Info--&gt;
https://www.archive.org
{{list subpages|Alandmanson|User}}

===[[African Arthropods|Project: African Arthropods]]===

;[[African Arthropods/Chelicerates|African Chelicerates]]
:Arachnids and sea spiders — No sub-pages yet.
;[[African Arthropods/Crustaceans|African Crustaceans]]
:Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods — No sub-pages yet.
;[[African Arthropods/Hexapods|African Hexapods]]
:[[African Arthropods/Insects|African Insects]]
:* '''[[African Arthropods/Diptera|Diptera]]''' 
:**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]]
:* '''[[African Arthropods/Hymenoptera|Hymenoptera]]'''
:**[[African Arthropods/Chalcidoidea|African Chalcidoidea]]
:***[[African Arthropods/Encyrtidae|African Encyrtidae]]
:***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]]
:***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]]
:***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]]
:**[[African Arthropods/Aculeata|African Aculeata]]
:***[[African Arthropods/Eumeninae|African potter wasps]]
:***[[African Arthropods/Philanthus|South African species of Philanthus]]
:* '''[[African Arthropods/Lepidoptera|Lepidoptera]]'''
;[[African Arthropods/Myriapods|African Myriapods]]
:Centipedes, Millipedes, Pauropodans, Symphylans — No sub-pages yet.&lt;br&gt;&lt;br&gt;

;Arthropods in South Africa
:[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]]
:[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']]
:[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]]

&lt;br&gt;

===To Do===
Microgastrine cocoons in a net: &lt;br&gt;
* http://www.waspweb.org/Chalcidoidea/Eupelmidae/Eupelminae/Eupelmus/Eupelmus/Eupelmus_species_2.htm
* https://commons.wikimedia.org/wiki/File:Microgastrinae_cocooncocoon_iNat_42943906.jpg
* https://www.inaturalist.org/observations/38150348
* https://www.inaturalist.org/observations/144355729
* https://www.inaturalist.org/observations/39807090
* https://www.inaturalist.org/observations/145817446&lt;br&gt;


[[Crop_production_in_KwaZulu-Natal|Project: Crop_production_in_KwaZulu-Natal]]

[[Crop production in KwaZulu-Natal Annotated Bibliography]]

[[Information for smallholders in KwaZulu-Natal]]

[[Crop_production_in_KwaZulu-Natal/Climate-smart_Agriculture|Climate-smart Agriculture in KZN]]

[[Plant propagation]]&lt;br&gt;
&lt;br&gt;


[[Animal Phyla/Arthropoda]]&lt;br&gt;

[[:Category:Animals]]&lt;br&gt;

[[:Category:Zoology]]&lt;br&gt;

[[:Category:Entomology]]</text>
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      <comment>/* Comments */ Contribution of lived experience, personal views and references related to the topic.</comment>
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      <text bytes="14850" sha1="qh31drp0af3ry0r2dyszntm3vjylki5" xml:space="preserve">==Comments==

Hi! I have lived experience of growing up with domestic violence. When I was younger, I used to think it was 'all one way' i.e. gendered (My Dad always abusing my Mum and me - because we were female and the weaker of the 'two' sexes in our household). All of the language used in our house was derogatory about women, and my Dad would openly call us typically gendered names. An example of this is my Dad often referring to my Mum as &quot;ole girl' ,especially when he was angry. Then it used to get really personal and he would use language that really targeted her personally - to make sure it hurt. One of my most violent memories, is actually not physical at all, but verbal (as in verbally abusive). When we sat together to eat a meal, my father would go around the table and 'rank' each family member in order of their intelligence or status in the family. He would always be ranked first, and then my brothers in age order, then me (the eldest of all the children, and the only girl) and lastly, my Mum. It really hurt, but we dare not show it because he was (and still is) a very violent man. This has really stuck with me all of my life (I'm now in my fifties), but I never really stopped to think about why and how it has manifested itself so deeply in my psyche, until one evening when my kids were at the dinner table and really picking on each other. I absolutely lost my temper with them, and sent them all to bed early because they kept arguing about it only being a joke, and insisting on me taking it all too seriously. With retrospect, I was reacting to my father, vicariously through my children .... just 40 years later. My thoughts around this are primarily about the gendered view of violence, what it looks like and what it considered acceptable behaviour for men and women. I note that you mentioned some of the neuroscience behind men's violent behaviour towards women, and I'd like to add some more perspective from my experience.
There are so many gendered assumptions that we grow up with that lead us to unconsciously 'understand' or 'make allowances' for domestic violence - particularly male to female violence. But I really think its a combination of many factors including wider societal social rules and patriarchal beliefs that encourage male dominance and female subordination, neuroscientific factors (like hormones levels and neurotransmitters such as dopamine); as well as, other risk factors like intergenerational illness such as alcoholism, early experiences of maladaptive attachment patterns, exposure to deviant behaviours, and personality traits. Any of these factors alone might be enough to produce a violent streak in anybody, but the combination of one and another can be disabling for the entire family unit. 
I think to explain this, Dutton's (2006) Nested Ecological Model is a good place to start. Check it out at https://www.sciencedirect.com/science/article/abs/pii/S0272735811001097, and also Dixon and Kevan's (2018) study on risk factors associated with male perpetration of intimate partner violence, for more info https://pubmed.ncbi.nlm.nih.gov/21851805/. The info on risk profiles really resonated with me, and is relevant to both yours and my book chapters.--[[User:SoSilverLibby|SoSilverLibby]] ([[User talk:SoSilverLibby|discuss]] • [[Special:Contributions/SoSilverLibby|contribs]]) 14:16, 28 November 2024 (UTC) Libby--[[User:SoSilverLibby|SoSilverLibby]] ([[User talk:SoSilverLibby|discuss]] • [[Special:Contributions/SoSilverLibby|contribs]]) 14:16, 28 November 2024 (UTC)

Hello! this is a really interesting topic  Good job on all your work so far. I noticed you only had one external source so i found this article that could be interesting (link here: https://www.verywellmind.com/domestic-abuse-why-do-they-do-it-62639 ). Good luck finishing your chapter :)
[[User:U3203008|U3203008]] ([[User talk:U3203008|discuss]] • [[Special:Contributions/U3203008|contribs]])

An empirical explanation for the motivation of domestic violence against women with evidence from  Harway &amp; O’Neil(1999) notes that biological explanation of men's violence is due to genetics, brain dysfunction factors, endocrine and neurotransmitter. So early aggressive behaviour is predicted in later aggressive acts including physical aggression, criminal behaviour, spouse or child abuse. Other associations of motivating behaviour explained (Yllo, 2001) and (O'NEIL &amp; HARWAY, 1997) is alcohol abuse, coercive communication and anger expression.--[[User:SihTosam|SihTosam]] ([[User talk:SihTosam|discuss]] • [[Special:Contributions/SihTosam|contribs]]) --[[User:SihTosam|SihTosam]] ([[User talk:SihTosam|discuss]] • [[Special:Contributions/SihTosam|contribs]]) 08:46, 25 August 2021 (UTC)

O'NEIL, J., &amp; HARWAY, M. (1997). A Multivariate Model Explaining Men's Violence Toward Women. Violence Against Women, 3(2), 182-203. https://doi.org/10.1177/1077801297003002005

O’Neil, J. M., &amp; Harway, M. (1999). What causes men’s violence against women? Sage Publications.

Yllo, K. (2001). What Causes Men's Violence against Women?: What Causes Men's Violence against Women?. American Anthropologist, 103(2), 574-575. https://doi.org/10.1525/aa.2001.103.2.574

Hi, here is a TED talk about 'Why domestic violence victims don't leave' which might be interesting and useful for your book chapter:https://www.youtube.com/watch?v=V1yW5IsnSjo. Kind regards --[[User:U3196787|U3196787]] ([[User talk:U3196787|discuss]] • [[Special:Contributions/U3196787|contribs]]) 08:45, 14 October 2021 (UTC)

Hey there, here's an Australian government resource full of statistics regarding rates of domestic violence here which might be useful to include in your chapter; Australian Institute of Health and Welfare 2019. Family, domestic and sexual violence in Australia: continuing the national story 2019. Cat. no. FDV 3. Canberra: AIHW. https://www.aihw.gov.au/getmedia/b0037b2d-a651-4abf-9f7b-00a85e3de528/aihw-fdv3-FDSV-in-Australia-2019.pdf.aspx?inline=true --[[User:U3187208|U3187208]] ([[User talk:U3187208|discuss]] • [[Special:Contributions/U3187208|contribs]]) 01:39, 17 October 2021 (UTC)

Hi, I really enjoyed reading this chapter and noticed as this is a late social contribution that perhaps you are looking for some external links. This is a Ted talk on &quot;A Mile in Her Shoes: Changing perspective on domestic violence&quot;: https://www.youtube.com/watch?v=wLNa6qwVpbA&amp;ab_channel=TEDxTalks hope this is useful! --[[User:Eilish Ritchie|Eilish Ritchie]] ([[User talk:Eilish Ritchie|discuss]] • [[Special:Contributions/Eilish Ritchie|contribs]]) 07:26, 17 October 2021 (UTC)

Hey! I really enjoyed reading your topic. It can be such a hard thing to comment on due to the sensitive nature of the topic. I though you might find this external like useful https://www.youtube.com/watch?v=V1yW5IsnSjo Goodluck! [[https://en.wikiversity.org/wiki/User:Margaret_Minikin?veaction=edit|Margaret Minikin]] 17/10/21 6:41 (UTC) 

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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:40, 18 September 2021 (UTC)
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 09:10, 11 November 2021 (UTC)
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:47, 19 November 2021 (UTC)</text>
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  <page>
    <title>Talk:Motivation and emotion/Book/2021/Coercive control motivation in relationships</title>
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      <id>2689187</id>
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      <timestamp>2024-11-28T14:39:29Z</timestamp>
      <contributor>
        <username>SoSilverLibby</username>
        <id>2947631</id>
      </contributor>
      <comment>Commented on enjoying the chapter and suggested some reference materials for support.</comment>
      <origin>2689187</origin>
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      <text bytes="12963" sha1="3uedgku008qdnhc9c1przgr9jlno56n" xml:space="preserve">I really enjoyed reading your Book Chapter - (especially the info on self-determination theory). I have been working on a similar chapter this year (2024) and thought you could consider thinking about adding some resources to your section on 'What can be done about it?' Perhaps some links to resources for victim-survivors such as The Domestic Violence Crisis Service https://dvcs.org.au/ or The Women: Choice and Change Program, which is run through Karalika and Relationships Australia   https://www.relationshipsnsw.org.au/group-workshops/womens-choice-change/. I have recently participated in this group, and it helped me tremendously as a victim-survivor of parents in a IPV relationship, and then after leaving a 20 year marriage, which with hindsight - which mirrored theirs. (A good plug supporting Bandura's Social Learning theory!!). Some links for perpetrators would be good too, such as https://dcj.nsw.gov.au/children-and-families/family-domestic-and-sexual-violence/support-programs/perpetrator-interventions.html and its not just for men!


--[[User:SoSilverLibby|SoSilverLibby]] ([[User talk:SoSilverLibby|discuss]] • [[Special:Contributions/SoSilverLibby|contribs]]) 14:38, 28 November 2024 (UTC)Libby --[[User:SoSilverLibby|SoSilverLibby]] ([[User talk:SoSilverLibby|discuss]] • [[Special:Contributions/SoSilverLibby|contribs]]) 14:38, 28 November 2024 (UTC)
=== Coercive control article ===
Check out this article regarding coercive control in intimate relationships - it discusses intention and motivation as well. doi.org/10.1016/j.avb.2017.08.003
 --[[User:U3194769|U3194769]] ([[User talk:U3194769|discuss]] • [[Special:Contributions/U3194769|contribs]]) 04:28, 25 August 2021 (UTC)
An interesting topic! I am not sure what angle you need but I found another source that may be of use by Emma Katz, 2019 (https://doi.org/10.1177/1077801218824998). --[[User:U3167879|U3167879]] ([[User talk:U3167879|discuss]] • [[Special:Contributions/U3167879|contribs]]) 08:59, 17 September 2021 (UTC)

This is such an interesting topic, I really enjoyed reading your chapter! Just a suggestion but I saw you have included one small quiz at the very end of your chapter. I think it might be useful for you to incorporate small quizzes throughout your page to make it more interactive and engaging? Great job through! [[User:U3187813|U3187813]] ([[User talk:U3187813|discuss]] • [[Special:Contributions/U3187813|contribs]]) 11:38, 17 October 2021 (UTC)

== Heading casing ==
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| {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:03, 18 September 2021 (UTC)
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:03, 18 September 2021 (UTC)
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## [https://apastyle.apa.org/style-grammar-guidelines/capitalization/diseases-disorders-therapies Do not capitalise the names of disorders, therapies, theories, etc.].
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:07, 4 November 2021 (UTC)
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# The presentation makes effective use of animated slides.
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# The amount of text presented per slide makes it easy to read and listen at the same time.
# The presentation is very well produced.
|7=
&lt;!-- Meta-data --&gt;
# The presentation uses an accurate combination of the chapter title and sub-title within the maximum 100-character limit for YouTube videos.
# A brief written description of the presentation is provided. Consider expanding.
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-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:34, 18 November 2021 (UTC)</text>
      <sha1>3uedgku008qdnhc9c1przgr9jlno56n</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2689189</id>
      <parentid>2688893</parentid>
      <timestamp>2024-11-28T14:54:03Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2689189</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="b2e51hrf2yo58jwkvfsl155km1o4ylp" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241127.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>b2e51hrf2yo58jwkvfsl155km1o4ylp</sha1>
    </revision>
    <revision>
      <id>2689195</id>
      <parentid>2689189</parentid>
      <timestamp>2024-11-28T14:57:47Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2689195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="h9p4muks01917marfde2z4bwwn0iaf0" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241128.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>h9p4muks01917marfde2z4bwwn0iaf0</sha1>
    </revision>
  </page>
  <page>
    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
    <id>285384</id>
    <revision>
      <id>2689252</id>
      <parentid>2689127</parentid>
      <timestamp>2024-11-28T23:49:06Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Linking Libraries */</comment>
      <origin>2689252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3249" sha1="94lnidbgksk4fkc80vbv0jopzv9l6zn" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241128.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20240724.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20230701.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])

* Shared Libraries ([[Media:LIB.2A.Shared.20241129.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>94lnidbgksk4fkc80vbv0jopzv9l6zn</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods</title>
    <ns>0</ns>
    <id>286872</id>
    <revision>
      <id>2689242</id>
      <parentid>2688344</parentid>
      <timestamp>2024-11-28T20:20:48Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2689242</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7098" sha1="mw5a0rgzycxtqp2qatfd2pxv0wct152" xml:space="preserve">This is an informal learning project for [[User:Alandmanson|Alandmanson]] and anyone that wishes to join in. See [[Talk:African_Arthropods|Discuss: African Arthropods project]].&lt;br&gt;

{{Navigation
|title = African Arthropods Project
|body  = 
;[[African Arthropods/Chelicerates|African Chelicerates]]
::No sub-pages yet
;[[African Arthropods/Crustaceans|African Crustaceans]]
::No sub-pages yet
;[[African Arthropods/Hexapods|African Hexapods]]
:[[African Arthropods/Insects|African Insects]]
:* '''[[African Arthropods/Diptera|Diptera]]''' 
:**[[African Arthropods/Acalyptrate flies|Acalyptrate flies]]
:* '''[[African Arthropods/Hymenoptera|Hymenoptera]]'''
:**[[African Arthropods/Chalcidoidea|African Chalcidoidea]]
:***[[African Arthropods/Encyrtidae|African Encyrtidae]]
:***[[African Arthropods/Afrotropical Encyrtidae Key|Key to the genera of Afrotropical Encyrtidae]]
:***[[African Arthropods/Chalcid wasps with branched antennae|African chalcid wasps with branched antennae]]
:***[[African Arthropods/Wasps associated with plant galls|Wasps associated with plant galls]]
:**[[African Arthropods/Aculeata|African Aculeata]]
:***[[African Arthropods/Eumeninae|African potter wasps]]
:***[[African Arthropods/Philanthus|South African species of Philanthus]]
:* '''[[African Arthropods/Lepidoptera|Lepidoptera]]'''
;[[African Arthropods/Myriapods|African Myriapods]]
::No sub-pages yet
}}

The extant Arthropoda of Africa can be subdivided into four Subphyla (and about 15 Classes). This classification is that followed by iNaturalist (July 2022).

[[African_Chelicerates|African Chelicerates]] - Including mites, harvestmen, solifuges, spiders, tailless whip scorpions, and sea spiders.&lt;br&gt;
 
[[African Crustaceans]] - Including branchiopods, barnacles, crabs, lobsters, crayfish, shrimp, fish lice, tongue worms, and ostracods.&lt;br&gt;

[[African Hexapods]] - Including springtails and [[African Arthropods/Insects|insects]].&lt;br&gt;
 
[[African Myriapods]] - Including centipedes, millipedes, pauropodans, and symphylans.&lt;br&gt;
 

== Subphylum Chelicerata ==
* Class [[w:Arachnida|Arachnida]] — Arachnids
&lt;gallery mode=&quot;packed&quot; heights=&quot;200&quot;&gt;
Velvet Christmas Spider by anagoria.jpg|[[w:Mite|Mites]]
Opiliones male IMG 9246s.jpg|[[w:Opiliones|Harvestmen]]
Solpugema00.jpg|[[w:Solifugae|Solifuges]]
Portia schultzi 57013020.jpg|[[w:Spider|Spiders]]
Damon annulatipes.jpg|[[w:Amblypygi|Tailless whip scorpions]]
&lt;/gallery&gt;
* Class [[w:Pycnogonida|Pycnogonida]] — Sea Siders or Pycnogonids
&lt;gallery mode=packed heights=200&gt;
Nymphon signatum 13403396.jpg|[[w:Sea spider|Sea Spiders]]
&lt;/gallery&gt;

== Subphylum Crustacea ==

* Class [[w:Branchiopoda|Branchiopoda]] — Branchiopods
&lt;gallery mode=packed heights=200&gt;
Branchiopoda Anostraca Branchipodopsis 2014 01 25 4802s.JPG|[[w:Anostraca|Fairy shrimps]]
&lt;/gallery&gt;
* Class [[w:Hexanauplia|Hexanauplia]] — Barnacles and Copepods
&lt;gallery mode=packed heights=200&gt;
Octomeris angulosa - inat 34781589.jpg|[[w:Barnacle|Barnacles]]
Cancerilla oblonga (10.3897-AfrInvertebr.57.9775) Figure 2.jpg|[[w:Copepoda|Copepods]]
&lt;/gallery&gt;
* Class [[w:Malacostraca|Malacostraca]] — Malacostracans, including crabs, lobsters, crayfish, shrimp, krill, prawns, woodlice, amphipods, and mantis shrimp
&lt;gallery mode=packed heights=200&gt;
Tuberculate crab (Plagusia depressa subsp. tuberculata).jpg|[[w:Decapoda|Crabs]]
Marioniscus spatulifrons.jpg|[[w:Isopoda|Isopods]]
&lt;gallery mode=packed heights=200&gt;
Mantis shrimp at Sodwana Bay, South Africa (3059956183).jpg|[[w:Hoplocarida|Mantis shrimps]]
&lt;/gallery&gt;
* Class [[w:Ichthyostraca|Ichthyostraca]] — Includes [[w:Branchiura|Branchiura]], fish lice and [[w:Pentastomida|Pentastomida]], tongue worms
&lt;gallery mode=packed heights=200&gt;
Genus Argulus Fish Louse Rob Taylor.jpg|[[w:Branchiura|Fish lice]]
&lt;/gallery&gt;
* Subclass [[w:Mystacocarida|Mystacocarida] — Mystacocaridans
&lt;gallery mode=packed heights=200&gt;
Mystacocarida-scale250um.jpg|[[w:Mystacocarida|Mystacocarids]]
&lt;/gallery&gt;
* Class [[w:Ostracoda|Ostracoda]] — Ostracods
&lt;gallery mode=packed heights=200&gt;
Ostracoda Botswana Robert Taylor 2020 c.jpg|[[w:Ostracoda|Ostracods]]
Ostracoda Botswana Robert Taylor 2020 e.jpg
&lt;/gallery&gt;

== Subphylum Hexapoda ==
* Class [[w:Entognatha|Entognatha]] — Entognathans, including springtails
&lt;gallery mode=packed heights=200&gt;
Gracilentulus_nr._floridanus_(YPM_IZ_098960)_(cropped).jpeg|[[w:Protura|Coneheads]]
Campodea fragilis 01.JPG|[[w:Diplura|Two-pronged bristletails]]
Slender Springtail iNat 105960417 a.jpg|[[w:Entomobryomorpha|Slender springtails]]
Plump Springtail iNat 105831052 -1.jpg|[[w:Poduromorpha|Plump springtails]]
Globular springtail iNat 112688442 a.jpg|[[w:Symphypleona|Globular springtails]]
&lt;/gallery&gt;

* Class [[w:Insecta|Insecta]] — [[African Arthropods/Insects|Insects]]
&lt;gallery mode=packed heights=200&gt;
African_Monarch_(Danaus_chrysippus_aegyptius)_(17389277322).jpg|[[w:Lepidoptera|Butterflies and moths]]
Dicronorrhina derbyana subsp derbyana, wyfie, Pretoria, a.jpg|[[w: Coleoptera|Beetles]]
Peltophorum africanum 1DS-II 6699.jpg|[[w: Hymenoptera|Ants, bees, wasps, and sawflies]]
Cotton Stainer (Dysdercus nigrofasciatus) (13951713711).jpg|[[w: Hemiptera|True bugs, hoppers, aphids, and allies]]
Green blowfly.jpg|[[w: Diptera |Flies]]
&lt;/gallery&gt;

== Subphylum Myriapoda ==
* Class [[w:Chilopoda|Chilopoda]] — Centipedes
&lt;gallery mode=packed heights=200&gt;
Very_pretty_centipede_that_fell_into_the_swimming_pool_yesterday._Beautiful_but_nasty,_Esther_got_stung_by_a_baby_and_it_was_not_nice,_I_suppose_this_one_could_have_an_interesting_bite._(8204823865).jpg|[[w:Scolopendromorpha|Tropical centipedes]]
Blue-legged Centipede (Ethmostigmus trigonopodus) (12681235843).jpg|[[w:Scolopendromorpha|Tropical centipedes]]
House centipede - Sri Lanka - 01.jpg|[[w:Scutigeromorpha|House centipedes]]
&lt;/gallery&gt;
* Class [[w:Diplopoda|Diplopoda]] — Millipedes
&lt;gallery mode=packed heights=200&gt;
Millipede,_South_Africa_(40435620062).jpg|[[w:Chilognatha|Chilognatha]]
&lt;/gallery&gt;
* Class [[w:Pauropoda|Pauropoda]] — Pauropodans
&lt;gallery mode=packed heights=200&gt;
Pauropodid (8701483114).jpg|[[w:Tetramerocerata|Tetramerocerata]]
&lt;/gallery&gt;
* Class [[w:Symphyla|Symphyla]] — Symphylans
&lt;gallery mode=packed heights=200&gt;
2022 04 23 Hanseniella Pietermaritzburg.jpg|[[w:Scutigerellidae|Scutigerellidae]]
&lt;/gallery&gt;

== Arthropods in South Africa ==
[[African Arthropods/Ferncliffe Nature Reserve|Ferncliffe Nature Reserve]]&lt;br&gt;
[[African Arthropods/Arthropods on ''Ficus burkei''|Arthropods on ''Ficus burkei'']]&lt;br&gt;
[[African Arthropods/Hymenoptera of South Africa|Hymenoptera of South Africa]]

== See Also ==
* [[Animal Phyla/Arthropoda]]
* [[Wikipedia: Arthropoda]]
* [[Wikipedia: Africa]]
* [[Wikipedia: Afrotropical realm|Wikipedia: Afrotropical biogeographic realm]]
* [https://www.palaeontologyonline.com/articles/2015/fossil-focus-cambrian-arthropods/?doing_wp_cron=1704092366.8973550796508789062500 Evolution of Arthropods - palaeontologyonline.com]&lt;br&gt;
&lt;br&gt;


[[Category:Animals]]
[[Category:Zoology]]
[[Category:Entomology]]
[[Category:Arthropoda]]
[[Category:African Arthropods]]</text>
      <sha1>mw5a0rgzycxtqp2qatfd2pxv0wct152</sha1>
    </revision>
  </page>
  <page>
    <title>User:Jtneill/Presentations/Using open wikis for teaching and learning</title>
    <ns>2</ns>
    <id>304085</id>
    <revision>
      <id>2689207</id>
      <parentid>2654026</parentid>
      <timestamp>2024-11-28T15:49:58Z</timestamp>
      <contributor>
        <ip>2600:387:15:1637:0:0:0:8</ip>
      </contributor>
      <comment>Dummies</comment>
      <origin>2689207</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8309" sha1="dhaak7wzd7ficgioexmv015xdevrvco" xml:space="preserve">{{title|Using open wikis for teaching and learning}}

&lt;div style=&quot;text-align: center&quot;&gt;
[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University]] OF DUMMIES

[https://ascilite.org/get-involved/sigs/learning-design-sig/ Australian Society for Computers in Learning in Tertiary Education (ASCILITE) Learning Design Special Interest Group] Webinar&lt;br&gt;Friday 15 March 2024, Online

[https://docs.google.com/presentation/d/1dP2n-LPmGSZAR0F3pUwW2ClVJFr1zGocUqpTBa4NsBw/edit?usp=sharing Slides]&lt;br&gt;(Google)

[https://youtu.be/kzXHOF-RhmU?si=4An0MMA4xmjYnddn&amp;t=217 Recording]&lt;br&gt;(YouTube; 53:37 mins including Q&amp;A)

[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ LinkedIn post]

[https://twitter.com/jtneill/status/1768516693553565884 X post]
&lt;/div&gt;

==Abstract==
A [[wiki]] is the simplest webpage that anyone can edit. The aim of this presentation is to illustrate the educative potential of using open wikis in education. Open wikis offer several pedagogical advantages and practical affordances for teaching and learning, yet are surprisingly underutilised. The best-known wikis are supported by the Wikimedia Foundation, including Wikipedia and its sister projects such as Wikiversity. These wikis can be used to curate open educational resources and conduct open research, as well as to engage students in learning how to work collaboratively to contribute to the knowledge commons. Student wiki projects can be conducted in any discipline and adapted across educational levels. This presentation will explore the educative potential of involving students in real-world wiki projects for learning and assessment, with case study examples.

==Bio==
James is an Assistant Professor in Psychology at the University of Canberra. James is a keen proponent of open educational practices and use of wikis for curriculum development and student learning projects.

==Aims==
* Explain why wikis are good for education
* Highlight wiki affordances
* Showcase Wikimedia Foundation platform esp. Wikiversity and Wikibooks

==Questions==
I invite you to share about your wiki experiences and curiousities. Here's a few prompts that may be useful:
# What experiences have you had with wikis? My experience is learning 
# Have you edited any wikis?no
# Do you have a Wikimedia user account?no 
# What educational wiki projects have you contributed to?I don't remember but is not the first time
# What is your university's approach to using wikis?iam only in grade 8 I just need guidance
# What do you advise teaching staff about using wikis? Wikipedia is the good place to refresh mind
# What questions or comments do you have about using wikis in open education? Learning

==Premises==
* Knowledge-sharing is good
* Open education is good
* Wikis offer underutilised utility

==Wiki history==
* Internet www 1993
* WikiWikiWeb 1995 (1st wiki)
* Wikipedia 2001
* Wikibooks 2003
* Wikiversity 2006

==What is a wiki?==
* Simplest webpage for collaborative editing
* wikiwiki = quick/speedy in Hawaiian

==Wikimedia Foundation mission==
&lt;blockquote&gt;
&quot;to empower and engage people around the world to collect and develop educational content under a free license or in the public domain, and to disseminate it effectively and globally.&quot;[[w:Wikimedia Foundation#Mission|WMF mission] (Wikipedia)
&lt;/blockquote&gt;

==Wikimedia sister projects==
Diverse ecosystem of free, open, multilingual wiki projects
{{Sisterprojects}}

==Wikibooks==
[[File:Wikibooks simple book blue beige borders.svg|right|150px]]
*Wikibooks is for new ebooks
*Wikisource is for pre-existing books
*Free alternative e.g., to [[w:Pressbooks|Pressbooks]]

==Wikiversity==
* Pages: 149,043
* Languages: 17
* Active users: 764 (210 on English Wikiversity)
* Admins 55 (11 on English Wikiversity)

For more detail, see [[w:Wikiversity#Languages|Wikiversity languages]] (Wikipedia)

==What is Wikiversity?==
&lt;blockquote&gt;
“Wikiversity is a Wikimedia Foundation project devoted to learning resources, learning projects, and research for use in all levels, types, and styles of education from pre-school to university, including professional training and informal learning. We invite teachers, students, and researchers to join us in creating open educational resources and collaborative learning communities.”
[[Wikiversity:Main Page|Source] (Wikiversity)
&lt;/blockquote&gt;

==Wikiversity self-tour==
[[File:Wikiversity Logo.png|right|150px]]
Some ways to explore:
*[[Help:Guides|Guided tours]]
*[[Category:Portals|Portals]]
*[[Wikiversity:Featured#Current featured content|Featured projects]]
*[[Wikiversity:Browse|Browse]]
*[[Special:RandomRootpage|Random]]

==Teachers can ...==
* Create, edit, and fork informal and formal open educational resources (e.g., course materials and learning activities) on any topic, size, and level
* Engage students in contributing to learning projects

==Researchers can ...==
* Develop research projects
* Seek peer-review
* Publish findings

==Students can ...==
* Access to read and view
* Create and edit
* Interact e.g.,
** Comment
** Debate
** Quizzes

==Teaching students to edit==
* Easy, fun, and experiential
* Students learn essential skills in 1 hour class:
** Create user account
** Edit user page
** Basic markup - bold, italics etc.
** Links - internal, [[w:Help:Interwiki linking|interwiki]], external
** Headings and table of contents
** Finding and embedding images

Example tutorial: [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing]]

==Administration and support==
* Local
** User - edit, move etc.
** Curator - delete, protect etc.
** Custodian (admin/sysop) - + user block
** Bureaucrat - +assign user permissions
** Check-user - IP address checking
* Meta
** Stewards - + global user block, spam + title blacklist, oversight
** Developer - + enable extensions, software changes

==Wiki affordances==
* Anyone can edit
* Policies, procedures, guidelines, decisions guided by community consensus
* Transparent editing history / version control
* Collaborative
* Highly stable

==LMS + Wikiversity==
; Learning management system (LMS)
* Institutional enrolments
* Assessment submission
* Marks
* Course announcements and discussion

; Wikiversity
* Generic open educational resources
** Lesson plans
** Class materials
** Assessment descriptions
* Student editing
* Interactive feedback

Use iFrame embedding, any webpage including a wiki page, can be embedded within an LMS e.g.,[https://unicanberra.instructure.com/courses/11508/pages/tutorial-03-physiological-needs

==Wiki challenges==
* Awareness of wikis is low
* Motivation to teach openly is weak
* Editing skills can be scary
* Institutional copyright policies often archaic
* Learning design / ed tech support lacking
* Local championing / communities of practice needed

At first glance, it seems like wikis &quot;can’t work&quot; - surely the dark side would win and it would turn into garbage, but it's a bit like seeing a bicycle for the first time and thinking no way could someone ride it without falling off.

The magic of wikis is that they do work because the critical mass of people willingly contributing to improving resources outweighs the relatively small amount of bad editing which tends to be quickly rectified.

==Example: Motivation and emotion==
[[Motivation and emotion]] is a 3rd psychology unit at the University of Canberra with an annual enrolment of approximately 150 students:
* [[Motivation and emotion|Home page]]
* [[Motivation and emotion/Book|Student-authored book chapters]]
* [[Motivation and emotion/Book/2023|2023 book table of contents]]
* [[Motivation and emotion/Book/2023/Actively open-minded thinking|Actively open-minded thinking]] (example chapter)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (case study article)

==Conclusion==
* Wikis provide a free, no cost, flexible, way of developing open educational curricula and learning activities
* Start small and learn
* Go forth and wikify

==See also==
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]]
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]]

[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
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&lt;div style=&quot;text-align: center&quot;&gt;
[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

Wikimedia Australia &quot;Using Wikimedia in Higher Education&quot; Webinar&lt;br&gt;Thursday 28 November 2024, Online

[https://docs.google.com/presentation/d/1uKiWbuae48OGDjih2Z5X8Kqh84Drw3WUDfUULn3IjxY/edit?usp=sharing Slides]&lt;br&gt;(Google)&lt;!--

[ Recording]&lt;br&gt;(YouTube; X:X mins including Q&amp;A)

[ LinkedIn post]

[ X post] --&gt;
&lt;/div&gt;

==Abstract==
The Wikimedia Foundation's mission is to make the sum of human knowledge freely available for all. Although Wikipedia is well known, it is only for encyclopedic information. The Wikimedia Foundation has other sister projects for other purposes. For example, Wikiversity provides an optimal, wiki-based platform for development and sharing of open educational resources. Wikiversity can be edited by anyone, including educators and students, making it an excellent platform for student-staff collaboration. Wikiversity can also be used for research.

==Bio==
James is an Assistant Professor in Psychology at the [[University of Canberra]]. James is a keen proponent of open educational practices and use of wikis for curriculum development and student learning projects.

==Values==
Arguably the highest value for academics is to contribute as openly as possible to the knowledge commons. Academics are (mostly) employed by public institutions to broker knowledge and support the education of society and its citizens.

However, academics are vulnerable to becoming institutionalised and operating largely within walled gardens and ivory towers. For example, it is easy to default to minimal sharing of knowledge through paywalled journals and closed learning management systems. This means that most citizens are unable to access much higher education knowledge.

Personally, I strive to practice [[open academia|open academia]]. This means sharing teaching, research, and service materials as openly and flexibly as possible. Ideally, this means using:
* open access
* open licensing
* open formats
* open governance/decision-making

;Premises
* Knowledge-sharing is good (or at least better than not sharing)
* Open education is good (or at least better than closed education)
* Wikis offer underutilised utility
* Wikiversity is a &quot;perfect&quot; platform for practicing open academia

==Wikis==
Wikis are the earliest and simplest openly editable webpages. [[w:WikiWikiWeb#Etymology|Wiki wiki is a Hawaiian term for quick]]. Wikis allow open communities to collaboratively construct and share information.

;Wiki history:
* 1993: Internet (www)
* 1995: WikiWikiWeb (the first wiki)
* 2001: Wikipedia (after some earlier prototypes)
* 2003: Wikibooks (for original work)
* 2006: Wikiversity (a spin off from Wikibooks for teaching, learning, and research)

==Wikimedia Foundation==
The non-profit [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) hosts a rich ecosystem of wiki-based sister projects which include Wikipedia for encyclopedic information and Wikiversity for education. Each of these projects is multi-lingual, with different instances for each language.

;[[w:Wikimedia Foundation#Mission|Wikimedia Foundation mission]]:

&lt;blockquote&gt;&quot;to empower and engage people around the world to collect and develop educational content under a free license or in the public domain, and to disseminate it effectively and globally.&quot;&lt;/blockquote&gt;

==Sister projects==
The Wikimedia Foundation hosts a diverse ecosystem of multilingual free and open wiki projects.

Most people are somewhat familiar with Wikipedia but are much less aware of the other projects.

A WMF user account is universal, so it works across all wikis and languages.

There are 15 [[w:Wikipedia:Wikimedia sister projects|sister projects]], including Wikipedia. They run on open-source MediaWiki software.

Each sister project has a specific purpose and focus. For example:
* [[w:|Wikipedia]], which is a wiki for encyclopedic information, but they are typically much less familiar with other projects such as 
* [[b:|Wikibooks]] (for authoring free books e.g., a textbooks)
* [[species:|Wikispecies]] (for flora and fauna taxonomic information)
* [[wikt:|Wiktionary]] (for definitions, synonyms etc.,) 
* [[n:|Wikinews]] (for news),
* [[c:|Wikimedia Commons]] (for images, audio, video etc)

Each of these projects offer are free, online organisations of information and knowledge that anyone can create, edit, and comment on.

Together, the sister projects provide more than the sum of their parts because they work together synergistically.

==Wikiversity==
The Wikiversity mission is to provide free and open teaching, learning, and research materials. Unlike, Wikipedia, original research is permitted.

Collectively, [https://wikiversity.org Wikiversity] consists of:
* ~150,000 pages in 17 languages (plus a multi-lingual hub for other languages)
* ~800 active users
* ~50 administrators (unpaid, volunteer)
{{Wikiversity:Main Page/Introduction}}
===Teaching===
Wikiversity can be used to host teaching materials, including lecture/tutorial/workshop notes.

Teachers can:
* Create, edit, and fork informal and formal open educational resources (e.g., course materials and learning activities) on any topic, size, and level
* Engage students in contributing to learning projects (e.g., for learning and assessment exercises) (e.g., [[Motivation and emotion/Book]])

Teaching students to edit:
* Easy, fun, and experiential
* Students learn essential skills in 1 hour class:
** Create user account
** Edit user page
** Basic markup - bold, italics etc.
** Links - internal, interwiki, external
** Headings and table of contents
** Finding and embedding images

See: [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing tutorial]]

===Research===
Original research is not allowed on Wikipedia; it is allowed (and encouraged) on Wikiversity. This research can range from informal to formal peer-reviewed (e.g., see [[WikiJournals]])

Researchers can:
* Develop research projects
* Seek peer-review
* Publish findings

===Administration and support===
* Local
** User - edit, move etc.
** Curator - delete, protect etc.
** Custodian (admin/sysop) - + user block
** Bureaucrat - +assign user permissions
** Check-user - IP address checking
* Meta
** Stewards - + global user block, spam + title blacklist, oversight
** Developer - + enable extensions, software changes

More info: [[Wikiversity:Support staff]]

===Example===
In the 3rd year undergraduate psychology unit, [[Motivation and emotion]], at the [[University of Canberra]], ~150 students per year learn how to collaboratively author 4,000 word online book chapters. To date, over 1,500 chapters have been created:

[[Motivation and emotion/Book|Browse the book]]

Each chapter tackles a unique topic and includes learning features such as:
* Links
* Images
* Tables
* Quizzes etc.

For more information about the principles and practices of this Wikiversity project, see:
{{Hanging indent|1=

{{User:Jtneill/Publications/2024/Collaborative}}

{{User:Jtneill/Publications/2024/Rich}}

}}

===LMS vs Wikiversity===
[[w:Learning management system|Learning management systems]] are for:
* Institutional enrolments
* Assessment submission
* Marks
* Course announcements and discussion

Wikiversity is for:
* [[w:Open educational resources|Open educational resources]]
** Lesson plans
** Class materials
** Assessment guidelines
** Student editing
** Discussion/feedback

===Challenges===
* Awareness of wikis is low
* Motivation to teach openly is weak
* Developing editing skills can be scary
* Institutional copyright policies often archaic
* Learning design / ed tech support lacking
* Local championing / communities of practice needed

===Lessons learned===
* Clarify/articulate values/teaching philosophy (establish the andragogical foundations)
* Identify a knowledge commons goal
* Scaffold the task (break it into chunks)
* Teach essential/basic skills
* Cultivate a curious, supportive learning environment (e.g., have a go, share, support)
* Start small and build up over time
* Share about experiences

==Conclusion==
* Wikis = free, no cost, flexible, way of developing open educational curricula and learning activities
* Start small to learn
* Go forth and wikify

&lt;!--

==See also==
--&gt;
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
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&lt;div style=&quot;text-align: center&quot;&gt;
[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://wikimedia.org.au/wiki/Using_Wikiversity_and_WikiLearn_for_teaching_and_learning Using Wikiversity and WikiLearn for teaching and learning], Wikimedia Australia&lt;br&gt;Thursday 28 November 2024, Online

[https://docs.google.com/presentation/d/1uKiWbuae48OGDjih2Z5X8Kqh84Drw3WUDfUULn3IjxY/edit?usp=sharing Slides]&lt;br&gt;(Google)&lt;!--

[ Recording]&lt;br&gt;(YouTube; X:X mins including Q&amp;A)

[ LinkedIn post]

[ X post] --&gt;
&lt;/div&gt;

==Abstract==
The Wikimedia Foundation's mission is to make the sum of human knowledge freely available for all. Although Wikipedia is well known, it is only for encyclopedic information. The Wikimedia Foundation has other sister projects for other purposes. For example, Wikiversity provides an optimal, wiki-based platform for development and sharing of open educational resources. Wikiversity can be edited by anyone, including educators and students, making it an excellent platform for student-staff collaboration. Wikiversity can also be used for research.

==Bio==
James is an Assistant Professor in Psychology at the [[University of Canberra]]. James is a keen proponent of open educational practices and use of wikis for curriculum development and student learning projects.

==Values==
Arguably the highest value for academics is to contribute as openly as possible to the knowledge commons. Academics are (mostly) employed by public institutions to broker knowledge and support the education of society and its citizens.

However, academics are vulnerable to becoming institutionalised and operating largely within walled gardens and ivory towers. For example, it is easy to default to minimal sharing of knowledge through paywalled journals and closed learning management systems. This means that most citizens are unable to access much higher education knowledge.

Personally, I strive to practice [[open academia|open academia]]. This means sharing teaching, research, and service materials as openly and flexibly as possible. Ideally, this means using:
* open access
* open licensing
* open formats
* open governance/decision-making

;Premises
* Knowledge-sharing is good (or at least better than not sharing)
* Open education is good (or at least better than closed education)
* Wikis offer underutilised utility
* Wikiversity is a &quot;perfect&quot; platform for practicing open academia

==Wikis==
Wikis are the earliest and simplest openly editable webpages. [[w:WikiWikiWeb#Etymology|Wiki wiki is a Hawaiian term for quick]]. Wikis allow open communities to collaboratively construct and share information.

;Wiki history:
* 1993: Internet (www)
* 1995: WikiWikiWeb (the first wiki)
* 2001: Wikipedia (after some earlier prototypes)
* 2003: Wikibooks (for original work)
* 2006: Wikiversity (a spin off from Wikibooks for teaching, learning, and research)

==Wikimedia Foundation==
The non-profit [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) hosts a rich ecosystem of wiki-based sister projects which include Wikipedia for encyclopedic information and Wikiversity for education. Each of these projects is multi-lingual, with different instances for each language.

;[[w:Wikimedia Foundation#Mission|Wikimedia Foundation mission]]:

&lt;blockquote&gt;&quot;to empower and engage people around the world to collect and develop educational content under a free license or in the public domain, and to disseminate it effectively and globally.&quot;&lt;/blockquote&gt;

==Sister projects==
The Wikimedia Foundation hosts a diverse ecosystem of multilingual free and open wiki projects.

Most people are somewhat familiar with Wikipedia but are much less aware of the other projects.

A WMF user account is universal, so it works across all wikis and languages.

There are 15 [[w:Wikipedia:Wikimedia sister projects|sister projects]], including Wikipedia. They run on open-source MediaWiki software.

Each sister project has a specific purpose and focus. For example:
* [[w:|Wikipedia]], which is a wiki for encyclopedic information, but they are typically much less familiar with other projects such as 
* [[b:|Wikibooks]] (for authoring free books e.g., a textbooks)
* [[species:|Wikispecies]] (for flora and fauna taxonomic information)
* [[wikt:|Wiktionary]] (for definitions, synonyms etc.,) 
* [[n:|Wikinews]] (for news),
* [[c:|Wikimedia Commons]] (for images, audio, video etc)

Each of these projects offer are free, online organisations of information and knowledge that anyone can create, edit, and comment on.

Together, the sister projects provide more than the sum of their parts because they work together synergistically.

==Wikiversity==
The Wikiversity mission is to provide free and open teaching, learning, and research materials. Unlike, Wikipedia, original research is permitted.

Collectively, [https://wikiversity.org Wikiversity] consists of:
* ~150,000 pages in 17 languages (plus a multi-lingual hub for other languages)
* ~800 active users
* ~50 administrators (unpaid, volunteer)
{{Wikiversity:Main Page/Introduction}}
===Teaching===
Wikiversity can be used to host teaching materials, including lecture/tutorial/workshop notes.

Teachers can:
* Create, edit, and fork informal and formal open educational resources (e.g., course materials and learning activities) on any topic, size, and level
* Engage students in contributing to learning projects (e.g., for learning and assessment exercises) (e.g., [[Motivation and emotion/Book]])

Teaching students to edit:
* Easy, fun, and experiential
* Students learn essential skills in 1 hour class:
** Create user account
** Edit user page
** Basic markup - bold, italics etc.
** Links - internal, interwiki, external
** Headings and table of contents
** Finding and embedding images

See: [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing tutorial]]

===Research===
Original research is not allowed on Wikipedia; it is allowed (and encouraged) on Wikiversity. This research can range from informal to formal peer-reviewed (e.g., see [[WikiJournals]])

Researchers can:
* Develop research projects
* Seek peer-review
* Publish findings

===Administration and support===
* Local
** User - edit, move etc.
** Curator - delete, protect etc.
** Custodian (admin/sysop) - + user block
** Bureaucrat - +assign user permissions
** Check-user - IP address checking
* Meta
** Stewards - + global user block, spam + title blacklist, oversight
** Developer - + enable extensions, software changes

More info: [[Wikiversity:Support staff]]

===Example===
In the 3rd year undergraduate psychology unit, [[Motivation and emotion]], at the [[University of Canberra]], ~150 students per year learn how to collaboratively author 4,000 word online book chapters. To date, over 1,500 chapters have been created:

[[Motivation and emotion/Book|Browse the book]]

Each chapter tackles a unique topic and includes learning features such as:
* Links
* Images
* Tables
* Quizzes etc.

For more information about the principles and practices of this Wikiversity project, see:
{{Hanging indent|1=

{{User:Jtneill/Publications/2024/Collaborative}}

{{User:Jtneill/Publications/2024/Rich}}

}}

===LMS vs Wikiversity===
[[w:Learning management system|Learning management systems]] are for:
* Institutional enrolments
* Assessment submission
* Marks
* Course announcements and discussion

Wikiversity is for:
* [[w:Open educational resources|Open educational resources]]
** Lesson plans
** Class materials
** Assessment guidelines
** Student editing
** Discussion/feedback

===Challenges===
* Awareness of wikis is low
* Motivation to teach openly is weak
* Developing editing skills can be scary
* Institutional copyright policies often archaic
* Learning design / ed tech support lacking
* Local championing / communities of practice needed

===Lessons learned===
* Clarify/articulate values/teaching philosophy (establish the andragogical foundations)
* Identify a knowledge commons goal
* Scaffold the task (break it into chunks)
* Teach essential/basic skills
* Cultivate a curious, supportive learning environment (e.g., have a go, share, support)
* Start small and build up over time
* Share about experiences

==Conclusion==
* Wikis = free, no cost, flexible, way of developing open educational curricula and learning activities
* Start small to learn
* Go forth and wikify

&lt;!--

==See also==
--&gt;
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
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&lt;div style=&quot;text-align: center&quot;&gt;
[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://wikimedia.org.au/wiki/Using_Wikiversity_and_WikiLearn_for_teaching_and_learning Using Wikiversity and WikiLearn for teaching and learning], Wikimedia Australia&lt;br&gt;Thursday 28 November 2024, Online

[https://docs.google.com/presentation/d/1uKiWbuae48OGDjih2Z5X8Kqh84Drw3WUDfUULn3IjxY/edit?usp=sharing Slides]&lt;br&gt;(Google)&lt;!--

[ Recording]&lt;br&gt;(YouTube; X:X mins including Q&amp;A)

[ LinkedIn post]

[ X post] --&gt;
&lt;/div&gt;

==Abstract==
The Wikimedia Foundation strives to make the sum of human knowledge freely available for all. Whilst Wikipedia is renowned as an encyclopedia, the Foundation also hosts a diverse ecosystem of sister projects designed for specific purposes. Among them, Wikiversity, stands out a dynamic platform for creating, sharing, and collaboratively editing open educational resources. By enabling contributions from educators, students, and researchers, Wikiversity fosters an inclusive and participatory approach to knowledge creation. As these resources are openly editable, anyone can contribute, allowing collaboration between educators, students, and researchers. This presentation explores the evolution, application, and transformative potential of Wikiversity as an platform for open teaching, learning, and research.

''Keywords'': open academia, open educational resources, collaborative learning, Wikimedia Foundation, Wikiversity

==Bio==
James is an Assistant Professor in Psychology at the [[University of Canberra]]. James is a keen proponent of open educational practices and use of wikis for curriculum development and student learning projects.

==Values==
Arguably the highest value for academics is to contribute as openly as possible to the knowledge commons. Academics are (mostly) employed by public institutions to broker knowledge and support the education of society and its citizens.

However, academics are vulnerable to becoming institutionalised and operating largely within walled gardens and ivory towers. For example, it is easy to default to minimal sharing of knowledge through paywalled journals and closed learning management systems. This means that most citizens are unable to access much higher education knowledge.

Personally, I strive to practice [[open academia|open academia]]. This means sharing teaching, research, and service materials as openly and flexibly as possible. Ideally, this means using:
* open access
* open licensing
* open formats
* open governance/decision-making

;Premises
* Knowledge-sharing is good (or at least better than not sharing)
* Open education is good (or at least better than closed education)
* Wikis offer underutilised utility
* Wikiversity is a &quot;perfect&quot; platform for practicing open academia

==Wikis==
Wikis are the earliest and simplest openly editable webpages. [[w:WikiWikiWeb#Etymology|Wiki wiki is a Hawaiian term for quick]]. Wikis allow open communities to collaboratively construct and share information.

;Wiki history:
* 1993: Internet (www)
* 1995: WikiWikiWeb (the first wiki)
* 2001: Wikipedia (after some earlier prototypes)
* 2003: Wikibooks (for original work)
* 2006: Wikiversity (a spin off from Wikibooks for teaching, learning, and research)

==Wikimedia Foundation==
The non-profit [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) hosts a rich ecosystem of wiki-based sister projects which include Wikipedia for encyclopedic information and Wikiversity for education. Each of these projects is multi-lingual, with different instances for each language.

;[[w:Wikimedia Foundation#Mission|Wikimedia Foundation mission]]:

&lt;blockquote&gt;&quot;to empower and engage people around the world to collect and develop educational content under a free license or in the public domain, and to disseminate it effectively and globally.&quot;&lt;/blockquote&gt;

==Sister projects==
The Wikimedia Foundation hosts a diverse ecosystem of multilingual free and open wiki projects.

Most people are somewhat familiar with Wikipedia but are much less aware of the other projects.

A WMF user account is universal, so it works across all wikis and languages.

There are 15 [[w:Wikipedia:Wikimedia sister projects|sister projects]], including Wikipedia. They run on open-source MediaWiki software.

Each sister project has a specific purpose and focus. For example:
* [[w:|Wikipedia]], which is a wiki for encyclopedic information, but they are typically much less familiar with other projects such as 
* [[b:|Wikibooks]] (for authoring free books e.g., a textbooks)
* [[species:|Wikispecies]] (for flora and fauna taxonomic information)
* [[wikt:|Wiktionary]] (for definitions, synonyms etc.,) 
* [[n:|Wikinews]] (for news),
* [[c:|Wikimedia Commons]] (for images, audio, video etc)

Each of these projects offer are free, online organisations of information and knowledge that anyone can create, edit, and comment on.

Together, the sister projects provide more than the sum of their parts because they work together synergistically.

==Wikiversity==
The Wikiversity mission is to provide free and open teaching, learning, and research materials. Unlike, Wikipedia, original research is permitted.

Collectively, [https://wikiversity.org Wikiversity] consists of:
* ~150,000 pages in 17 languages (plus a multi-lingual hub for other languages)
* ~800 active users
* ~50 administrators (unpaid, volunteer)
{{Wikiversity:Main Page/Introduction}}
===Teaching===
Wikiversity can be used to host teaching materials, including lecture/tutorial/workshop notes.

Teachers can:
* Create, edit, and fork informal and formal open educational resources (e.g., course materials and learning activities) on any topic, size, and level
* Engage students in contributing to learning projects (e.g., for learning and assessment exercises) (e.g., [[Motivation and emotion/Book]])

Teaching students to edit:
* Easy, fun, and experiential
* Students learn essential skills in 1 hour class:
** Create user account
** Edit user page
** Basic markup - bold, italics etc.
** Links - internal, interwiki, external
** Headings and table of contents
** Finding and embedding images

See: [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing tutorial]]

===Research===
Original research is not allowed on Wikipedia; it is allowed (and encouraged) on Wikiversity. This research can range from informal to formal peer-reviewed (e.g., see [[WikiJournals]])

Researchers can:
* Develop research projects
* Seek peer-review
* Publish findings

===Administration and support===
* Local
** User - edit, move etc.
** Curator - delete, protect etc.
** Custodian (admin/sysop) - + user block
** Bureaucrat - +assign user permissions
** Check-user - IP address checking
* Meta
** Stewards - + global user block, spam + title blacklist, oversight
** Developer - + enable extensions, software changes

More info: [[Wikiversity:Support staff]]

===Example===
In the 3rd year undergraduate psychology unit, [[Motivation and emotion]], at the [[University of Canberra]], ~150 students per year learn how to collaboratively author 4,000 word online book chapters. To date, over 1,500 chapters have been created:

[[Motivation and emotion/Book|Browse the book]]

Each chapter tackles a unique topic and includes learning features such as:
* Links
* Images
* Tables
* Quizzes etc.

For more information about the principles and practices of this Wikiversity project, see:
{{Hanging indent|1=

{{User:Jtneill/Publications/2024/Collaborative}}

{{User:Jtneill/Publications/2024/Rich}}

}}

===LMS vs Wikiversity===
[[w:Learning management system|Learning management systems]] are for:
* Institutional enrolments
* Assessment submission
* Marks
* Course announcements and discussion

Wikiversity is for:
* [[w:Open educational resources|Open educational resources]]
** Lesson plans
** Class materials
** Assessment guidelines
** Student editing
** Discussion/feedback

===Challenges===
* Awareness of wikis is low
* Motivation to teach openly is weak
* Developing editing skills can be scary
* Institutional copyright policies often archaic
* Learning design / ed tech support lacking
* Local championing / communities of practice needed

===Lessons learned===
* Clarify/articulate values/teaching philosophy (establish the andragogical foundations)
* Identify a knowledge commons goal
* Scaffold the task (break it into chunks)
* Teach essential/basic skills
* Cultivate a curious, supportive learning environment (e.g., have a go, share, support)
* Start small and build up over time
* Share about experiences

==Conclusion==
* Wikis = free, no cost, flexible, way of developing open educational curricula and learning activities
* Start small to learn
* Go forth and wikify

&lt;!--

==See also==
--&gt;
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
      <sha1>890ntpj1efv8asl5a4voolyhhez3bje</sha1>
    </revision>
    <revision>
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      <text bytes="8922" sha1="kngkwpe2l0qaffvx2n5f0w6khl5xmh4" xml:space="preserve">{{title|Wikiversity for teaching, learning, and research}}

&lt;div style=&quot;text-align: center&quot;&gt;
[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://wikimedia.org.au/wiki/Using_Wikiversity_and_WikiLearn_for_teaching_and_learning Using Wikiversity and WikiLearn for teaching and learning], [https://wikimedia.org.au/ Wikimedia Australia]&lt;br&gt;Thursday 28 November 2024, Online

[https://docs.google.com/presentation/d/1uKiWbuae48OGDjih2Z5X8Kqh84Drw3WUDfUULn3IjxY/edit?usp=sharing Slides]&lt;br&gt;(Google)&lt;!--

[ Recording]&lt;br&gt;(YouTube; X:X mins including Q&amp;A)

[ LinkedIn post]

[ X post] --&gt;
&lt;/div&gt;

==Abstract==
The Wikimedia Foundation strives to make the sum of human knowledge freely available for all. Whilst Wikipedia is renowned as an encyclopedia, the Foundation also hosts a diverse ecosystem of sister projects designed for specific purposes. Among them, Wikiversity, stands out a dynamic platform for creating, sharing, and collaboratively editing open educational resources. By enabling contributions from educators, students, and researchers, Wikiversity fosters an inclusive and participatory approach to knowledge creation. As these resources are openly editable, anyone can contribute, allowing collaboration between educators, students, and researchers. This presentation explores the evolution, application, and transformative potential of Wikiversity as an platform for open teaching, learning, and research.

''Keywords'': open academia, open educational resources, collaborative learning, Wikimedia Foundation, Wikiversity

==Bio==
James is an Assistant Professor in Psychology at the [[University of Canberra]]. James is a keen proponent of open educational practices and use of wikis for curriculum development and student learning projects.

==Values==
Arguably the highest value for academics is to contribute as openly as possible to the knowledge commons. Academics are (mostly) employed by public institutions to broker knowledge and support the education of society and its citizens.

However, academics are vulnerable to becoming institutionalised and operating largely within walled gardens and ivory towers. For example, it is easy to default to minimal sharing of knowledge through paywalled journals and closed learning management systems. This means that most citizens are unable to access much higher education knowledge.

Personally, I strive to practice [[open academia|open academia]]. This means sharing teaching, research, and service materials as openly and flexibly as possible. Ideally, this means using:
* open access
* open licensing
* open formats
* open governance/decision-making

;Premises
* Knowledge-sharing is good (or at least better than not sharing)
* Open education is good (or at least better than closed education)
* Wikis offer underutilised utility
* Wikiversity is a &quot;perfect&quot; platform for practicing open academia

==Wikis==
Wikis are the earliest and simplest openly editable webpages. [[w:WikiWikiWeb#Etymology|Wiki wiki is a Hawaiian term for quick]]. Wikis allow open communities to collaboratively construct and share information.

;Wiki history:
* 1993: Internet (www)
* 1995: WikiWikiWeb (the first wiki)
* 2001: Wikipedia (after some earlier prototypes)
* 2003: Wikibooks (for original work)
* 2006: Wikiversity (a spin off from Wikibooks for teaching, learning, and research)

==Wikimedia Foundation==
The non-profit [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) hosts a rich ecosystem of wiki-based sister projects which include Wikipedia for encyclopedic information and Wikiversity for education. Each of these projects is multi-lingual, with different instances for each language.

;[[w:Wikimedia Foundation#Mission|Wikimedia Foundation mission]]:

&lt;blockquote&gt;&quot;to empower and engage people around the world to collect and develop educational content under a free license or in the public domain, and to disseminate it effectively and globally.&quot;&lt;/blockquote&gt;

==Sister projects==
The Wikimedia Foundation hosts a diverse ecosystem of multilingual free and open wiki projects.

Most people are somewhat familiar with Wikipedia but are much less aware of the other projects.

A WMF user account is universal, so it works across all wikis and languages.

There are 15 [[w:Wikipedia:Wikimedia sister projects|sister projects]], including Wikipedia. They run on open-source MediaWiki software.

Each sister project has a specific purpose and focus. For example:
* [[w:|Wikipedia]], which is a wiki for encyclopedic information, but they are typically much less familiar with other projects such as 
* [[b:|Wikibooks]] (for authoring free books e.g., a textbooks)
* [[species:|Wikispecies]] (for flora and fauna taxonomic information)
* [[wikt:|Wiktionary]] (for definitions, synonyms etc.,) 
* [[n:|Wikinews]] (for news),
* [[c:|Wikimedia Commons]] (for images, audio, video etc)

Each of these projects offer are free, online organisations of information and knowledge that anyone can create, edit, and comment on.

Together, the sister projects provide more than the sum of their parts because they work together synergistically.

==Wikiversity==
The Wikiversity mission is to provide free and open teaching, learning, and research materials. Unlike, Wikipedia, original research is permitted.

Collectively, [https://wikiversity.org Wikiversity] consists of:
* ~150,000 pages in 17 languages (plus a multi-lingual hub for other languages)
* ~800 active users
* ~50 administrators (unpaid, volunteer)
{{Wikiversity:Main Page/Introduction}}
===Teaching===
Wikiversity can be used to host teaching materials, including lecture/tutorial/workshop notes.

Teachers can:
* Create, edit, and fork informal and formal open educational resources (e.g., course materials and learning activities) on any topic, size, and level
* Engage students in contributing to learning projects (e.g., for learning and assessment exercises) (e.g., [[Motivation and emotion/Book]])

Teaching students to edit:
* Easy, fun, and experiential
* Students learn essential skills in 1 hour class:
** Create user account
** Edit user page
** Basic markup - bold, italics etc.
** Links - internal, interwiki, external
** Headings and table of contents
** Finding and embedding images

See: [[Motivation and emotion/Tutorials/Wiki editing|Wiki editing tutorial]]

===Research===
Original research is not allowed on Wikipedia; it is allowed (and encouraged) on Wikiversity. This research can range from informal to formal peer-reviewed (e.g., see [[WikiJournals]])

Researchers can:
* Develop research projects
* Seek peer-review
* Publish findings

===Administration and support===
* Local
** User - edit, move etc.
** Curator - delete, protect etc.
** Custodian (admin/sysop) - + user block
** Bureaucrat - +assign user permissions
** Check-user - IP address checking
* Meta
** Stewards - + global user block, spam + title blacklist, oversight
** Developer - + enable extensions, software changes

More info: [[Wikiversity:Support staff]]

===Example===
In the 3rd year undergraduate psychology unit, [[Motivation and emotion]], at the [[University of Canberra]], ~150 students per year learn how to collaboratively author 4,000 word online book chapters. To date, over 1,500 chapters have been created:

[[Motivation and emotion/Book|Browse the book]]

Each chapter tackles a unique topic and includes learning features such as:
* Links
* Images
* Tables
* Quizzes etc.

For more information about the principles and practices of this Wikiversity project, see:
{{Hanging indent|1=

{{User:Jtneill/Publications/2024/Collaborative}}

{{User:Jtneill/Publications/2024/Rich}}

}}

===LMS vs Wikiversity===
[[w:Learning management system|Learning management systems]] are for:
* Institutional enrolments
* Assessment submission
* Marks
* Course announcements and discussion

Wikiversity is for:
* [[w:Open educational resources|Open educational resources]]
** Lesson plans
** Class materials
** Assessment guidelines
** Student editing
** Discussion/feedback

===Challenges===
* Awareness of wikis is low
* Motivation to teach openly is weak
* Developing editing skills can be scary
* Institutional copyright policies often archaic
* Learning design / ed tech support lacking
* Local championing / communities of practice needed

===Lessons learned===
* Clarify/articulate values/teaching philosophy (establish the andragogical foundations)
* Identify a knowledge commons goal
* Scaffold the task (break it into chunks)
* Teach essential/basic skills
* Cultivate a curious, supportive learning environment (e.g., have a go, share, support)
* Start small and build up over time
* Share about experiences

==Conclusion==
* Wikis = free, no cost, flexible, way of developing open educational curricula and learning activities
* Start small to learn
* Go forth and wikify

&lt;!--

==See also==
--&gt;
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
      <sha1>kngkwpe2l0qaffvx2n5f0w6khl5xmh4</sha1>
    </revision>
  </page>
  <page>
    <title>Conformal symmetry, its motivations, its applications</title>
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      <id>2689200</id>
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      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
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== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A map &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
To get an idea on the existence of conformal transformations, let us count the equations and unknowns in the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt;, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;. If &lt;math&gt;d=\dim(M)&lt;/math&gt;, we have 
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Map &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, we have &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns. For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

[[Category: CFT course]]</text>
      <sha1>o5k8wrchvmvmhzxhjxbak4keryrbnzy</sha1>
    </revision>
    <revision>
      <id>2689202</id>
      <parentid>2689200</parentid>
      <timestamp>2024-11-28T15:25:49Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
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== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^d&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

[[Category: CFT course]]</text>
      <sha1>a7e7ve1k551ki5z8p8h001d15pziivi</sha1>
    </revision>
    <revision>
      <id>2689214</id>
      <parentid>2689202</parentid>
      <timestamp>2024-11-28T16:13:50Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
      <comment>/* Conformal invariance */</comment>
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== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

[[Category: CFT course]]</text>
      <sha1>b35dy793jqcdhxs43ofujwgpwdd7m85</sha1>
    </revision>
    <revision>
      <id>2689267</id>
      <parentid>2689214</parentid>
      <timestamp>2024-11-29T08:30:39Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
      <comment>/* Exercises */</comment>
      <origin>2689267</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4414" sha1="qi8ghrlxicrcu0tu8zwq15rmnxhgidq" xml:space="preserve">
== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

=== COGS: The conformal group of flat space ===

Consider the Euclidean space &lt;math&gt;\mathbb{R}^d&lt;/math&gt; with the flat metric &lt;math&gt;g_{\mu\nu}=\delta_{\mu\nu}&lt;/math&gt;, and the Minkovskian space &lt;math&gt;\mathbb{R}^{d+1,1}&lt;/math&gt; with coordinates &lt;math&gt;Y=\left(y^\mu,y^-,y^+\right)&lt;/math&gt; with &lt;math&gt;\mu = 1,2,\dots, d&lt;/math&gt; and the flat metric &lt;math&gt;ds^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ &lt;/math&gt;.
Consider the diffeormorphisms
:&lt;math&gt; \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d &amp; \to &amp; \mathbb{R}^{d+1,1} \\ x^\mu &amp;\mapsto &amp; \left(x^\mu,\|x\|^2,1\right) \end{array}\right.  \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} &amp; \to &amp; \mathbb{R}^{d+1,1} \\ Y &amp;\mapsto &amp; \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{\|y\|^2}{y^+},1\right) \end{array}\right.
&lt;/math&gt;
# Check that &lt;math&gt;\varphi&lt;/math&gt; is an isometry.

[[Category: CFT course]]</text>
      <sha1>qi8ghrlxicrcu0tu8zwq15rmnxhgidq</sha1>
    </revision>
    <revision>
      <id>2689268</id>
      <parentid>2689267</parentid>
      <timestamp>2024-11-29T08:35:19Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
      <comment>/* COGS: The conformal group of flat space */</comment>
      <origin>2689268</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4669" sha1="l1i5j6zsnzmv0o593mpf34p58hl5bui" xml:space="preserve">
== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

=== COGS: The conformal group of flat space ===

Consider the Euclidean space &lt;math&gt;\mathbb{R}^d&lt;/math&gt; with the flat metric &lt;math&gt;g_{\mu\nu}=\delta_{\mu\nu}&lt;/math&gt;, and the Minkovskian space &lt;math&gt;\mathbb{R}^{d+1,1}&lt;/math&gt; with coordinates &lt;math&gt;Y=\left(y^\mu,y^-,y^+\right)&lt;/math&gt; with &lt;math&gt;\mu = 1,2,\dots, d&lt;/math&gt; and the flat metric &lt;math&gt;\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ &lt;/math&gt;.
Consider the diffeormorphisms
:&lt;math&gt; \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d &amp; \to &amp; \mathbb{R}^{d+1,1} \\ x^\mu &amp;\mapsto &amp; \left(x^\mu,\|x\|^2,1\right) \end{array}\right.  \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} &amp; \to &amp; \mathbb{R}^{d+1,1} \\ Y &amp;\mapsto &amp; \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{\|y\|^2}{y^+},1\right) \end{array}\right.
&lt;/math&gt;
# Check that &lt;math&gt;\varphi&lt;/math&gt; is an isometry. Is &lt;math&gt;\psi&lt;/math&gt; an isometry? Is it a conformal transformation?
# Show that the restriction of &lt;math&gt;\psi&lt;/math&gt; to the light cone &lt;math&gt;\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\middle| \|Y\|^2 = 0\right\}&lt;/math&gt; is a conformal transformation.


[[Category: CFT course]]</text>
      <sha1>l1i5j6zsnzmv0o593mpf34p58hl5bui</sha1>
    </revision>
    <revision>
      <id>2689271</id>
      <parentid>2689268</parentid>
      <timestamp>2024-11-29T08:37:23Z</timestamp>
      <contributor>
        <username>Sylvain Ribault</username>
        <id>2127778</id>
      </contributor>
      <comment>/* COGS: The conformal group of flat space */</comment>
      <origin>2689271</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4672" sha1="3jhkzaxz0dcpmvdx4uyhekbn0yjn49r" xml:space="preserve">
== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

=== COGS: The conformal group of flat space ===

Consider the Euclidean space &lt;math&gt;\mathbb{R}^d&lt;/math&gt; with the flat metric &lt;math&gt;g_{\mu\nu}=\delta_{\mu\nu}&lt;/math&gt;, and the Minkowski space &lt;math&gt;\mathbb{R}^{d+1,1}&lt;/math&gt; with coordinates &lt;math&gt;Y=\left(y^\mu,y^-,y^+\right)&lt;/math&gt; with &lt;math&gt;\mu = 1,2,\dots, d&lt;/math&gt; and the flat metric &lt;math&gt;\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ &lt;/math&gt;.
Consider the diffeormorphisms
:&lt;math&gt; \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d &amp; \to &amp; \mathbb{R}^{d+1,1} \\ x^\mu &amp;\mapsto &amp; \left(x^\mu,\|x\|^2,1\right) \end{array}\right.  \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} &amp; \to &amp; \mathbb{R}^{d+1,1} \\ Y &amp;\mapsto &amp; \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{\|y\|^2}{y^+},1\right) \end{array}\right.
&lt;/math&gt;
# Check that &lt;math&gt;\varphi&lt;/math&gt; is an isometry. Is &lt;math&gt;\psi&lt;/math&gt; an isometry? Is it a conformal transformation?
# Show that the restriction of &lt;math&gt;\psi&lt;/math&gt; to the light cone &lt;math&gt;\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}&lt;/math&gt; is a conformal transformation.


[[Category: CFT course]]</text>
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      <comment>/* COGS: The conformal group of flat space */</comment>
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== Conformal invariance == 

=== Conformal transformations ===

On a given space or spacetime &lt;math&gt;M&lt;/math&gt; with coordinates &lt;math&gt;x^\mu&lt;/math&gt;, distances are defined using a metric &lt;math&gt;g_{\mu\nu}&lt;/math&gt;. In particular, the length of an infinitesimal vector &lt;math&gt;v^\mu&lt;/math&gt; is &lt;math&gt;\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}&lt;/math&gt;. If we know distances, we can also compute angles. The angle &lt;math&gt;\theta&lt;/math&gt; between two infinitesimal vectors &lt;math&gt;v^\mu,w^\mu&lt;/math&gt; obeys 
:&lt;math&gt;
\cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}}
&lt;/math&gt;
For &lt;math&gt;f:M\to M&lt;/math&gt; a diffeomorphism, we define the [[w:pullback]] &lt;math&gt;f^*g&lt;/math&gt; of the metric by 
:&lt;math&gt;
 (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu}
&lt;/math&gt;
equivalently &lt;math&gt;(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu&lt;/math&gt;. 
A diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; is called an '''isometry''' if it preserves distances, equivalently if 
:&lt;math&gt;f^* g = g&lt;/math&gt;.
It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;, the metrics &lt;math&gt;g_{\mu\nu}&lt;/math&gt; and &lt;math&gt;\lambda g_{\mu\nu}&lt;/math&gt; define the same angles. 
So &lt;math&gt;f&lt;/math&gt; is a conformal transformation if it preserve the metric up to a scalar factor,
:&lt;math&gt;
\exists \lambda,\ f^* g = \lambda g
&lt;/math&gt;
The set of conformal transformations is called the '''conformal group''' associated to &lt;math&gt;M,g&lt;/math&gt;.

Let us indicate how many functions on &lt;math&gt;M&lt;/math&gt; are needed to parametrize the objects &lt;math&gt;\lambda, f,g&lt;/math&gt;, if &lt;math&gt;d=\dim(M)&lt;/math&gt;:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Object 
! Number of functions on &lt;math&gt;M&lt;/math&gt;
|-
| Function &lt;math&gt;\lambda:M\to \mathbb{R}&lt;/math&gt;
| &lt;math&gt;1&lt;/math&gt;
|- 
| Diffeomorphism &lt;math&gt;f:M\to M&lt;/math&gt; 
| &lt;math&gt;d&lt;/math&gt;
|-
| Metric &lt;math&gt;g&lt;/math&gt; on &lt;math&gt;M&lt;/math&gt;
| &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt;
|}
Therefore, given two metrics &lt;math&gt;g_1,g_2&lt;/math&gt;, the condition &lt;math&gt;f^* g_1 = \lambda g_2&lt;/math&gt; that they are conformally equivalent involves &lt;math&gt;\frac{d(d+1)}{2}&lt;/math&gt; equations for &lt;math&gt;d+1&lt;/math&gt; unknowns.
For &lt;math&gt;d\leq 2&lt;/math&gt;, there is always a solution (under reasonable assumptions): in particular any metric on &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is conformally flat, in other words there always exist [[w:isothermal coordinates]]. For &lt;math&gt;d\geq 3&lt;/math&gt;, two metrics are in general not equivalent modulo rescaling. 


The counting is also valid in the case &lt;math&gt;g_1=g_2&lt;/math&gt;, which informs us on the existence of conformal transformations:
{| class=&quot;wikitable&quot; style=&quot;font-size:small; text-align:center;&quot;
|-
! Dimension 
! Conformal transformations
|-
| &lt;math&gt;d=1&lt;/math&gt;
| Any &lt;math&gt;f&lt;/math&gt; 
|- 
| &lt;math&gt;d=2&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M&lt;/math&gt;
|-
| &lt;math&gt;d\geq 3&lt;/math&gt;
| Some &lt;math&gt;f&lt;/math&gt;, depending on &lt;math&gt;M,g&lt;/math&gt;
|}

d=2 need more precise statements. Riemann surfaces, moduli. 

case of flat metric, dilations etc.

not conformal example. &lt;math&gt;z^2&lt;/math&gt;? Not conformal at a point. 

conformal symmetry: inv. under conf. transfo.

=== Conformal symmetry and gravitation ===

Now the metric is dynamical as well.

2d already special

String theory (while we are at it)

== Scale invariance and conformal invariance == 

== Fixed points of the renormalization group ==

== Applications ==

== Exercises ==

=== COGS: The conformal group of flat space ===

Consider the Euclidean space &lt;math&gt;\mathbb{R}^d&lt;/math&gt; with the flat metric &lt;math&gt;g_{\mu\nu}=\delta_{\mu\nu}&lt;/math&gt;, and the Minkowski space &lt;math&gt;\mathbb{R}^{d+1,1}&lt;/math&gt; with coordinates &lt;math&gt;Y=\left(y^\mu,y^-,y^+\right)&lt;/math&gt; with &lt;math&gt;\mu = 1,2,\dots, d&lt;/math&gt; and the flat metric &lt;math&gt;\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ &lt;/math&gt;.
Consider the diffeormorphisms
:&lt;math&gt; \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d &amp; \to &amp; \mathbb{R}^{d+1,1} \\ x^\mu &amp;\mapsto &amp; \left(x^\mu,\|x\|^2,1\right) \end{array}\right.  \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} &amp; \to &amp; \mathbb{R}^{d+1,1} \\ Y &amp;\mapsto &amp; \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right.
&lt;/math&gt;
# Check that &lt;math&gt;\varphi&lt;/math&gt; is an isometry. Is &lt;math&gt;\psi&lt;/math&gt; an isometry? Is it a conformal transformation?
# Show that the restriction of &lt;math&gt;\psi&lt;/math&gt; to the light cone &lt;math&gt;\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}&lt;/math&gt; is a conformal transformation.


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      <text bytes="74654" sha1="s0bwnshpkg75x6tnme1xny8tdcw7ypc" xml:space="preserve">=Bal Poudré at Warwick Castle=
==Overview==
A bal poudré was held at Warwick Castle on Friday, 1 February 1895, with [[Social Victorians/People/Warwick|Countess Warwick]] dressed as Marie Antoinette. [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was part of the house party as well as attending the ball,&lt;ref&gt;&quot;Court Circular.&quot; ''Times'', 2 Feb. 1895, p. 10. ''The Times Digital Archive'', http://tinyurl.galegroup.com/tinyurl/AHQju3. Accessed 20 June 2019.&lt;/ref&gt;&lt;ref&gt;“Grand Bal Poudre at Warwick Castle.” ''Midland Daily Telegraph'' [now in BNA: ''Coventry Evening Telegraph''] 1 February 1895, Friday: 3 [of 4], Col. 4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000337/18950201/021/0003 (accessed July 2019).&lt;/ref&gt; as was &quot;Mr [[Social Victorians/People/Craven|Caryl Craven]], to whom so many thanks are due for the able way in which he assisted his charming hostess in carrying out her scheme, Mr Craven being quite an authority on eighteenth century French art and dress.&quot;&lt;ref&gt;&quot;The Warwick Bal Poudre.&quot; ''The Queen, The Lady's Newspaper'' 09 February 1895 Saturday: 38 [of 80], Col. 2c [of 3] – 39, Col. 3c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18950209/233/0038.&lt;/ref&gt;

Daisy, Countess Warwick dressed as Marie Antoinette for the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress]] ball as well.

Miss Beatrice Fitzherbert &quot;wore a beautiful pearl necklace, with large diamond pendant, and two diamond sprays, all of which were given by George IV. to Mrs Fitzberbert.&quot;&lt;ref name=&quot;:0&quot;&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt; (6, 4b)

Lady Peel wore a costume that, the Leamington Spa Courier says, &quot;was designed by Madame Eloffe, dressmaker to Marie Antoinette.&quot;&lt;ref name=&quot;:0&quot; /&gt; (6, 6a)

==Logistics==
* Friday, 1 February 1895
* Warwick Castle
* Hosts, Countess and the Earl of Warwick
==Related Events==
* 

== Main Newspaper Report ==
The ''Leamington Spa Courier'' had the definitive story in the next issue, a week after the event:&lt;blockquote&gt;THE GRAND BAL POUDRE AT WARWICK CASTLE.

A profound impression has been created throughout the country by the enormously successful ''bal poudre'' given by the Earl and Countess of Warwick at Warwick Castle on Friday night last week, a lengthy, but — owing to the exigencies of the occasion — necessarily incomplete record of which appeared in our issue on the following morning. It is conceded on all hands that it was unmistakably the most splendidly organised and artistically perfect function of the kind that has been given during the present century, and certainly unexampled in the annals of the county. In times past, the historic fortlace has been the scene of many gay and festive ''re-unions'', but history gives no mention of one that in any way equalled in point of completeness of detail or magnificence that in which some 400 of the present Earl and Countess’s guests participated on Friday night. For the nonce, the prosaic modern gave place to the splendour of the past, and the luxurious and gorgeous conditions which prevailed at the Tuilleries during the glories of the regime of Louis XV., and the reign of his ill-fated successor and his beautiful consort, were revived in all their sparkling radiance, thus creating a pageant of unrivalled grandeur and beauty, and one that will be long retained in the recollections of those who took part in or were permitted to gaze upon it. Moreover, never, perhaps, had the old Castle had within its walls such a notable company, including as it did, some of the most distinguished personages of the day, connections of Royalty, Ambassadors, of foreign Powers, Dukes and Duchesses, Earls and Countesses, Lords and Ladies, representatives of the three great services of the State, Statesman, lawyers, and other ornaments of the highest and most aristocratic circles. The long suite of rooms, with the abundance of rich and historic art treasures therein contained, was most tastefully and effectively decorated, and the gilded and brocaded furniture and lovely fittings were arranged so as to form a replica of the interior of the Tuilleries at the period which the Countess had, with consummate judgment, selected for representation. Wide-spreading palms were placed at different points, and rare flowers of brilliant hues from Cannes and other parts of Southern Europe were seen on every hand. The whole was brilliantly illuminated by innumerable wax candles affixed to crystal chandeliers, in addition to the rays of the electric light, emitted from glow-lamps so constructed as to simulate candles, and having the bulbs hidden by delicately tinted shades; and when the guests in their picturesque costumes perambulated the apartments, the spectacle afforded was unique and enthralling. The most striking scenes, however, were witnessed in the Cedar Drawing-room when the dancing was in full operation, and again when the richly-dressed and white-wigged guests sat down to supper in the grand Banquetting-hall.

The guests commenced to arrive about half-past ten, and carriages continued to roll up until close upon midnight. The traffic was directed by the same staff of police who were in attendance at the recent concert, and these were also assisted by the Commissionaires. Under the instructions of the House Steward, Inspector Hall and his men guided the traffic most skilfully, notwithstanding that the entrance to the courtyard beneath the gateway and barbican is very narrow. Precaution had been taken to fix a large number of lamps along the approaches to the Castle, to minimise the danger of an accident. The guests did not use the grand entrance under the porch, but entered by the door at the other end beyond the chapel, over which a large striped awning served as a porch and a crush room, the interior being decorated with flowering and foliage plants, and splendidly lighted by pendant lamps. They then passed through the armour passage to the centre State Drawing-room, adjoining the ballroom, where they were received by the Countess, the train of whose lovely and charming costume, a la Marie Antoinette, was borne by her little daughter, Lady Marjorie Greville, and a young companion, Miss Hamilton, who were attired as imitation China shepherdesses in white broche silk, and large white satin hats, trimmed with roses and long ostrich feathers, and carried wands. Dancing commenced at a quarter to 11, the music being supplied by Herr Wurms’ Viennese band, the members of which wore the dress of the period carried out in white and gold, and were ensconced in an orchestra formed in one of the arched windows. The ball was opened by a quadrille, in which one set was made up of the Countess and Count Demyn (the Austro-Hungarian Ambassador), the Duchess of Sutherland and [[Social Victorians/People/de Soveral|M. de Soverel]] [sic] (the Portuguese Ambassador), Princess Henry of Pless and the Earl of Warwick, and Prince Henry of Pless and Lady Feodora Sturt. The programmes of dance music were in book form, bearing a miniature medallion of Marie Antoinette on the one side, and Warwick Castle, set in a little Louis Quinze frame, on the other. The scene, while the dancing was in full swing, was replete with animation and splendid beauty. The infinite variety of costumes, flashing diamonds and other jewels, and a brilliance of colour ever changing with wondrous rapidity, as the dancers advanced and receded, or mingled in the crowd, backed by the cedar pannelling with the light falling from the candelabra and incandescent electric lights upon the fine Vandyck family portraits hanging round it, the large crystal chandeliers pending from the white and gold ceiling, and standing in each corner of the room and on either side of the great marble and alabaster mantelpiece, made up a picture at once quaint, full of life, animation, and picturesque beauty. Shortly after midnight, three trumpeters, correctly dressed in the gold-bedecked uniform of English heralds of the time of Louis XV., took up a position at the entrance to the banquetting hall and gave the signal that supper was served, by blowing a fanfare. Thereupon, a procession was formed, the Conntess [sic] of Warwick leading the way with Count Deym. M. de Soverel followed with the Duchess of Sutherland; then came the Earl of Warwick with Princess Henry of Pless, Prince Henry with Lady Feodora Sturt, the Earl of Lonsdale with Mrs Arthur Paget, and Lord Kenyon with Mrs Miller-Munday [sic for Mundy]. This party of 12 seated themselves at the centre table, other guests occupying the round and oval tables, about 14 in number, and each laid for eight. Special arrangements had been made for the serving of ''souper'', which was supplied entirely from the Castle kitchens, and it was originally intended, when it was thought the number of guests would not exceed 240, that all should sit down together. But the applications from those who wished to be included in what may, without exaggeration, be termed an historical event in the social functions of Warwickshire, were so very numerous that it was necessary to divide the company into two sections. On the centre table was an imposing display of the handsome gold and silver family plate, including a celebrated gold cup modelled by Benvenuto Cellini, and the floral embellishments consisted of choice flowers from Cannes and magnificent orchids from Trentham. The round and oval tables were also handsomely decorated with silver plate. The meal was a truly sumptuous one, and the menu, which was printed inside a little white and gold Louis XV. screen, having a picture of Warwick Castle on one side and “Souper, Février 1, 1895,” on the reverse, included some triumphs of the culinary art. The dessert comprised strawberries, apricots, grapes, pineapples, and other fruits rare and expensive at this season of the year,[sic] The hall, with its shining coats of mail, the magnificent Beauvais tapestry forming portieres and hanging from the gallery, the massive silver candelabra on the tables, and the immense ecclesiastical candlesticks standing on the floor and bearing torches which towered far above the heads of the guests, constituted a truly marvellous sight, and one upon which the eye never tired to dwell. To render the picture more complete, the servants, who flitted about attending upon the wants of the guests, were clothed in the livery of the period, some in white and gold and red velvet, and others in sabre suits of black, all wearing knee-breeches, silk stockings, and white wigs. The staff of servants at the Castle was quite inadequate to carry out the various duties which devolved upon them in consequence of the ball, and Mr J. Hall (the House-steward) consequently found it necessary to engage a special staff of first-class waiters from London. A few privileged persons, to whom tickets had previously been issued, were admitted to the long narrow passage in the thickness of the wall near the roof, which was discovered at the time of the disastrous fire in 1871, to which access is gained through the oak and carnation rooms. After supper, dancing was resumed, and continued with unabated vigour until considerably after four o’clock, the Countess remaining during the whole of that time with her guests.

One of the guests made an unfortunate miss of the train which cost him a good deal of inconvenience, and his host and hostess some anxiety. The gentleman in question was taken as one of a large house party in the county to Warwick, and there was a special train chartered from Milverton to carry the guests back to a station near the host's residence. A short time before his party were returning home, the guest went into the smoke-room at the Castle, and though the heard the name of his host called, he thought it was the company of a lady of the same name who were wanted. In the end the gentleman was left behind, and then he drove to Leamington, but could get no train from there. All the hotels were [Col. a/b] full, and so it was no use to apply there for a bed. The consequence was he had to spend the night in a waiting room either at Leamington or Milverton Station, getting back to the country house on Saturday. He will long remember Lady Warwick’s ball.

Letters have been received from guests expressing satisfaction in regard to the efficient way in which the police carried out their duties. It may be stated that the numerous alterations and renovations at the Castle, especially in regard to the private apartments, have been carried out by Messrs Bertram and Sons, the great upholsterers, of Dean-street, Soho.

As one of the lady guests was alighting from her carriage at the Castle, on Friday night, a large diamond and turquoise ornament, valued at 200 guineas, became detached from her hair, and fell to the ground. The loss was quickly discovered, and, fortunately, the costly ornament was recovered intact.

Viscount Dungarven, whe was to have formed one of Mr W. M. Low’s party, was prevented by unforseen circumstances from attending the ball. Mr Perry (Bitham House) was also prevented attending by illness.

LIST OF GUESTS.

It has been found impossible to obtain a complete list of the names of the guests owing to the fact that the presentation of tickets was dispensed with, and we have, therefore, been compelled to rely on extraneous sources for information. The following is a list of the names of a large number of those present at the function:—

PRINCES [init caps large, rest sm] — Francis of Teck and Henry of Pless (Viscomte de Bragabene [Bragelonne?].)

PRINCESS — Henry of Pless (Adrienne Lecouvreur.)

EARLS — Clonmell (modern Court dress), Rosslyn (Duc de Nemours), Lonsdale (M. de Copinson, Keeper of the Koyal [sic] Stud, Louis XV.), and Chesterfield (Court costume.)

DUKE of Manchester (Marquis de Grammont.)

DUCHESS of Sutherland (Queen of Louis XV.)

MARQUIS of Hertford (Court costume.)

MARCHIONESS of Hertford (Court costume.)

COUNT Paiffy (costume Louis XV.)

COUNTESSES Cairns (Duchess de Bouillon) and Rosslyn (Marchande Coquette.)

COMTESSE Ahlefeldt-Laurvig (Dame de la temps Louis XVI.)

VISCOUNTS — Southwell (Court dress of the period), and Clifden (Court dress of the period ).

HIS EXCELLENCY the Portuguese Minister (Mousquetaire of the 2ud [sic] Company of the Royal Household, Louis XV.)

HIS EXCELLENCY the Austro-Hungarian Ambassador (English Court dress.)

LORDS — Burford (Mousquetaire), Cecil Manners (Court dress of the period), Churchill (Mons. de Brissac), Kenyon (officer of the Regiment du Roi, Louis XVI.), Clifton (officer of the Guards, Louis XV.), Lovat (Comte d’Artagnas), Richard Neville (Duc de Lauzun), Frederick Hamilton; Royston (Souis Brigadier of Mousquetaires, Louis XVI.), Grey de Wilton (gentlemen temps Louis XV.). and Doneraile (modern Court dress.)

LADIES — Norreys (Paysaune Galante), Ann Murray (Madame de Pompadour), Waller (Comtesse d’Artois), Peel (costume 1787), Chetwode, Angela St. Clair Erskine (Lady Mary Campbell), Eva Greville (Polichinelle, Louis (XV.), G. Petre, Feodora Sturt (Madame la Marquise de la Pompadour), Gerard (Duchess de Pognac), Edith Seymour (Lady of the reign of Louis XVI.), Mordaunt (Princess de Lambelle), and Churchill (French Marquise in the time of Louis XV.)

SIRS — Algernon Osborne (civilian costume, Louis XV.), Archibald Edmonstone (Mousquetuaire), Francis Burdett, Charles Mordaunt (gentleman of the time of Louis XVI.), and F. Peel.

HONOURABLES — Mrs Louis Greville (dress of the period), Dudley Ward (Mousquetaire), C. Finch (gentleman of the period of Louis XV.), Captain Alwyn Greville (Mousquetaire), Mr and Mrs Chandos Leigh, Mrs Alwyn Greville (Dame de la Court Louis XV.), Captain Hedworth Lambton (Courtier of Louis XVI.), Humphrey Sturt, M.P. (Abbé Bouvet), Mrs E. Lyon (''à la'' Watteau), Mrs Dudley Ward, B. W. H. Stoner (Mousquetaire, Louis XVI.), Sidney Greville (officer of the Regiment of the Swiss Guards), Louis Greville (Mousquetaire, Regiment de Provence, Louis XV.), [[Social Victorians/People/Keppel|George Keppel]] (Mousquetaire), [[Social Victorians/People/Keppel|Mrs George Keppel]] (lady, t[i]me of Louis XVI.), Malcolm Lyon, Mrs Herbert Dormer (costume, Louis XV.), Mrs Frank Parker, and Cecil Freemantle (Court dress of the period).

BARONS — Macar (Court dress of the period) and Schimmelpennick Van der Oye (Court dress).

BARONESS — Schimmelpennick Van der Oye (costume Louis XVI.)

GENERAL — Arbuthnot (Court dress of the period).

COLONELS — Paulet and Mildmay Willson, C.B. (Scots Guards).

MAJORS — Armstrong (modern Court dress), Norris Fosbery (Mousquetaire), and Alston.

CAPTAINS — Molesworth (Mousquetaire), J. Barry (costume, Louis XVI.), Somerset (Mousquetaire), Brinkley (Court costume of the period), East (Mousquetaire), Granville (Mousquetaire), Cowan, Lafone, Keighly-Peach, Bruce Hamilton, H. Welman (Court dress), Grant, Towers Clark (Court dress), Allfrey (Court dress, Louis XV.), and Oxley, 60th Rifles.

MESDAMES — Armstrong (French Marquise), Gerald Arbuthnot (Court Dame), Armitage (Dame de la temps Louis XV.), H. Allfrey (Marquise, temps Louis XV.), Brinkley (Court Dame, Louis XV.), Frank Bibby (Lady of the Court of Louis XVI.), Everard Browne (Court Dame), Beech, Aubrey Cartwright, Chamberlayne (Court dress), Cowan, Cove-Jones, H. Chamberlain, George Cartland, Cartwright (Court Dame), J. S. Dugdale (Court lady), Blanche Drummond, Lindsay Eric-Smith (Pompadour costume), Fosbery, Wilson Fitzgerald, Fairfax-Lucy (Marquise Louis XV.), Granville, Graham, Gaskell (Grande Dame), Hulton (Court costume), Harvey Drummond, Irwin, Joliffe (Watteau), Edward Lucas (Lady of Court Louis XVI.), Morton P. Lucas (Court Lady, Louis XV.), E. Little, Leslie (of Balquhain), Lakin (Madame Roland), W. M. Low, Leslie (a la Watteau), Beresford Melville (Dame de la Court), J. Menzies (Duchess d’Angoulêne), Molesworth (costume Louis XVI.), Basil Montgomery (Marquise), Miller-Munday [sic] (Marie Therese, Queen of the Sicilies), Robert O. Milne (Dame de la Louis XVI.), Norris, Osborne (Madame de Pompadour), Arthur Paget (Duchess d’Orleans), Paulet, Ramsden (Madame de Colonne), Arthur Somerset (Shepherdess a la Watteau), Smythe, L. Gay Scott, Beauchamp Scott (Lady of time of Louis XVI.), Shaw, Sanders, Fred Shaw, S. C. Smith (Marquise temps Louis XVI.), Tree (Lady Louis XVI. period), Thursby-Pelham (Court dress), Tower (Duchess de Polignac), Towers-Clark (Lady of the Court of Louis XV.)[,] Francis Williams, Wheatley (Lady of time of Louis XV.), West [(]Court costume Louis XV.), and Francis Williams (Louis XV. costume).

MADEMOISELLES—Allfrey, Armstrong (a laWatteau), Bromley Davenport, N. Booker (English lady of the Court of King George III.), Booker (Lady of time of Louis XV.), Muriel Bell, Nora Battye, Decapell Brooke, Spender Clay (Mdme Lamballe), Carleton (Watteau, Louis XV.), Chetwode, Anna Cassel, Carruthers (costume of the period Louis XV.), Champion, Hugh Drummond (Court costume), Constance Dormer (costume, Louis XV.), Beatrice Fitzherbert (Court Dame), Lucy (Mademoiselle de Montmirail), Granville, Gaskell (costume, Louis XV.), Hodgson (a la Watteau), Gladys Hankey (Marquese, reign Louis XV.), Irwin, Keighly-Peach akin (Dame de la Cours, Louis XV.), Lakin (a la Watteau), Lister-Kaye, Violet Leigh (Mdlle. de Chévreuse), Murray, Miller-Mundy (Court dress), J. Menzies (Duchess d’ Angoulêne), Naylor (Lady of the Court of Louis XV.), Nicol, Osborne (Mdme. de Pompadour, in garden dress), Perry, Constance Peel, Ramsden (Mdlle. de Colonne), E. N. Ramsden (Mdlle. de Coloane), Rushton (Lady of the Court of Louis XV), C. Starkey, Cicely Dudley Smith (Court dress, Louis XVI.), May Sanders (Louis XVI. costume), Cornwallis West (Mdlle. de la Court), Muriel Wilson (English costume of the period, Louis XV. and XVI.), Fleetwood Wilson (Lady, time Louis XVI.), and Waller (Fille de la Comtesse d’Artoix).

MESSIEURS — G. A. Arbuthnot (modern Court dress), W. C. Alston (Infanterie Regiment de Forés), Allfrey, J. Arkwright, W. Armstrong, J. P. Arkwright, Robertson Aikman, Frank Bibby, Bromley-Davenport, Brinckman, P. B. Vander Byl (Mousquetaire), Beaumont, 60th Rifles, L. Bethell (Mousquetaire), Bainbridge, A. E. Batchelor (Garde de la Porte), Everard Browne (gentleman, temps Louis XV.), R. Barnes, Battye, F. C. Hunter Blair (Mousquetaire), Beech (Garde au corps du Roi), C. B. Clutterbuck (Mousequetaire), Cassel (modern Court dress), Collings (modern court dress), Felix Cassel, Caryl Craven (military uniform of the period), Aubrey Cartwright, Chamberlayne, Bertram Chaplin, Cove-Jones, E. S. Chattock, H. Chamberlain, Drummond Chaplin (Court dress), G. Cartland, J. S. Dugdale (Recorder), M. Farquahar (Mousquetaire), Cyril Foley (officer du corps du Roi Pologne Stanilas), Kenneth Foster (Mousquetaire), S. M. Fraser (Mousquetaire), Fairfax-Lucy (Colonel George Lucy), J. S. Forbes (Mousquetaire), B. J. Fitzgerald (Mousquetaire), Francis Fitz-Herbert (Fusilier du Roi), J. B. Fitz-Herbert (gentleman, temps Louis XV.), Wilson Fitzgerald, R. Flower (modern Court dress), Francis, Flower (modern Court dress), Granville, J, Grenfell, Graham, G. de J. Hamilton (Mousequetaire), E. Harrington (Mousquetaire), Hutton, Head, H. T. Hickman (Court dress), Percival Hodgson (Court dress), Irwin, Joliffe, Joostens (Diplomatic Court dress), M. T. Kennard (Maison du Roi), F. Laycock (officer of Pondicherry Regiment), Morton P. Lucas (gentleman of the period), R. W. Lindsay (Court dress, late 18th century), E. Little, Lister-Kay, Lakin (modern Court dress), R. Lakin (the Duc de Brissac), Richard Lant (modern court dress), John Lant (M. Vauthier), H. G. Lakin (the Marquis de Breze), W. M. Low (David Garrick), Meyrick, Murray (Mousquetaire), C. de Murietta (Marechal Saxe), J. Moncrieffe (gentleman of the period), H. Mordaunt (gentleman, temps. Louis XV.), T. J. Meyrick (gentleman, temps. XV.), F. Menzies, H. Molesworth, Basil Montgomery (Courtier, Louis XV.), John Monckton, R. O. Milne (Chevan-le’ ger de la Garde du Roi, Louis XVI.), H. du C. Norris (Court dress, Louis XV.), J. Norris (Marquis of France), Norton (Louis XVI. costume), C. S. Paulet (modern Court dress), Quinton-Dick (Mousquetaire), Arthur Paget, Ralph Paget (dress of Louis XV.), Oswald Petre (modern Court dress), George Peel, W. R. W. Peel, G. R. Powell (Court dress), Mark J. Paget (gentleman of period of Louis XV.), Ramsden (Mons. de Calonne), L. G. Scott (Mousquetaire), H. Spender Clay (Court dress of the period), Smythe, Shaw, S. Sanders (Mousquetaire), F. Shaw, S. O. Smith (modern court dress), M. Oswald Smith (gentleman of the Court of Louis XVI.), Cameron Skinner, W. L. Thursby (Mousequetaire), C. J. H. Tower (officier Gardes Suisses), Tree, Thursby-Pelham, Tower (officier Gardes Suisses), J. H. Wheatley (modern Court dress), Read Walker (officier d'Infanterie), Francis Williams, Montague Wood, West, Gordon Wood and Anthony White.

DESCRIPTION OF COSTUMES.

Appended are descriptions of the chief costumes worn:—

EARL OF WARWICK.

Field Marshal, Louis XVI. — Military coat with the long skirts of the period, having turned-back revers of white cloth, laced, after the military fashion, with gold, white knee-breeches and silk stockings. The cravat and ruffles were of lace. A white wig in Louis XVI. style, and a three-cornered black beaver hat with gold braid all round the brim, which was edged with small white ostrich plumes, completed a handsome and artistic costume. A sword was worn in a swordbelt of the period.

COUNTESS OF WARWICK.

Marie Antoinette — Gala costume. Rich brocade dress, with a ground of a delicate tint of pearl, with a suggestion of pink in it, the design roses in gold, with gold foliage, lilies in white, some small blue flowers and clusters of pink blossoms, with bright old- world green as foliage. The skirt was quite plain, and the [b/c] bodice drawn into shaped points at the hips, so that it sat right out at either side. It was full, and yard on the ground at the back, The bodice was finished with points back and front, and was cut with absolute perfection. Round the shoulders were full soft folds of gold-flecked French silk muslin edged with beautiful gold lace. The sleeves were plain and tight to the elbow, whence they were finished with triple frills of the gold-flecked muslin, each bordered with gold lace, and with ruffles falling from beneath the frills of point d’Alençon lace. The frills were headed with bands of gold embroidery. At the back, suspended from both shoulders by gold cords, was a beautiful Court mantle of deep rich blue velvet, not so pale as turquoise nor so strong as the shade we call Royal, but a bright lovely colour. This was embroidered all over with a raised design of fleur de lys in dull and burnished gold, and was lined with the same blue velvet. The hair was dressed high with a magnificently embroidered head-dress. Her ladyship wore the Warwick family diamonds round her neck as a collar, a turquoise velvet cap clasped with jewels on her white coiffure and a bandeau of family jewels under her cap. Her court mantle was fastened at the shoulders with a tiara of diamonds widened out so as to clasp the cloak from shoulder to shoulder.

THE COMTESSSE AHLEFELDT.

Dame de Ia temps Louis XVI. — cream silk petticoat, with front of real ha[n]d-worked silk embroidery, done in the time of Louis XVI., the design being convolvulus and other flowers wrought in dull pink, blue, and green silks, and feathers tied with true lover’s knots. There was a bright shell pink tunic-shaped overdress, with Watteau back, edged with real Brussels, white silk stomacher, large pink bows in front and on each hip; and wreath of pink roses.

MRS GERALD ARBUTHNOT.

Brocade gown with Watteau back and paniers, cerise satin petticoat, studded with large blue satin bows, cerise velvet stomacher, fechu of Brussels lace; head-dress, cap of cerise velvet and blue plumes.

MRS ARMITAGE.

Dame de la temps Louis XV. — Light grey satin dress brocaded with bunches of cyclamen, roses with green leaves, and ornamented with velvet to match, and groups of yellow and cyclamen roses.

MRS ARMITAGE (KIRROUGHTEN).

Lady of the Court of Louis XVI. — Bodice and short train, with Watteau plait of pale heliotrope and green brocade, with large revers of heliotrope satin, and bodice trimmed with petunia velvet and petunia and yellow roses and lace; petticoat of heliotrope satin and lace flounces, diamond and sapphire ornaments; hair ''poudré'', with heliotrope feathers.

MISS ARMSTRONG.

''A la'' Watteau. — White watered silk, brocaded with stripes and clusters of roses. The front of the petticoat was draped with blue chiffon, and edged at the bottom with pink roses, bodice with blue satin bows in front, and on the shoulders and neck, and sleeves trimmed with full white chiffon and pink roses, powdered hair, wreath of roses and blue bow.

MRS ARMSTRONG. French Marquise. — Handsome cream coloured real old brocade with black velvet front, and trimmings of very old point de Venice, and festoons of pink roses.

MRS EVERARD BROWNE.

Brocaded satin with silver stripes.

MRS FRANK BIBBY.

Lady of the Court of Louis XVI. — White satin dress, with skirt draped with old lace, pink chiffon sash embroidered with silver; diamond buttons on corsage.

CAPTAIN BRINKLEY (WARWICK).

Court dress of the period — Claret-coloured coat and knee-breeches, white silk embroidered waistcoat, white silk stockings and old paste buckled shoes, Louis XVI. wig, and Court sword.

MRS BRINKLEY.

Marquise du Deffant — Train of white brocade embroidered in roses and forget-me-nots, the paniers lined and turned back with green satin and guipure; petticoat of pink satin and old Honiton lace, trimmed with pink roses; Louis XV. wig, with roses and diamonds.

THE EARL OF BURFORD.

Military costume of Louis XV.’s time — White cloth with pale blue facings, trimmed handsomely and effectively with gold.

MISS MURIEL BELL.

Princesse de Lamballe — Pink satin brocade, white petticoat, pink roses; hair ''poudré''.

MISS N. BATTYE (LONDON).

English dress of the period — Light blue satin dress[,] lace fichu, large black velvet hat with white ostrich feathers.

MISS DE CAPELL BROOKE.

Lady of the time of Louis XV. — Pink figured silk, over white satin skirt, edged with gold gimp, Watteau back; hair powdered.

MR BEECH.

Garde du Corps du Roi — Crimson coat, white facings, and gold lace.

MRS BEECH.

Madame de La Fayette — Old brocade, with lace and crimson roses and black velvet bows.

MR F. C. HUNTER BLAIR (LEAMINGTON).

Mousquetaire Uniform Louis XV., in white, scarlet, and gold.

CAPTAIN JIM BARRY (LONDON).

Mousquetaire Louis XV. — Black knee-breeches, light blue coat and waistcoat, faced with white, and trimmed with gold lace.

MR CARYL CRAVEN.

Mousquetaire, Louis XVI. — White and gold.

THE EARL OF CHESTERFIELD.

Court costume — Coat of pale blue corded silk, the cuffs, pocket flaps, and fronts all richly wrought with gold, while the buttons were old paste and amethysts. The knee-breeches were blue silk, and the blue silk stockings were clocked with gold, and Court shoes were worn, with diamond buckles, The waistcoat was of yellow satin, brocaded with pink rosebuds, and having old paste and amethyst buttons. A jabot of old lace was pinned with a diamond brooch, and the ruffles were of similar lace. A white wig was worn with a three-cornered gold-laced and white-plumed hat.

COUNTESS CAIRNS.

Duchess de Bouillon — Dress of light-hued satin, with relief of pink diamonds, and pink roses in the hair.

MR CHAPLIN.

Court suit of green silk velvet, with embroidered vest and white wig.

LORD CHURCHILL.

Court costume — Blue brocade, with steel buttons and knot of ribbon, fringed with silver on one shoulder; white satin waistcoat and blue knee-breeches.

LORD CLIFTON.

Officer of the Guards, Louis XV. — Coat of pale green cloth, turned back with crimson, and laced with gold.

MISS SPENDER CLAY.

Madme. Lamballe — A pretty pink and white brocade dress in the style of Louis XV., and with large hat, trimmed with ostrich plumes and roses. She carried a white wand surmounted by roses.

MISS CARLETON.

Watteau, Louis XV. — Blue silk brocaded dress, with little pink roses, and pink satin petticoat with Watteau pleat, hair arranged with pink wreath of roses and pink feathers.

MRS CARTWRIGHT.

White satin dress, trimmed with sable and point de gaze lace; musseline de soie fichu edged with lace, and caught up with clusters of pink roses.

MISS CHAMPION (NORFOLK).

Dress of old brocade; petticoat of Rose de Barri satin, trimmed with pearls and lace; lace fichu, large rose hat with plumes, and pearl ornaments.

MRS CHAMBERLAYNE (STONEY THORPE).

Marquise Louis XV. — Pink satin petticoat, yellow flowered silk bodice, and train from the shoulders; pink satin ribbon and diamonds in the powdered hair.

MISS CARUTHERS (WARDINGTON, BANBURY).

Short-waisted dress of period Louis XV.; yellow brocade over yellow satin petticoat; old lace and roses.

MR BERTRAM CHAPLIN.

Period Louis XVI. — White satin coat and breeches, pink satin waistcoat.

MR QUENTIN DICK.

Officer of the Household of Louis XV. — White cloth uniform, faced with blue, and braided with gold.

MISS DRUMMOND (SHERBOURNE HOUSE).

A blue brioche, brocaded with pink roses and leaves, and gaily trimmed with pink Banksia roses, petticoat of white satin, flounced with lovely Brussels lace, bodice [sic] of bioche silk, with white front trimmed with roses and old-fashioned gauze, necklace of roses and pearls, and wreath of roses.

HON. MRS HUBERT DORMER (LONDON).

Court dress Louis XVI. — Petticoat of pink satin, point lace flounce; overdress of dark red satin, in paniers, looped with red and pink roses, diamond and pearl ornaments.

MR J. S. DUGDALE, Q.C.

Recorder’s Court dress of the period — Black silk gown with lace ruffles, black silk stockings, buckle shoes, and full bottomed wig.

MRS J. S. DUGDALE.

A very handsome bright blue silk brocaded with white, and made ''à la'' Pompadour, with white satin front trimmed with dark fur, the bodice made with pearl trimmings, and a white muslin fichu tied at one side under a bunch of pink roses, hair dressed with blue feathers, wreath of pink roses, and a tiara of diamonds in front.

[Col. c/d]

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ONSTANCE&lt;/small&gt; D&lt;small&gt;ORMER&lt;/small&gt; (H&lt;small&gt;ASTINGS&lt;/small&gt;.)

Marquise Louis XVI. — Gown of white silk brocaded with roses, Watteau back. Pearl ornaments.

L&lt;small&gt;ADY&lt;/small&gt; A&lt;small&gt;NGELA&lt;/small&gt; S&lt;small&gt;T&lt;/small&gt;. C&lt;small&gt;LAIR&lt;/small&gt; E&lt;small&gt;RSKINE&lt;/small&gt;.

Lady Mary Campbell — White muslin costume, with broad blue silk sash.

S&lt;small&gt;IR&lt;/small&gt; A&lt;small&gt;RCHIBALD&lt;/small&gt; E&lt;small&gt;DMONSTONE&lt;/small&gt;.

Mousquetaire — White cloth uniform, faced with blue and showing a blue waistcoat, the whole having a large amount of silver military braiding.

M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;INDSAY&lt;/small&gt; E&lt;small&gt;RIC&lt;/small&gt;-S&lt;small&gt;MITH&lt;/small&gt; (E&lt;small&gt;LFINSWARD&lt;/small&gt;, H&lt;small&gt;AYWARD'S&lt;/small&gt; H&lt;small&gt;EATH&lt;/small&gt;. [sic no paren]

Pompadour dress, period Louis XVI. — Yellow brocade, and white satin petticoat.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;AIRFAX&lt;/small&gt;-L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt;).

A Marquise. — Rich white brocade dress, with blue and straw brocade saque, edged with Brussels lace, and Brussels lace flounce, Vandycked round, petticoat with pink roses, lace ruffles and fichu, and pink roses and diamonds in the hair completed the costume.

M&lt;small&gt;R&lt;/small&gt; F&lt;small&gt;AIRFAX&lt;/small&gt;-L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt;).

Colonel George Lucy — Red lilac-coloured cloth suit, Court dress of the period, edged with silver lace, and belonged to Colonel G. Lucy in 1744; silk stockings of the same colour, high-heeled shoes, with diamond buckles, and knee buckles, lace ruffles, and cravat.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;ASIL&lt;/small&gt; J. F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Mousquetaire — Uniform of dark green cloth, faced with tan, and trimmed with silver, old point d’Alencon ruffles, tan silk sash, and cross belt of tan and silver.

M&lt;small&gt;R&lt;/small&gt; B. F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Mousquetaire — White uniform, with orange velvet facings braided with gold, crossbelt of white and gold, a yellow sash, and the high black leather boots of the period.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;-F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Dress of the real old brocade of Louis XV.’s reign.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. C&lt;small&gt;LEMENT&lt;/small&gt; F&lt;small&gt;INCH&lt;/small&gt;.

Gentleman of the period Louis XV. — Coat of blue watered silk, with silver trimming, satin breeches to match, white satin vest, and black hat decked with silver.

M&lt;small&gt;ISS&lt;/small&gt; B&lt;small&gt;EATRICE&lt;/small&gt; F&lt;small&gt;ITZHERBERT&lt;/small&gt;.

Dress of pale blue satin, lined with pink, with pink roses on the corsage, Louis XVI. period. She wore a beautiful pearl necklace, with large diamond pendant, and two diamond sprays, all of which were given by George IV. to Mrs Fitzberbert.

M&lt;small&gt;AJOR&lt;/small&gt; F&lt;small&gt;OSBERY&lt;/small&gt; (W&lt;small&gt;ARWICK&lt;/small&gt;).

Mousquetaire — Claret-coloured tunic, with salmon-colour cuffs, lace ruffles, &amp;c.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;OSBERY&lt;/small&gt; (W&lt;small&gt;ARWICK&lt;/small&gt;).

Marquise — Pink satin petticoat, covered with lace, grey and pink brocaded bodice and train, pink roses and ostrich plume in powdered hair.

L&lt;small&gt;ADY&lt;/small&gt; G&lt;small&gt;ERARD&lt;/small&gt;.

Duchess de Pognac — Dress of pale blue brocade, decked with small roses, with front of pink satin; fichu of muslin and lace, and stomacher of lace and roses.

L&lt;small&gt;ADY&lt;/small&gt; E&lt;small&gt;VA&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Polichinelle, Louis XV. — White satin gown ornamented at the bottom with a trelliswork of silver, studded with small pink roses; corsage to correspond, and fastened across the stomacher by large diamond hooks and eyes.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. S&lt;small&gt;IDNEY&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Officer of the Regiment of Swiss Guards — Coat of pale blue cloth, nearly bordering on green — quite a turquoise shade. The revers were white, and turned back from a white waistcoat braided with gold. The braiding was continued down the white revers of the coat and on the skirts; white satin knee-breeches, silk stockings, Court shoes, white wig, and three-cornered hat, trimmed with gold braid and white ostrich feathers, completed one of the most effective of military attires. A sword was, of course, worn.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. L&lt;small&gt;OUIS&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Mousquetaire, Louis XV. — Claret-coloured coat, laced with gold over white; a white silk sash, sword-belt of red cloth with gold, white knee-breeches, Court shoes, silk stockings, and the wig and three-cornered hat of the time.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;OUIS&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Dress of the period — Petticoat of deep rose-petal pink satin, with a full flounce of white lace headed by trails of roses; over-dress of white satin, brocaded with a design of roses and lined with pale-green satin, pointed bodice showing a pink vest laced across, and ruffles and fichu of Mechlin lace to correspond with the flounce. The hair was powdered and dressed high, with an ornament of roses and diamonds at one side.

C&lt;small&gt;APTAIN THE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. A&lt;small&gt;LWYN&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Mousquetaire — Coat of scarlet cloth, cuffs and fronts turned back with white and laced with gold, and broad red silk sash, white knee-breeches, silk stockings, and Court shoes.

THE H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;LWYN&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Dame de la Court Louis XV. — Over-dress of pink mirror velvet bordered with dark fur, opening over a front of cream satin, long pink velvet sleeves with roses and fichu of fine old lace on the corsage; hair dressed ''a la'' Princess Lambale.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;ARKELL&lt;/small&gt;.

Grande Dame — Blue shot-satin dress adorned with point, d’Alençon lace, veiled with silver tissue under white gauze and tied up by a wide blue chiffon sash caught at the arm-holes with diamond buttons; ornaments, enamelled medallions set in diamonds.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;ASKELL&lt;/small&gt;.

Gainsborough costume — White satin and blue chiffon, Louis XVI.; old diamond necklace.

M&lt;small&gt;ISS&lt;/small&gt; G&lt;small&gt;ASKELL&lt;/small&gt;.

Costume, Louis XV. — Blue and pink costume of that period, with a very large black velvet hat, trimmed with blue feathers.

M&lt;small&gt;ISS&lt;/small&gt; G&lt;small&gt;ORDON&lt;/small&gt;.

All in white, lined with blue satin, the front of the bodice made of fine muslin, caught up with small pink roses; and a little wreath of pink roses in the hair, and diamonds.

M&lt;small&gt;ARCHIONESS OF&lt;/small&gt; H&lt;small&gt;ERTFORD&lt;/small&gt;.

Lady time of Louis XVI. — Black velvet dress and train, white satin front covered with old point lace; long pointed bodice with lace fichu, long velvet sleeves lined with white satin, front of dress covered with diamonds. White full-dress wig, with lace lappets and diamonds.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;RNEST&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt; (G&lt;small&gt;ROVE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;.)

Marquise Louis XVI. — Overdress of light green satin brocaded with pink roses and faced with pale pink satin over white satin petticoat, with lace flounce beaded with pink roses. Pearl and diamond ornaments.

E&lt;small&gt;RNEST&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt; (G&lt;small&gt;ROVE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

English Court dress — Black velvet, point lace ruffles.

M&lt;small&gt;R&lt;/small&gt; H. T. H&lt;small&gt;ICKMAN&lt;/small&gt;.

Court dress, time of Louis XV. — Black velvet coat, knee-breeches, trimmed with white lace.

M&lt;small&gt;RS&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt;.

Court costume — Dress with paniers of pale-green brocade over a white satin petticoat having a flounce of lace headed by roses.

M&lt;small&gt;ISS&lt;/small&gt; H&lt;small&gt;ODGSON&lt;/small&gt;.

''A la'' Watteau — Sang de boeuf coloured satin petticoat, trimmed with old lace, caught up with roses; a corset and polonaise of rose figured satin, the latter trimmed with deep revers of green satin; white wig; ornaments, pearls and diamonds.

M. J&lt;small&gt;OOSTENS&lt;/small&gt; (B&lt;small&gt;ELGIAN&lt;/small&gt; L&lt;small&gt;EGATION&lt;/small&gt;, L&lt;small&gt;ONDON&lt;/small&gt;).

Courtier, Louis XVI. — White satin knee breeches, claret velvet coat and waistcoat, point lace ruffles.

M&lt;small&gt;RS&lt;/small&gt; H. J&lt;small&gt;OLIFFE&lt;/small&gt; (G&lt;small&gt;OLDICOTE&lt;/small&gt;).

Marquise of Louis XV. — Blue silk brocaded dress with pink roses, the petticoat of pink satin trimmed with white lace and pink roses, and the over-dress turned back with green satin edged with gold embroidery.

L&lt;small&gt;ORD&lt;/small&gt; K&lt;small&gt;ENYON&lt;/small&gt;.

Officier of the Regiment du Roi, Louis XVI. — Handsome dress of white cloth faced with pale-blue and laced with gold.

T&lt;small&gt;HE&lt;/small&gt; [[Social Victorians/People/Keppel|H&lt;small&gt;ON&lt;/small&gt;. G&lt;small&gt;EORGE&lt;/small&gt; K&lt;small&gt;EPPEL&lt;/small&gt;]] (2, Wilton Crescent, London).

Mousequetaire — White cloth, with an exquisitely jewelled Order around his neck.

T&lt;small&gt;HE&lt;/small&gt; [[Social Victorians/People/Keppel|H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;EORGE&lt;/small&gt; K&lt;small&gt;EPPEL&lt;/small&gt;]].

Lady, time Louis XVI. — Gown of shell pink satin, pointed bodice, with full paniers, of antique brocade of the real deep rose shade known as du Barri sewn with silver thread and bouquets of roses. Full petticoat, of dull creamy-tinted satin, with a deep band round it of silver tissue embroidered with garlands of small leafless roses. The sleeves had long ruffles of old lace. The hair was powdered and dressed elaborately and high, with three rose du Barri feathers in it and a little cap of lace. The shoes were of pink satin, with diamond buckles.

T&lt;small&gt;HE&lt;/small&gt; E&lt;small&gt;ARL OF&lt;/small&gt; L&lt;small&gt;ONSDALE&lt;/small&gt;.

M. de Capuisan, Keeper of the Royal Stud, Louis XV. — Coat and knee-breeches were of ruby velvet, richly wrought with gold and with rare and valuable paste buttons on the former, while the vest was of pearl-white satin edged with very beautiful embroidery, white silk stockings, Court shoes with diamond buckles, lace ruffles and jabot with diamond brooch, jewelled hilted Court sword, and white wig with three-cornered hat with gold lace and white plumes.

L&lt;small&gt;ORD&lt;/small&gt; L&lt;small&gt;OVAT&lt;/small&gt;.

Comte d’Artagnas — Military costume of the period in white, faced with blue and laced with gold; an embroidered pouch slung from his belt, embroidered in gold, silk stockings, Court shoes, white wig, sword, and three-cornered hat.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

Madenoiselle de Montmirail — White satin petticoat, with deep flounce of Brussels lace, caught up with pompom; pink roses; witite brocade saque, laced with pearls; lace ruffles and fichu; large black velvet hat and plumes.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;AKIN&lt;/small&gt;.

Watteau costume — White satin brocade with white satin petticoat, festooned with roses.

[Col. 4–5]

T&lt;small&gt;HE&lt;/small&gt; P&lt;small&gt;ORTUGUESE&lt;/small&gt; M&lt;small&gt;INISTER&lt;/small&gt; (D&lt;small&gt;ON&lt;/small&gt; L&lt;small&gt;OUIE DE&lt;/small&gt; L&lt;small&gt;OUVERAL&lt;/small&gt; [Soveral]).

Mousquetaire of the 2nd Company of the Royal Household, Louis XV. — Scarlet, laced with gold and relieved with white, high black Mousquetaire boots, a plastron [sic] embroidered with the Royal arms, white wig, three-cornered hat gold-laced and white-plumed, sword.

M&lt;small&gt;R&lt;/small&gt; R&lt;small&gt;ICHARD&lt;/small&gt; L&lt;small&gt;ANT&lt;/small&gt; (N&lt;small&gt;AILCOTE&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;, C&lt;small&gt;OVENTRY&lt;/small&gt;).

Present day Court dress, with Louis XV. white wig.

H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;HANDOS&lt;/small&gt; L&lt;small&gt;EIGH&lt;/small&gt;.

The Duchesse de Polignac, period Louis XVI. — Petticoat of pale pink brocade, with corsage and train of sapphire blue velvet and lace fichu.

M&lt;small&gt;ISS&lt;/small&gt; V&lt;small&gt;IOLET&lt;/small&gt; L&lt;small&gt;EIGH&lt;/small&gt;.

Mdlle. de Chévreuse, period Louis XV. — Petticoat of white satin, with lattice work of pink roses, corsage with paniers and Watteau plait of sxy blue satin, lined with pale pink satin; powdered hair, with small wreath of roses, pearls, and white plume.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;RNEST&lt;/small&gt; L&lt;small&gt;ITTLE&lt;/small&gt; (&lt;small&gt;OF&lt;/small&gt; N&lt;small&gt;EWBOLD&lt;/small&gt; P&lt;small&gt;ACEY&lt;/small&gt;).

Lady of the reign of Louis XV. — Train of brocade, in white and purple, over dress of satin, trimmed with old lace and pink roses.

M&lt;small&gt;R&lt;/small&gt; L&lt;small&gt;INDSAY&lt;/small&gt; (R&lt;small&gt;ED&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;, B&lt;small&gt;ARFORD&lt;/small&gt;).

Court dress late 18th century, composed of black velvet.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;DWARD&lt;/small&gt; L&lt;small&gt;UCAS&lt;/small&gt; (15, L&lt;small&gt;ENNOX&lt;/small&gt; G&lt;small&gt;ARDENS&lt;/small&gt;, L&lt;small&gt;ONDON&lt;/small&gt;)

Lady of Court Louis XVI. — Pink brocade, with green satin petticoat.

M&lt;small&gt;RS&lt;/small&gt; M&lt;small&gt;ORTON&lt;/small&gt; P. L&lt;small&gt;UCAS&lt;/small&gt; (T&lt;small&gt;HE&lt;/small&gt; O&lt;small&gt;AKS&lt;/small&gt;).

Court lady, Louis XV. — Black velvet bodice and train; and white satin petticoat trimmed Brussels lace and roses.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;ORTON&lt;/small&gt; P. L&lt;small&gt;UCAS&lt;/small&gt;.

Gentleman of the period — Black velvet Court dress, trimmed with steel, white satin waistcoat, and knee- breeches.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt;-K&lt;small&gt;AYE&lt;/small&gt;.

Period Louis XVI. — Blue silk brocade, white petticoat, pink roses.

M&lt;small&gt;R&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt;-K&lt;small&gt;AYE&lt;/small&gt;.

Period Louis XVI. — Plum-coloured velvet coat and breeches, brocaded satin waistcoat.

M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;ESLIE AND&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; E. L&lt;small&gt;YON&lt;/small&gt;.

A la Watteau — Dresses of white and gold flowered brioche, with plain white satin fronts handsomely painted, the design being large, full blown pink roses and butterflies; hip panniers, and the bodices were of gold brioche, with white satin fronts trimmed with lace, large pearls, and Cairngorm jewels, neck ruffles edged with pearls and gold, and aigrettes and velvet bow of pink and gold.

C&lt;small&gt;APTAIN THE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. H&lt;small&gt;EDWORTH&lt;/small&gt; L&lt;small&gt;AMBTON&lt;/small&gt;.

Courtier of Louis XVI. — Coat of bronze satin, richly-embroidered knee-breeches, and richly-embroidered waistcoat of pearl white corded silk, lace ruffles and jabot, and all the details to correspond.

D&lt;small&gt;UKE&lt;/small&gt; &lt;small&gt;OF&lt;/small&gt; M&lt;small&gt;ANCHESTER&lt;/small&gt;.

Marquis de Grammont — White satin knee-breeches, white silk stockings, shoes with paste buckles, a coat of real old Louis XVI. brocade, with a design of feathers in gold on a cream-coloured background and of pink rose sprays, the fronts, pockets, and cuffs all richly wrought in gold, and with fine old paste buttons. He had also an exquisite real lace jabot, fastened with an antique diamond brooch, a white satin waistcoat finely emboidered, white wig, and black three-cornered hat.

S&lt;small&gt;IR&lt;/small&gt; C. M&lt;small&gt;ORDAUNT&lt;/small&gt;, B&lt;small&gt;ART&lt;/small&gt;.

Gentleman of the period — Coat and knee-breeches of black velvet, with waistcoat of black broché, all three trimmed with cut steel buttons; lace ruffles and necktie; white wig, three-cornered hat, gold-headed cane finished with red and green ribbons; black silk stockings, and shoes with silver buckles.

L&lt;small&gt;ADY&lt;/small&gt; M&lt;small&gt;ORDAUNT&lt;/small&gt; (W&lt;small&gt;ALTON&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Princess de Lamballe — Bodice and train of rich white brocade, trimmed with lace and wreaths of small pink roses, the train being edged with them all round; over pink satin petticoat, ornamented with bows of lace, lace flounce, and vandykes of pink roses. Hair powdered, over a cushion with curls in the neck, with wreath of pink roses, and loops of pink ribbon: White satin shoes, with pink heels and pink rosettes. Row of pearls round neck.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;ASIL&lt;/small&gt; M&lt;small&gt;ONTGOMERY&lt;/small&gt;.

Mousquetaire — Uniform of white cloth and Royal blue velvet, embroidered in gold.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;ONCRIEFFE&lt;/small&gt;.

Gentleman of the period — Coat of pale ''vieux'' rose brocade trimmed with silver, satin breeches to match, sea-green satin waistcoat lightly worked over in silver, ruffles and jabot of white lace.

M&lt;small&gt;R&lt;/small&gt; R&lt;small&gt;OBERT&lt;/small&gt; O. M&lt;small&gt;ILNE&lt;/small&gt;.

Chevau léger de la Garde de Roi, Louis XVI. — Scarlet cloth coat, faced with gold lace, knee-breeches, three-cornered hat with ostrich feathers, large Hessian boots.

M&lt;small&gt;R&lt;/small&gt; J. M&lt;small&gt;ONCRIEFFE&lt;/small&gt;.

Courtier Louis XVI. — Coat of yellow brocade, heliotrope knee-breeches, and gold laced heliotrope waistcoat. The coat had paste buttons, and the details as to wig, sword, ruffles, shoes, stockings, and three-cornered hat, were all correctly carried out.

M&lt;small&gt;R&lt;/small&gt; H. M&lt;small&gt;ORDAUNT&lt;/small&gt; (W&lt;small&gt;ALTON&lt;/small&gt;).

Courtier of period Louis XV. — Coat of mauve brocade, full skirted and embroidered with gold, the waistcoat of mauve satin embroidered in floral design, knee breeches of shot manve and gold, silk stockings, Court shoes, powdered hair, lace ruffles, jabot and sword made up a costume, correct in every particular, of one of Louis XVI.’s courtiers.

M&lt;small&gt;RS&lt;/small&gt; H&lt;small&gt;ERBERT&lt;/small&gt; M&lt;small&gt;OLESWORTH&lt;/small&gt; (D&lt;small&gt;EVONPORT&lt;/small&gt;, D&lt;small&gt;EVON&lt;/small&gt;).

Court dress of Louis XVI. — Yellow satin gown in paniers, trimmed with light blue velvet and pink roses. Diamond ornaments.

C&lt;small&gt;APTAIN&lt;/small&gt; H&lt;small&gt;ERBERT&lt;/small&gt; M&lt;small&gt;OLESWORTH&lt;/small&gt; (F&lt;small&gt;IELD&lt;/small&gt; A&lt;small&gt;RTILLERY&lt;/small&gt;, D&lt;small&gt;EVONPORT&lt;/small&gt;, D&lt;small&gt;EVON&lt;/small&gt;).

Mousquetaire, Louis XV. — Claret velvet costume, slashed with pink satin.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;URRAY&lt;/small&gt; (D&lt;small&gt;IDDINGTON&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Short eau-de-nil satin skirt, trimmed band of passementarie, bodice and paniers of pink and white broché, looped up with pink satin bows; trimmed ruching of mousselaine-de-soie over pink satin; white wig; pink roses; ornaments, pearls and diamonds.

L&lt;small&gt;ADY&lt;/small&gt; A&lt;small&gt;NN&lt;/small&gt; M&lt;small&gt;URRAY&lt;/small&gt;.

Madam de Pompadour (after La Tour). — Dress was of white and gold, brocaded with large sprays of coloured flowers trimmed with ruching of vieux rose satin and lace; white wig; lace cap, with lappets and bow; ornaments, pearls and emeralds.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;ILLER&lt;/small&gt;-M&lt;small&gt;UNDAY&lt;/small&gt; [sic].

Pink dress with fichu.

M&lt;small&gt;RS&lt;/small&gt; M&lt;small&gt;ILLER&lt;/small&gt;-M&lt;small&gt;UNDAY&lt;/small&gt; [sic] (Shipley Hall, Derby).

Court dress — Petticoat of yellow satin, draped with white silk muslin edged with little pink roses; over-bodice low and double-breasted, of blue velvet with miniature and paste buttons, and revers faced with pale-hued brocade and finished with lace; headdress of Marie Theresè, Queen the Sicilies, a silk handkerchief tied in a knot at one side, very like the portrait of Madame le Brun in the Louvre Gallery done by herself; also plumes and jewels.

M&lt;small&gt;RS&lt;/small&gt; J. M&lt;small&gt;ENZIES&lt;/small&gt;.

Duchess d’Angoulêne — Dress of pink satin in a loose full skirt, with a frill round the hem, a white muslin bodice, and a blue sash. Her bonnet, one of the period, was of blue velvet, with purple and blue ostrich plumes.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;ERESFORD&lt;/small&gt; M&lt;small&gt;ELVILLE&lt;/small&gt;.

Dress of green satin, lined with rose silk, and trimmed with roses and Venetian point lace.

L&lt;small&gt;ORD&lt;/small&gt; R&lt;small&gt;ICHARD&lt;/small&gt; N&lt;small&gt;EVILLE&lt;/small&gt;.

Duc de Lanzun — Military coat of violet cloth, faced with white and braided with silver, violet silk stockings and knee-breeches, a white satin gold-embroidered waistcoat, silver epaulets, a sword in sword-belt, and all details to correspond.

MR H. D&lt;small&gt;U&lt;/small&gt; C. N&lt;small&gt;ORRIS&lt;/small&gt;.

Marquis of France, temps Louis XVI. — Coat and breeches rose silk, large diamond buttons to waistcoat, and superb diamond buckles on shoes. Sword hilt jewelled in steel.

M&lt;small&gt;R&lt;/small&gt; J. N&lt;small&gt;ORRIS&lt;/small&gt;.

Marquis of France — Green brocade velvet coat and breeches, white satin waistcoat, diamond buckles to shoes.

M&lt;small&gt;RS&lt;/small&gt; O&lt;small&gt;SBORNE&lt;/small&gt;.

Madame de Pompadour. — White silk dress, flowered with dandelions, and made with a Watteau back faced with pink, the petticoat of Rose du Barri silk with |ace flounces and jewelled trimmings, bodice also trimmed with lace and jewels, and long puffed tulle sleeves, drawn with narrow black ribbons; feathers in hair; gold and jewelled girdle.

M&lt;small&gt;ISS&lt;/small&gt; O&lt;small&gt;SBORNE&lt;/small&gt;.

Madame de Pompadour — In garden dress, pink petticoat with bands of black velvet at the edge, and overdress of white, brocaded with small rosebuds, large black velvet bow on one side under large pink roses, bodice ''en suite'' with lace and roses, and black bow on the right shoulder, and large white chip hat, wreathed in roses, and with black velvet and roses under the brim.

S&lt;small&gt;IR&lt;/small&gt; A&lt;small&gt;LGERNON&lt;/small&gt; O&lt;small&gt;SBORN&lt;/small&gt;, C&lt;small&gt;HICKSANDS&lt;/small&gt; P&lt;small&gt;RIORY&lt;/small&gt;.

Civilian costume — Black velvet, with steel buttons, lace cravat, white wig and sword.

P&lt;small&gt;RINCESS&lt;/small&gt; H&lt;small&gt;ENRY OF&lt;/small&gt; P&lt;small&gt;LESS&lt;/small&gt;.

Adrienne Lecouvreur — A handsome gown of ivory satin, the underskirt full, and embroidered, nearly half a yard deep, all round with gold, turquoise, amethyst, topaz, and briiliant jewelling. On the hip, the upper skirt was caught back with turquoise blue satin similarly embroidered. There were scalloped embroidered basques. Hair high and poudré; ornaments, diamond and turquoise.

P&lt;small&gt;RINCE&lt;/small&gt; H&lt;small&gt;ENRY OF&lt;/small&gt; P&lt;small&gt;LESS&lt;/small&gt;.

Vicomte de Bragebone — Green uniform of an Officer of the Guard of Louis XVI., faced with scarlet and relieved with white, the whole elaborately braided with gold. [Col 5c–6a]

C&lt;small&gt;OUNT&lt;/small&gt; P&lt;small&gt;ALFFY&lt;/small&gt; (A&lt;small&gt;USTRAILIAN&lt;/small&gt; E&lt;small&gt;MBASSY&lt;/small&gt;).

Court dress—Black velvet, point lace ruffles.

M&lt;small&gt;ISS&lt;/small&gt; P&lt;small&gt;ERRY&lt;/small&gt; (B&lt;small&gt;ITHAM&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;).

Louis XV. period. — Blue brocade, white satin petticoat, with beautiful deep old BrusseIs lace flounce.

L&lt;small&gt;ADY&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt;.

Lady of 1787 — This costume, which was '''designed by Madame Eloffe, dressmaker to Marie Antoinette''', was composed of a white satin skirt, waistcoat and sleeves of blue satin; the back of the bodice, puffs on the sleeves, and loose train being of striped canary yellow and mauve brocade; a fichu of white satin lace, and frills of lace on the sleeves. A white wig with long falling curls, and a very high turban of lace on a blue bandeau, with feathers and flowers on the left side; ornaments, diamonds and emeralds.

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ONSTANCE&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt;.

Bergère of about 1771 — Petticoat of pink satin, with festoons of lace, hooped up with pink and yellow roses; bodice and puffed top skirt of white brocade and pink roses, and large puffed sleeves; Tuscany straw Gainsborough hat, with high pink and white feathers; hair ''poudre''.

M&lt;small&gt;R&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; P&lt;small&gt;ETRE&lt;/small&gt; (W&lt;small&gt;HITLEY&lt;/small&gt; A&lt;small&gt;BBEY&lt;/small&gt;, C&lt;small&gt;OVENTRY&lt;/small&gt;).

Court dress (present day), with wig.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;RANK&lt;/small&gt; P&lt;small&gt;ARKER&lt;/small&gt;.

Costume from a picture by Sir Joshua Reynolds — Grey brocade, white muslin fichu and cap; powdered hair.

M&lt;small&gt;ISS&lt;/small&gt; P&lt;small&gt;RACH&lt;/small&gt;.

Louis XV., Dame de Cours — White satin bodoce [sic] and tunic, jewelled over blue petticoat, trimmed with lace and roses, coffieur of the period.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RTHUR&lt;/small&gt; P&lt;small&gt;AGET&lt;/small&gt;.

Duchess d’Orleans — A white satin dress bordered with sable and richly embroidered with steel. She wore diamond ornaments across the top of her bodice, down the front of the dress, and round her neck, while in her white hair were three black ostrich plumes. Her dress sleeves were bound with roses.

E&lt;small&gt;ARL OF&lt;/small&gt; R&lt;small&gt;OSSLYN&lt;/small&gt;.

Duc de Nemours — Uniform of a Colonel of Dragoons, period Louis XVI.; French grey and cherry colour, elaborately braided with gold.

L&lt;small&gt;ORD&lt;/small&gt; R&lt;small&gt;OYSTON&lt;/small&gt;.

Souis Brigadier of Mousquetaires, Louis XVI. — White coat with gold-laced cuffs and collar, a tabard of sapphire blue edged with gold and white embroidery, a diamond Maltese cross suspended on the breast, white knee-breeches and silk stockings, Court shoes, white wig, and three-cornered hat.

T&lt;small&gt;HE&lt;/small&gt; &lt;small&gt;DUCHESS OF&lt;/small&gt; S&lt;small&gt;UTHERLAND&lt;/small&gt;.

Marie Leczinski, the wife of Louis XV. — White satin dress wrought handsomely with silver, and a regal robe, or mantle, of ruby velvet embroidered with gold. The front of the long pointed bodice was fairly ablaze with jewels — rubies, emeralds, and diamonds — of the Sutherland family collection. Along rivière of very large brilliants was used to loop up the Court mantle at the shoulders, a pointed diadem was worn on the forehead, and a large pearl ornament with pendant pear-shaped pearls at one side. The coiffure was low, in Louis XV. style.

L&lt;small&gt;ADY&lt;/small&gt; J&lt;small&gt;ANE&lt;/small&gt; S&lt;small&gt;EYMOUR&lt;/small&gt;.

Lady, time of Louis XVI. — Biue quilted satia petticoat, white satin over-skirt and bodice trimmed with broad Valenciennes lace and pale-pink roses; hair powdered, with chaplet of roses and pearl ornaments.

L&lt;small&gt;ADY&lt;/small&gt; F&lt;small&gt;EODORA&lt;/small&gt; S&lt;small&gt;TURT&lt;/small&gt;.

Madame la Marquise de la Pompadour — White satin costume, with the over-dress full and plain, and opening on a similar under-dress. The sleeves were of satin to the elbow, and finished with ruffles of point d’Alençon lace, while lace was arranged round the top of the bodice. The hair was worn high and ''poudré'', with a black cockade, the badge of the Queen’s Household, at one side. She wore a black satin cap sewn with diamonds and having a diamond aigrette.

B&lt;small&gt;ARONESS&lt;/small&gt; S&lt;small&gt;CHIMMELPENNINCK VON DER&lt;/small&gt; O&lt;small&gt;YE&lt;/small&gt;

Pompadour costume—Overdress of dark green satin en train, Watteau back, over petticoat of white satin, brocaded with flowers; pearl and diamond ornaments.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;AURICE&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt; (H&lt;small&gt;AMMINWOOD&lt;/small&gt;, E&lt;small&gt;AST&lt;/small&gt; G&lt;small&gt;RINSTEAD&lt;/small&gt;).

Gentleman of the period of Louis XVI., copied from an old family picture.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;UY&lt;/small&gt; S&lt;small&gt;COTT&lt;/small&gt; (H&lt;small&gt;OCKLEY&lt;/small&gt;, M&lt;small&gt;ARTON&lt;/small&gt;).

Lady of the time of Louis XV. — Pink satin skirt, with Brussels lace and roses, white brocade body and overskirt, white wig, with pink roses and feather.

M&lt;small&gt;RS&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt; (T&lt;small&gt;HE&lt;/small&gt; L&lt;small&gt;AWN&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;). 

Lady of the Court of Louis XV. — A handsome gown of Louis XV. period, made of yellow brocade, lined and turned back with pale mauve satin and gold embroidery, over a petticoat of moss-green satin, with flounces of Brussels lace, and trails of westeria; bodice of yellow brocade, with fischu [sic] of white chiffon, fastened with green velvet bow and diamonds.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;AY&lt;/small&gt; S&lt;small&gt;ANDERS&lt;/small&gt; (S&lt;small&gt;NITTERFIELD&lt;/small&gt;).

Court dress of the period of Louis XVI. — White silk gown trimmed with gold, pink satin petticoat with revers of old point lace; hair ''poudré''; muslin fichu and pink roses.

MR S. S&lt;small&gt;ANDERS&lt;/small&gt; (S&lt;small&gt;NITTERFIELD&lt;/small&gt;).

Mousquetaire Corps du Garde de Louis XVI. — White frock coat with silver lace facings, blue waistcoat and breeches, white silk stockings, white wig, three-cornered hat, and sword.

M&lt;small&gt;ISS&lt;/small&gt; C. S&lt;small&gt;TARKEY&lt;/small&gt; (N&lt;small&gt;OTTS&lt;/small&gt;).

Dress of white brocade, lined with pink silk; pink silk petticoat, trimmed with old lace and wreaths of roses; hair ''poudré'', with pink roses; pearl ornaments.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;EAUCHAMP&lt;/small&gt; S&lt;small&gt;COTT&lt;/small&gt;.

Period Louis XVI. — White and mauve satin brocade, trimmed with lace and fur, white wig.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RTHUR&lt;/small&gt; S&lt;small&gt;OMERSET&lt;/small&gt;.

Shepherdess ''à la'' Watteau — Pompadour style, the dress and hat being trimmed with pink roses, and her crook tied with similar flowers.

M&lt;small&gt;R&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt;.

Gentleman of the Court, Louis XVI. — Dark-striped yellow coat, with needlework design in flowers in shaded silk, embroidered cream silk waistcoat, and pale-green satin breeches.

C&lt;small&gt;APTAIN&lt;/small&gt; S&lt;small&gt;OMERSET&lt;/small&gt;.

Mousquetaire — Gold cloth with broad gold-bued silk sash.

H&lt;small&gt;ON&lt;/small&gt;. B. W. H. S&lt;small&gt;TONOR&lt;/small&gt;.

Mousquetaire, Louis XVI. — Knickerbockers of a dark blue, with a doublet to match, having a cross emblazoned on the breast, and deep point lace collar, white satin coat skirts almost like a simulation of armour below his doublet, and a long military cloak of French grey cloth lined with scarlet, three-cornered hat, white wig, sword, silk stockings, and Court shoes.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;HURSBY&lt;/small&gt;-P&lt;small&gt;ELHAM&lt;/small&gt;.

White satin, with white roses and lace.

M&lt;small&gt;ISS&lt;/small&gt; V&lt;small&gt;IOLET&lt;/small&gt; L&lt;small&gt;OFTUS&lt;/small&gt; T&lt;small&gt;OTTENHAM&lt;/small&gt;.

Pink and white brocade over white satin, and pink roses in her powdered hair.

M&lt;small&gt;R&lt;/small&gt; T&lt;small&gt;OWER&lt;/small&gt;.

Officier Garde Suisse — Scarlet coat lined with white, blue facings, and three-cornered hat.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;OWER&lt;/small&gt;.

Duchesse de Polignac — White satin, with old lace and rose coloured plumes.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;REE&lt;/small&gt;.

Court Dame, Louis XV — Skirt of pink satin, with a tunic of most lovely silver tinsel brocade, having alternate stripes encllosing bunches of roses and baskets of flowers. Were the usual Watteau back and hip panniers, and the petticoat was arranged with a twist of chiffon above a frill of most beautiful cream lace, over these being a garland of pink roses, caught up on either side with a cluster of pink and cream feathers repeating to the top; the sleeves were made of brocade to the elbow, with hanging cream lace over pink chiffon and feathers on the shoulders, and the dress was completed by pink slippers with pink velvet bows, a Louis XV. fan of great beauty, and a staff of pink and green with green ribbons, roses, and feathers to match the dress. 

L&lt;small&gt;ORD&lt;/small&gt; G&lt;small&gt;REY DE&lt;/small&gt; W&lt;small&gt;ILTON&lt;/small&gt;.

Gentleman, temps Louis XV. — Coat of dark petunia velvet embroidered with gold, a white satin waistcoat elaborately gold laced, white silk stockings, jabot ruffles, wig, sword, hat, and shoes ''en suite''.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt; D&lt;small&gt;UDLEY&lt;/small&gt; W&lt;small&gt;ARD&lt;/small&gt;.

Mousquetaire — Uniform of dark blue cloth, with scarlet facings and elaborately braided with gold.

L&lt;small&gt;ADY&lt;/small&gt; W&lt;small&gt;ALLER&lt;/small&gt;.

Comtesse d’Artois — White satin quilted petticoat with a pearl at each corner of the pattern; gown of grey brocade lined with white satin; white lisse stomacher crossed by grey velvet bows fastened with diamonds.

M&lt;small&gt;ISS&lt;/small&gt; W&lt;small&gt;ALLER&lt;/small&gt;.

Fille de la Comtesse d’Artois — Gown of pale blue satin flecked with pink roses tied with ribbon, paniers and wreaths of roses. 

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ORNWALLIS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt;.

Mademoiselle de la Court — Costume after a picture by Roslin depicting a girl about to decorate the statue of Love. She wore an underdress of pale pink satin with gown of white satin with demi-train, lined with pink. The body was decked with tulle, and long tulle streamers were pendant from the sleeves; head-dress was roses and violets, with pink and white ribbons.

M&lt;small&gt;R&lt;/small&gt; W&lt;small&gt;ILLIAMS&lt;/small&gt;.

Black velvet Court dress, with Louis XV. wig.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;RANCIS&lt;/small&gt; W&lt;small&gt;ILLIAMS&lt;/small&gt; (W&lt;small&gt;ATCHBURY&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;).

Lady of the Court, Louis XV. — White satin bodice and train, with white satin petticoat trimmed with lace and pink roses.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt; (A&lt;small&gt;LSCOT&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

Lady of the Court of Louis XV. — A handsome gown of Louis XV. period, made of white broché silk, with bouquets of pink roses, over petticoat of rich pink satin with deep flounce of Brussels lace, caught up with trails of pink roses; bodice of same broché, with white chiffon fichu fastened in front with pink velvet bow and diamonds. [6, Col. 6c – 7, Col. 1a]

M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;ARTLAND&lt;/small&gt;.

Lady of the court, Louis XV. — Green brocade bodice and skirt; white satin petticoat trimmed with old lace and roses; white Leghorn hat with roses.

M&lt;small&gt;ISS&lt;/small&gt; F&lt;small&gt;LEETWOOD&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;.

Lady of the period — Black peau de soie silk, the Watteau and over-skirt lined with white, pointed panniers, skirt caught up with roses, while black roses wee worn on the powdered hair. A splendid stomacher of diamonds and emeralds was worn.

L&lt;small&gt;IEUTENANT&lt;/small&gt;-C&lt;small&gt;OLONEL&lt;/small&gt; M&lt;small&gt;ILDMAY&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, C.B. (S&lt;small&gt;COTS&lt;/small&gt; G&lt;small&gt;UARDS&lt;/small&gt;).

Guardsman, 1790 — Red tunic, gold braid, [9R?] on buttons, white gaiters coming above the knee, black garters, wig, and three-cornered hat. 

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;HEATLEY&lt;/small&gt; (B&lt;small&gt;ERKSWELL&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Lady of Louis XV. Court — Bodice and train of pale green and pearl coloured striped brocade, with bunches of pink roses; petticoat and corsage of pale green satin, embroidered in pearls, and high collar of lace; diamond ornaments, a wig with pink roses and diamond stars.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;ILLIE&lt;/small&gt; L&lt;small&gt;OW&lt;/small&gt; (W&lt;small&gt;ELLESBOURNE&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;).

Duchess of Gainsborough — Dress copied exactly from the portrait of the Hon. Mrs Graham, in brocade, with pink satin petticoat, a big black hat trimmed with white plumes, with diamonds.

M&lt;small&gt;R&lt;/small&gt; W. M. L&lt;small&gt;OW&lt;/small&gt;.

David Garrick — Costume worn by the actor, Mr Richard Wyndham, when impersonating that character, of rich purple vevet coat, purple satin waistcoat and knee-breeches, steel buttons, purple silk stockings, diamond buckles, black three-cornered hat, and steel sword.&lt;ref&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt;&lt;/blockquote&gt;

==Anthology==

====Quote Intro====
The day of the ball, the ''Coventry Evening Telegraph'' published the following:&lt;blockquote&gt;GRAND BAL POUDRE AT WARWICK CASTLE.

Writing this morning our Warwick representative says: Warwick Castle will tonight be the scene of a memorable spectacle, the Earl and Countess of Warwick having invited about four hundred guests to a ''bal poudre'', in which the costumes were to be of the style of the Louis XIV. and XV. period. The event has been looked forward to with considerable interest by the ''élite'' of the fashionable world, on account of the prominent position occupied by the Countess in society. Great preparations were made the Castle, the greater portion of which has been most lavishly decorated in the light and airy French style of the period. The dancing will take place in the Cedar drawing-room, the adjoining rooms having been set apart as retiring rooms. Supper will be served in the Great Hall, where the whole of the guests will be able to sit down together. The decorations have been carried out under the personal supervision of the hostess, who has received the valuable assistance and advice of Mr. Caryll Craven. The dance music will be supplied by Worm's famous &quot;White Viennese&quot; Band, while Johnson's (Manchester) Band will discourse in the supper room. The hostess will be dressed as &quot;Mary Antoinette,&quot; Queen of Louis XVI. Her costume will be of rose-coloured brocade with a gold pattern, and a sky-blue velvet train embroidered with gold fleur-de lys. Lady Warwick's relative, the Duchess of Sutherland, will appear as the wife of Louis XV. in a costume of white and silver with crimson velvet train and silver fleur-de-lys. Lord Warwick will be in the dress of a military officer of the period, while Prince Francis of Teck has signified his intention of appearing in the uniform of &quot;the Royals&quot; (of the period). Owing to the demise of Lord Randolph Churchill, the Duke of Marlborough will not be present.

The house party at the Castle included the Austro-Hungarian Ambassador, the Portuguese Minister, Prince Francis of Teck, Prince and Princess Henry of Pless and Miss Cornwallis West, Duchess of Sutherland and Lady Angela St. Clair Erskine, Duke of Manchester, Earl and Countess of Rosslyn, Earl of Lonsdale, Earl of Burford, Earl of Chesterfield, Countess Cairns, Lord Clifden, Lord Kenyon, Lady Gerard, Lord Grey de Wilton, Lord Royston, Lord Lovat, Lady Norreys, Lady Eva Greville, Lord Richard Neville, Hon H. and Lady Fedora Sturt, Hon. H. Stonor, Captain the Hon. Hedworth Lambton, Mr. F. Menzies and Miss Muriel Wilson, Miss Naylor, Mr. Arthur Paget, Mr. Cyril Foley, Mr. C. de Murietta, and Mr. Layoock.

The following accepted invitations to the ball, and most of them brought parties with them, the guests numbering in all about four hundred:— The Earl and Countess of Aylesford, Mr. and Mrs. Aubrey Cartwright, Mr. and Mrs. J. Stratford Dugdale, Mr. and Mrs. Chamberlayne, Sir C. and Lady Mordaunt, Mr. and Mrs. Smythe, Lord and Lady Hertford, Lady and Miss Waller, Mr. J. and Mr. J. P. Arkwright, Mr. and Mrs. M. Lucas, Mr. and Mrs. J. B. Dugdale (18), Mr., Lady Anne, and Miss Murray, Captain and Mrs. Brinkley, Mr. and Mrs. Guy Scott, Mr., Mrs., and Miss Irwin, Mr. and Miss Perry, Major and Mrs. Fosbery, Mr. Lindsay, Mr. R. Paget, Sir A. and Lady Hodgson, Mrs. Beauchamp Scott, Major and Mrs. Norris, Mr. and Mrs. Tree, Mr., Mrs. and Miss Granville, Mr. and Mrs. Joliffe, Captain and Mrs. Osborne, Mr. and Mrs. E. Little, Mr., Mrs., and Miss Lakin, Officers 6th Reg. District, Mr. Batchelor, Hon. Mrs. and Miss Chandos Leigh, Colonel and Mrs. Paulet, Mr. F. Hunter Blair, Mr. J. Alston, Mr. and Mrs. Hutton, Captain and Mrs. Cowan, Mr. and Mrs. Williams, Mr. and the Misses Allfrey, Mr. and Mrs. Wilson Fitzgerald, Mr. and Mrs. W. Allfrey, Mrs. and Miss Drummond, Mr. and Mrs. Shaw, Mr. R. and Mr. J. Lant, Mr. and Mrs. Sanders and party, Captain Lafone, Sir F. and Lady Peel, Captain and Mrs. Keighly-Peach, Miss Nicol and party, Mr. and Mrs. Fred Shaw, Mr. and Mrs. Cove Jones, Mr. and Lady G. Petre, Mr. R. Barnes, Mr. and Mrs. Wheatley, Mr., Mrs., and Miss Ramsden, Mr. and Mrs. W. M. Low, Mrs. Basil Montgomery, Mr. and Mrs. Thursby-Pelham, Mr. and Mrs. H. Chamberlain, Mr. Francis, Mr. and Mrs. Graham, Mr. and Mrs. West, Mr. and Mrs. Sam Smith, Officers 17th Lancers.&lt;ref&gt;&quot;Grand Bal Poudre at Warwick Castle.&quot; ''Coventry Evening Telegraph'' 01 February 1895, Friday: 3 [of 4], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000337/18950201/021/0003.&lt;/ref&gt;&lt;/blockquote&gt;The report from the ''Morning Post'' the next day:&lt;blockquote&gt;The Countess of Warwick's Bal Poudré at Warwick Castle last night was attended by a company of nearly 400 guests, and was a brilliant success. The magnificent suite of apartments was superbly decorated with choice flowers, while the many treasures of antiquity and historic interest which the Castle contains were displayed in the various rooms. The choice of costume was restricted to the period covering the reigns of Louis XV. and Louis XVI., with powdered hair or white wigs, but gentlemen were given the option of appearing in English Court dress with Louis XV. wigs. The Countess of Warwick, who represented Marie Antoinette, wore a dress of rose-coloured material brocaded with gold, with a train of sky-blue velvet, embroidered with fleur-de-lis. The Earl of Warwick was attired in a Maison du Roi costume of rich velvet, with gold and diamond buttons. Prince Francis of Teck wore the uniform of the period of his own regiment, the Royals. The Duchess of Sutherland, as the wife of Louis XV., was in a costume of white and silver, with a crimson velvet train embroidered with silver fleur-de-lis. Prince Henry of Pless wore a blue military dress of the period with red facings, while the Earl of Rosslyn donned the uniform of a Colonel of the reign of Louis XVI. The Hon. H. Sturt represented the Church of the period as an Abbé, and Mr. W. Low the stage as David Garrick. Amongst the other guests were the Austro-Hungarian Ambassador, the Portuguese Minister, Princess Henry of Pless, Lady Angela St. Clair Erskine, the Duke of Manchester, the Earl and Countess of Rosslyn, the Earl of Lonsdale, the Earl of Burford, the Earl of Chesterfield, Countess Cairns, Lord Clifden, Lord Kenyon, Lady Gerard, Lord Grey de Wilton, Lord Royston, Lord Lovat, Lady Norreys, Lady Eva Greville, Lord Richard Nevill, Lady Feodorowna Sturt, the Hon. S. Greville, the Hon. H. Stonor, Captain the Hon. Hedworth Lambton, Mrs. Menzies, Miss Muriel Wilson [sic no comma] Miss Naylor, Mr. Arthur Paget, Mr. Cyril Foley, Mr. C. de Murrieta, Mr. Caryl Craven, Mr. Kennard, and Mr. Laycock. The Countess of Aylesford brought a large party from Packington Hall. Herr Würm's White Viennese Band occupied the orchestra. Dancing commenced at nine o'clock, and at midnight the entire company sat down to supper in the large banqueting hall. The assembly was undoubtedly one of the most brilliant which has ever been gathered together within the walls of the historic Castle.&lt;ref&gt;&quot;Arrangements for This Day.&quot; ''Morning Post'' 02 February 1895, Saturday: 5 [of 10], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18950202/052/0005.&lt;/ref&gt;&lt;/blockquote&gt;

== Notes and Questions ==
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==References==</text>
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      <text bytes="76498" sha1="40kpvo08vuv3rrberbw6x2j86aax4bj" xml:space="preserve">=Bal Poudré at Warwick Castle=
==Overview==
A bal poudré was held at Warwick Castle on Friday, 1 February 1895, with [[Social Victorians/People/Warwick|Countess Warwick]] dressed as Marie Antoinette. [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was part of the house party as well as attending the ball,&lt;ref&gt;&quot;Court Circular.&quot; ''Times'', 2 Feb. 1895, p. 10. ''The Times Digital Archive'', http://tinyurl.galegroup.com/tinyurl/AHQju3. Accessed 20 June 2019.&lt;/ref&gt;&lt;ref&gt;“Grand Bal Poudre at Warwick Castle.” ''Midland Daily Telegraph'' [now in BNA: ''Coventry Evening Telegraph''] 1 February 1895, Friday: 3 [of 4], Col. 4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000337/18950201/021/0003 (accessed July 2019).&lt;/ref&gt; as was &quot;Mr [[Social Victorians/People/Craven|Caryl Craven]], to whom so many thanks are due for the able way in which he assisted his charming hostess in carrying out her scheme, Mr Craven being quite an authority on eighteenth century French art and dress.&quot;&lt;ref&gt;&quot;The Warwick Bal Poudre.&quot; ''The Queen, The Lady's Newspaper'' 09 February 1895 Saturday: 38 [of 80], Col. 2c [of 3] – 39, Col. 3c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18950209/233/0038.&lt;/ref&gt;

Daisy, Countess Warwick dressed as Marie Antoinette for the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress]] ball as well.

Miss Beatrice Fitzherbert &quot;wore a beautiful pearl necklace, with large diamond pendant, and two diamond sprays, all of which were given by George IV. to Mrs Fitzberbert.&quot;&lt;ref name=&quot;:0&quot;&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt; (6, 4b)

Lady Peel wore a costume that, the Leamington Spa Courier says, &quot;was designed by Madame Eloffe, dressmaker to Marie Antoinette.&quot;&lt;ref name=&quot;:0&quot; /&gt; (6, 6a)

==Logistics==
* Friday, 1 February 1895
* Warwick Castle
* Hosts, Countess and the Earl of Warwick
==Related Events==
* 2 July 1897, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]]

== Main Newspaper Report ==
The ''Leamington Spa Courier'' had the definitive story in the next issue, a week after the event:&lt;blockquote&gt;THE GRAND BAL POUDRE AT WARWICK CASTLE.

A profound impression has been created throughout the country by the enormously successful ''bal poudre'' given by the Earl and Countess of Warwick at Warwick Castle on Friday night last week, a lengthy, but — owing to the exigencies of the occasion — necessarily incomplete record of which appeared in our issue on the following morning. It is conceded on all hands that it was unmistakably the most splendidly organised and artistically perfect function of the kind that has been given during the present century, and certainly unexampled in the annals of the county. In times past, the historic fortlace has been the scene of many gay and festive ''re-unions'', but history gives no mention of one that in any way equalled in point of completeness of detail or magnificence that in which some 400 of the present Earl and Countess’s guests participated on Friday night. For the nonce, the prosaic modern gave place to the splendour of the past, and the luxurious and gorgeous conditions which prevailed at the Tuilleries during the glories of the regime of Louis XV., and the reign of his ill-fated successor and his beautiful consort, were revived in all their sparkling radiance, thus creating a pageant of unrivalled grandeur and beauty, and one that will be long retained in the recollections of those who took part in or were permitted to gaze upon it. Moreover, never, perhaps, had the old Castle had within its walls such a notable company, including as it did, some of the most distinguished personages of the day, connections of Royalty, Ambassadors, of foreign Powers, Dukes and Duchesses, Earls and Countesses, Lords and Ladies, representatives of the three great services of the State, Statesman, lawyers, and other ornaments of the highest and most aristocratic circles. The long suite of rooms, with the abundance of rich and historic art treasures therein contained, was most tastefully and effectively decorated, and the gilded and brocaded furniture and lovely fittings were arranged so as to form a replica of the interior of the Tuilleries at the period which the Countess had, with consummate judgment, selected for representation. Wide-spreading palms were placed at different points, and rare flowers of brilliant hues from Cannes and other parts of Southern Europe were seen on every hand. The whole was brilliantly illuminated by innumerable wax candles affixed to crystal chandeliers, in addition to the rays of the electric light, emitted from glow-lamps so constructed as to simulate candles, and having the bulbs hidden by delicately tinted shades; and when the guests in their picturesque costumes perambulated the apartments, the spectacle afforded was unique and enthralling. The most striking scenes, however, were witnessed in the Cedar Drawing-room when the dancing was in full operation, and again when the richly-dressed and white-wigged guests sat down to supper in the grand Banquetting-hall. (6, Col. 1a/1b)

The guests commenced to arrive about half-past ten, and carriages continued to roll up until close upon midnight. The traffic was directed by the same staff of police who were in attendance at the recent concert, and these were also assisted by the Commissionaires. Under the instructions of the House Steward, Inspector Hall and his men guided the traffic most skilfully, notwithstanding that the entrance to the courtyard beneath the gateway and barbican is very narrow. Precaution had been taken to fix a large number of lamps along the approaches to the Castle, to minimise the danger of an accident. The guests did not use the grand entrance under the porch, but entered by the door at the other end beyond the chapel, over which a large striped awning served as a porch and a crush room, the interior being decorated with flowering and foliage plants, and splendidly lighted by pendant lamps. They then passed through the armour passage to the centre State Drawing-room, adjoining the ballroom, where they were received by the Countess, the train of whose lovely and charming costume, a la Marie Antoinette, was borne by her little daughter, Lady Marjorie Greville, and a young companion, Miss Hamilton, who were attired as imitation China shepherdesses in white broche silk, and large white satin hats, trimmed with roses and long ostrich feathers, and carried wands. Dancing commenced at a quarter to 11, the music being supplied by Herr Wurms’ Viennese band, the members of which wore the dress of the period carried out in white and gold, and were ensconced in an orchestra formed in one of the arched windows. The ball was opened by a quadrille, in which one set was made up of the Countess and Count Demyn (the Austro-Hungarian Ambassador), the Duchess of Sutherland and [[Social Victorians/People/de Soveral|M. de Soverel]] [sic] (the Portuguese Ambassador), Princess Henry of Pless and the Earl of Warwick, and Prince Henry of Pless and Lady Feodora Sturt. The programmes of dance music were in book form, bearing a miniature medallion of Marie Antoinette on the one side, and Warwick Castle, set in a little Louis Quinze frame, on the other. The scene, while the dancing was in full swing, was replete with animation and splendid beauty. The infinite variety of costumes, flashing diamonds and other jewels, and a brilliance of colour ever changing with wondrous rapidity, as the dancers advanced and receded, or mingled in the crowd, backed by the cedar pannelling with the light falling from the candelabra and incandescent electric lights upon the fine Vandyck family portraits hanging round it, the large crystal chandeliers pending from the white and gold ceiling, and standing in each corner of the room and on either side of the great marble and alabaster mantelpiece, made up a picture at once quaint, full of life, animation, and picturesque beauty. Shortly after midnight, three trumpeters, correctly dressed in the gold-bedecked uniform of English heralds of the time of Louis XV., took up a position at the entrance to the banquetting hall and gave the signal that supper was served, by blowing a fanfare. Thereupon, a procession was formed, the Conntess [sic] of Warwick leading the way with Count Deym. M. de Soverel followed with the Duchess of Sutherland; then came the Earl of Warwick with Princess Henry of Pless, Prince Henry with Lady Feodora Sturt, the Earl of Lonsdale with Mrs Arthur Paget, and Lord Kenyon with Mrs Miller-Munday [sic for Mundy]. [6, Col. 1b–c] This party of 12 seated themselves at the centre table, other guests occupying the round and oval tables, about 14 in number, and each laid for eight. Special arrangements had been made for the serving of ''souper'', which was supplied entirely from the Castle kitchens, and it was originally intended, when it was thought the number of guests would not exceed 240, that all should sit down together. But the applications from those who wished to be included in what may, without exaggeration, be termed an historical event in the social functions of Warwickshire, were so very numerous that it was necessary to divide the company into two sections. On the centre table was an imposing display of the handsome gold and silver family plate, including a celebrated gold cup modelled by Benvenuto Cellini, and the floral embellishments consisted of choice flowers from Cannes and magnificent orchids from Trentham. The round and oval tables were also handsomely decorated with silver plate. The meal was a truly sumptuous one, and the menu, which was printed inside a little white and gold Louis XV. screen, having a picture of Warwick Castle on one side and “Souper, Février 1, 1895,” on the reverse, included some triumphs of the culinary art. The dessert comprised strawberries, apricots, grapes, pineapples, and other fruits rare and expensive at this season of the year,[sic] The hall, with its shining coats of mail, the magnificent Beauvais tapestry forming portieres and hanging from the gallery, the massive silver candelabra on the tables, and the immense ecclesiastical candlesticks standing on the floor and bearing torches which towered far above the heads of the guests, constituted a truly marvellous sight, and one upon which the eye never tired to dwell. To render the picture more complete, the servants, who flitted about attending upon the wants of the guests, were clothed in the livery of the period, some in white and gold and red velvet, and others in sabre suits of black, all wearing knee-breeches, silk stockings, and white wigs. The staff of servants at the Castle was quite inadequate to carry out the various duties which devolved upon them in consequence of the ball, and Mr J. Hall (the House-steward) consequently found it necessary to engage a special staff of first-class waiters from London. A few privileged persons, to whom tickets had previously been issued, were admitted to the long narrow passage in the thickness of the wall near the roof, which was discovered at the time of the disastrous fire in 1871, to which access is gained through the oak and carnation rooms. After supper, dancing was resumed, and continued with unabated vigour until considerably after four o’clock, the Countess remaining during the whole of that time with her guests.

One of the guests made an unfortunate miss of the train which cost him a good deal of inconvenience, and his host and hostess some anxiety. The gentleman in question was taken as one of a large house party in the county to Warwick, and there was a special train chartered from Milverton to carry the guests back to a station near the host's residence. A short time before his party were returning home, the guest went into the smoke-room at the Castle, and though the heard the name of his host called, he thought it was the company of a lady of the same name who were wanted. In the end the gentleman was left behind, and then he drove to Leamington, but could get no train from there. All the hotels were [6, Col. 1c–2a] full, and so it was no use to apply there for a bed. The consequence was he had to spend the night in a waiting room either at Leamington or Milverton Station, getting back to the country house on Saturday. He will long remember Lady Warwick’s ball.

Letters have been received from guests expressing satisfaction in regard to the efficient way in which the police carried out their duties. It may be stated that the numerous alterations and renovations at the Castle, especially in regard to the private apartments, have been carried out by Messrs Bertram and Sons, the great upholsterers, of Dean-street, Soho.

As one of the lady guests was alighting from her carriage at the Castle, on Friday night, a large diamond and turquoise ornament, valued at 200 guineas, became detached from her hair, and fell to the ground. The loss was quickly discovered, and, fortunately, the costly ornament was recovered intact.

Viscount Dungarven, whe was to have formed one of Mr W. M. Low’s party, was prevented by unforseen circumstances from attending the ball. Mr Perry (Bitham House) was also prevented attending by illness.

LIST OF GUESTS.

It has been found impossible to obtain a complete list of the names of the guests owing to the fact that the presentation of tickets was dispensed with, and we have, therefore, been compelled to rely on extraneous sources for information. The following is a list of the names of a large number of those present at the function:—

PRINCES [init caps large, rest sm] — Francis of Teck and Henry of Pless (Viscomte de Bragabene [Bragelonne?].)

PRINCESS — Henry of Pless (Adrienne Lecouvreur.)

EARLS — Clonmell (modern Court dress), Rosslyn (Duc de Nemours), Lonsdale (M. de Copinson, Keeper of the Koyal [sic] Stud, Louis XV.), and Chesterfield (Court costume.)

DUKE of Manchester (Marquis de Grammont.)

DUCHESS of Sutherland (Queen of Louis XV.)

MARQUIS of Hertford (Court costume.)

MARCHIONESS of Hertford (Court costume.)

COUNT Paiffy (costume Louis XV.)

COUNTESSES Cairns (Duchess de Bouillon) and Rosslyn (Marchande Coquette.)

COMTESSE Ahlefeldt-Laurvig (Dame de la temps Louis XVI.)

VISCOUNTS — Southwell (Court dress of the period), and Clifden (Court dress of the period ).

HIS EXCELLENCY the Portuguese Minister (Mousquetaire of the 2ud [sic] Company of the Royal Household, Louis XV.)

HIS EXCELLENCY the Austro-Hungarian Ambassador (English Court dress.)

LORDS — Burford (Mousquetaire), Cecil Manners (Court dress of the period), Churchill (Mons. de Brissac), Kenyon (officer of the Regiment du Roi, Louis XVI.), Clifton (officer of the Guards, Louis XV.), Lovat (Comte d’Artagnas), Richard Neville (Duc de Lauzun), Frederick Hamilton; Royston (Souis Brigadier of Mousquetaires, Louis XVI.), Grey de Wilton (gentlemen temps Louis XV.). and Doneraile (modern Court dress.)

LADIES — Norreys (Paysaune Galante), Ann Murray (Madame de Pompadour), Waller (Comtesse d’Artois), Peel (costume 1787), Chetwode, Angela St. Clair Erskine (Lady Mary Campbell), Eva Greville (Polichinelle, Louis (XV.), G. Petre, Feodora Sturt (Madame la Marquise de la Pompadour), Gerard (Duchess de Pognac), Edith Seymour (Lady of the reign of Louis XVI.), Mordaunt (Princess de Lambelle), and Churchill (French Marquise in the time of Louis XV.)

SIRS — Algernon Osborne (civilian costume, Louis XV.), Archibald Edmonstone (Mousquetuaire), Francis Burdett, Charles Mordaunt (gentleman of the time of Louis XVI.), and F. Peel.

[6, Col. 2a–b]

HONOURABLES — Mrs Louis Greville (dress of the period), Dudley Ward (Mousquetaire), C. Finch (gentleman of the period of Louis XV.), Captain Alwyn Greville (Mousquetaire), Mr and Mrs Chandos Leigh, Mrs Alwyn Greville (Dame de la Court Louis XV.), Captain Hedworth Lambton (Courtier of Louis XVI.), Humphrey Sturt, M.P. (Abbé Bouvet), Mrs E. Lyon (''à la'' Watteau), Mrs Dudley Ward, B. W. H. Stoner (Mousquetaire, Louis XVI.), Sidney Greville (officer of the Regiment of the Swiss Guards), Louis Greville (Mousquetaire, Regiment de Provence, Louis XV.), [[Social Victorians/People/Keppel|George Keppel]] (Mousquetaire), [[Social Victorians/People/Keppel|Mrs George Keppel]] (lady, t[i]me of Louis XVI.), Malcolm Lyon, Mrs Herbert Dormer (costume, Louis XV.), Mrs Frank Parker, and Cecil Freemantle (Court dress of the period).

BARONS — Macar (Court dress of the period) and Schimmelpennick Van der Oye (Court dress).

BARONESS — Schimmelpennick Van der Oye (costume Louis XVI.)

GENERAL — Arbuthnot (Court dress of the period).

COLONELS — Paulet and Mildmay Willson, C.B. (Scots Guards).

MAJORS — Armstrong (modern Court dress), Norris Fosbery (Mousquetaire), and Alston.

CAPTAINS — Molesworth (Mousquetaire), J. Barry (costume, Louis XVI.), Somerset (Mousquetaire), Brinkley (Court costume of the period), East (Mousquetaire), Granville (Mousquetaire), Cowan, Lafone, Keighly-Peach, Bruce Hamilton, H. Welman (Court dress), Grant, Towers Clark (Court dress), Allfrey (Court dress, Louis XV.), and Oxley, 60th Rifles.

MESDAMES — Armstrong (French Marquise), Gerald Arbuthnot (Court Dame), Armitage (Dame de la temps Louis XV.), H. Allfrey (Marquise, temps Louis XV.), Brinkley (Court Dame, Louis XV.), Frank Bibby (Lady of the Court of Louis XVI.), Everard Browne (Court Dame), Beech, Aubrey Cartwright, Chamberlayne (Court dress), Cowan, Cove-Jones, H. Chamberlain, George Cartland, Cartwright (Court Dame), J. S. Dugdale (Court lady), Blanche Drummond, Lindsay Eric-Smith (Pompadour costume), Fosbery, Wilson Fitzgerald, Fairfax-Lucy (Marquise Louis XV.), Granville, Graham, Gaskell (Grande Dame), Hulton (Court costume), Harvey Drummond, Irwin, Joliffe (Watteau), Edward Lucas (Lady of Court Louis XVI.), Morton P. Lucas (Court Lady, Louis XV.), E. Little, Leslie (of Balquhain), Lakin (Madame Roland), W. M. Low, Leslie (a la Watteau), Beresford Melville (Dame de la Court), J. Menzies (Duchess d’Angoulêne), Molesworth (costume Louis XVI.), Basil Montgomery (Marquise), Miller-Munday [sic] (Marie Therese, Queen of the Sicilies), Robert O. Milne (Dame de la Louis XVI.), Norris, Osborne (Madame de Pompadour), Arthur Paget (Duchess d’Orleans), Paulet, Ramsden (Madame de Colonne), Arthur Somerset (Shepherdess a la Watteau), Smythe, L. Gay Scott, Beauchamp Scott (Lady of time of Louis XVI.), Shaw, Sanders, Fred Shaw, S. C. Smith (Marquise temps Louis XVI.), Tree (Lady Louis XVI. period), Thursby-Pelham (Court dress), Tower (Duchess de Polignac), Towers-Clark (Lady of the Court of Louis XV.)[,] Francis Williams, Wheatley (Lady of time of Louis XV.), West [(]Court costume Louis XV.), and Francis Williams (Louis XV. costume).

[6, Col. 2b–c]

MADEMOISELLES—Allfrey, Armstrong (a laWatteau), Bromley Davenport, N. Booker (English lady of the Court of King George III.), Booker (Lady of time of Louis XV.), Muriel Bell, Nora Battye, Decapell Brooke, Spender Clay (Mdme Lamballe), Carleton (Watteau, Louis XV.), Chetwode, Anna Cassel, Carruthers (costume of the period Louis XV.), Champion, Hugh Drummond (Court costume), Constance Dormer (costume, Louis XV.), Beatrice Fitzherbert (Court Dame), Lucy (Mademoiselle de Montmirail), Granville, Gaskell (costume, Louis XV.), Hodgson (a la Watteau), Gladys Hankey (Marquese, reign Louis XV.), Irwin, Keighly-Peach akin (Dame de la Cours, Louis XV.), Lakin (a la Watteau), Lister-Kaye, Violet Leigh (Mdlle. de Chévreuse), Murray, Miller-Mundy (Court dress), J. Menzies (Duchess d’ Angoulêne), Naylor (Lady of the Court of Louis XV.), Nicol, Osborne (Mdme. de Pompadour, in garden dress), Perry, Constance Peel, Ramsden (Mdlle. de Colonne), E. N. Ramsden (Mdlle. de Coloane), Rushton (Lady of the Court of Louis XV), C. Starkey, Cicely Dudley Smith (Court dress, Louis XVI.), May Sanders (Louis XVI. costume), Cornwallis West (Mdlle. de la Court), Muriel Wilson (English costume of the period, Louis XV. and XVI.), Fleetwood Wilson (Lady, time Louis XVI.), and Waller (Fille de la Comtesse d’Artoix).

MESSIEURS — G. A. Arbuthnot (modern Court dress), W. C. Alston (Infanterie Regiment de Forés), Allfrey, J. Arkwright, W. Armstrong, J. P. Arkwright, Robertson Aikman, Frank Bibby, Bromley-Davenport, Brinckman, P. B. Vander Byl (Mousquetaire), Beaumont, 60th Rifles, L. Bethell (Mousquetaire), Bainbridge, A. E. Batchelor (Garde de la Porte), Everard Browne (gentleman, temps Louis XV.), R. Barnes, Battye, F. C. Hunter Blair (Mousquetaire), Beech (Garde au corps du Roi), C. B. Clutterbuck (Mousequetaire), Cassel (modern Court dress), Collings (modern court dress), Felix Cassel, Caryl Craven (military uniform of the period), Aubrey Cartwright, Chamberlayne, Bertram Chaplin, Cove-Jones, E. S. Chattock, H. Chamberlain, Drummond Chaplin (Court dress), G. Cartland, J. S. Dugdale (Recorder), M. Farquahar (Mousquetaire), Cyril Foley (officer du corps du Roi Pologne Stanilas), Kenneth Foster (Mousquetaire), S. M. Fraser (Mousquetaire), Fairfax-Lucy (Colonel George Lucy), J. S. Forbes (Mousquetaire), B. J. Fitzgerald (Mousquetaire), Francis Fitz-Herbert (Fusilier du Roi), J. B. Fitz-Herbert (gentleman, temps Louis XV.), Wilson Fitzgerald, R. Flower (modern Court dress), Francis, Flower (modern Court dress), Granville, J, Grenfell, Graham, G. de J. Hamilton (Mousequetaire), E. Harrington (Mousquetaire), Hutton, Head, H. T. Hickman (Court dress), Percival Hodgson (Court dress), Irwin, Joliffe, Joostens (Diplomatic Court dress), M. T. Kennard (Maison du Roi), F. Laycock (officer of Pondicherry Regiment), Morton P. Lucas (gentleman of the period), R. W. Lindsay (Court dress, late 18th century), E. Little, Lister-Kay, Lakin (modern Court dress), R. Lakin (the Duc de Brissac), Richard Lant (modern court dress), John Lant (M. Vauthier), H. G. Lakin (the Marquis de Breze), W. M. Low (David Garrick), Meyrick, Murray (Mousquetaire), C. de Murietta (Marechal Saxe), J. Moncrieffe (gentleman of the period), H. Mordaunt (gentleman, temps. Louis XV.), T. J. Meyrick (gentleman, temps. XV.), F. Menzies, H. Molesworth, Basil Montgomery (Courtier, Louis XV.), John Monckton, R. O. Milne (Chevan-le’ ger de la Garde du Roi, Louis XVI.), H. du C. Norris (Court dress, Louis XV.), J. Norris (Marquis of France), Norton (Louis XVI. costume), C. S. Paulet (modern Court dress), Quinton-Dick (Mousquetaire), Arthur Paget, Ralph Paget (dress of Louis XV.), Oswald Petre (modern Court dress), George Peel, W. R. W. Peel, G. R. Powell (Court dress), Mark J. Paget (gentleman of period of Louis XV.), Ramsden (Mons. de Calonne), L. G. Scott (Mousquetaire), H. Spender Clay (Court dress of the period), Smythe, Shaw, S. Sanders (Mousquetaire), F. Shaw, S. O. Smith (modern court dress), M. Oswald Smith (gentleman of the Court of Louis XVI.), Cameron Skinner, W. L. Thursby (Mousequetaire), C. J. H. Tower (officier Gardes Suisses), Tree, Thursby-Pelham, Tower (officier Gardes Suisses), J. H. Wheatley (modern Court dress), Read Walker (officier d'Infanterie), Francis Williams, Montague Wood, West, Gordon Wood and Anthony White.

DESCRIPTION OF COSTUMES.

Appended are descriptions of the chief costumes worn:—

E&lt;small&gt;ARL OF&lt;/small&gt; W&lt;small&gt;ARWICK&lt;/small&gt;.

Field Marshal, Louis XVI. — Military coat with the long skirts of the period, having turned-back revers of white cloth, laced, after the military fashion, with gold, white knee-breeches and silk stockings. The cravat and ruffles were of lace. A white wig in Louis XVI. style, and a three-cornered black beaver hat with gold braid all round the brim, which was edged with small white ostrich plumes, completed a handsome and artistic costume. A sword was worn in a swordbelt of the period.

C&lt;small&gt;OUNTESS OF&lt;/small&gt; W&lt;small&gt;ARWICK&lt;/small&gt;.

Marie Antoinette — Gala costume. Rich brocade dress, with a ground of a delicate tint of pearl, with a suggestion of pink in it, the design roses in gold, with gold foliage, lilies in white, some small blue flowers and clusters of pink blossoms, with bright old- world green as foliage. The skirt was quite plain, and the [6, Col. 2c–3a] bodice drawn into shaped points at the hips, so that it sat right out at either side. It was full, and yard on the ground at the back. The bodice was finished with points back and front, and was cut with absolute perfection. Round the shoulders were full soft folds of gold-flecked French silk muslin edged with beautiful gold lace. The sleeves were plain and tight to the elbow, whence they were finished with triple frills of the gold-flecked muslin, each bordered with gold lace, and with ruffles falling from beneath the frills of point d’Alençon lace. The frills were headed with bands of gold embroidery. At the back, suspended from both shoulders by gold cords, was a beautiful Court mantle of deep rich blue velvet, not so pale as turquoise nor so strong as the shade we call Royal, but a bright lovely colour. This was embroidered all over with a raised design of fleur de lys in dull and burnished gold, and was lined with the same blue velvet. The hair was dressed high with a magnificently embroidered head-dress. Her ladyship wore the Warwick family diamonds round her neck as a collar, a turquoise velvet cap clasped with jewels on her white coiffure and a bandeau of family jewels under her cap. Her court mantle was fastened at the shoulders with a tiara of diamonds widened out so as to clasp the cloak from shoulder to shoulder.

T&lt;small&gt;HE&lt;/small&gt; C&lt;small&gt;OMTESSSE&lt;/small&gt; A&lt;small&gt;HLEFELDT&lt;/small&gt;.

Dame de Ia temps Louis XVI. — cream silk petticoat, with front of real ha[n]d-worked silk embroidery, done in the time of Louis XVI., the design being convolvulus and other flowers wrought in dull pink, blue, and green silks, and feathers tied with true lover’s knots. There was a bright shell pink tunic-shaped overdress, with Watteau back, edged with real Brussels, white silk stomacher, large pink bows in front and on each hip; and wreath of pink roses.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;ERALD&lt;/small&gt; A&lt;small&gt;RBUTHNOT&lt;/small&gt;.

Brocade gown with Watteau back and paniers, cerise satin petticoat, studded with large blue satin bows, cerise velvet stomacher, fechu of Brussels lace; head-dress, cap of cerise velvet and blue plumes.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RMITAGE&lt;/small&gt;.

Dame de la temps Louis XV. — Light grey satin dress brocaded with bunches of cyclamen, roses with green leaves, and ornamented with velvet to match, and groups of yellow and cyclamen roses.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RMITAGE&lt;/small&gt; (K&lt;small&gt;IRROUGHTEN&lt;/small&gt;).

Lady of the Court of Louis XVI. — Bodice and short train, with Watteau plait of pale heliotrope and green brocade, with large revers of heliotrope satin, and bodice trimmed with petunia velvet and petunia and yellow roses and lace; petticoat of heliotrope satin and lace flounces, diamond and sapphire ornaments; hair ''poudré'', with heliotrope feathers.

M&lt;small&gt;ISS&lt;/small&gt; A&lt;small&gt;RMSTRONG&lt;/small&gt;.

''A la'' Watteau. — White watered silk, brocaded with stripes and clusters of roses. The front of the petticoat was draped with blue chiffon, and edged at the bottom with pink roses, bodice with blue satin bows in front, and on the shoulders and neck, and sleeves trimmed with full white chiffon and pink roses, powdered hair, wreath of roses and blue bow.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RMSTRONG&lt;/small&gt;. French Marquise. — Handsome cream coloured real old brocade with black velvet front, and trimmings of very old point de Venice, and festoons of pink roses.

[6, Col. 3a–b]

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;VERARD&lt;/small&gt; B&lt;small&gt;ROWNE&lt;/small&gt;.

Brocaded satin with silver stripes.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;RANK&lt;/small&gt; B&lt;small&gt;IBBY&lt;/small&gt;.

Lady of the Court of Louis XVI. — White satin dress, with skirt draped with old lace, pink chiffon sash embroidered with silver; diamond buttons on corsage.

C&lt;small&gt;APTAIN&lt;/small&gt; B&lt;small&gt;RINKLEY&lt;/small&gt; (W&lt;small&gt;ARWICK&lt;/small&gt;).

Court dress of the period — Claret-coloured coat and knee-breeches, white silk embroidered waistcoat, white silk stockings and old paste buckled shoes, Louis XVI. wig, and Court sword.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;RINKLEY&lt;/small&gt;.

Marquise du Deffant — Train of white brocade embroidered in roses and forget-me-nots, the paniers lined and turned back with green satin and guipure; petticoat of pink satin and old Honiton lace, trimmed with pink roses; Louis XV. wig, with roses and diamonds.

T&lt;small&gt;HE&lt;/small&gt; E&lt;small&gt;ARL OF&lt;/small&gt; B&lt;small&gt;URFORD&lt;/small&gt;.

Military costume of Louis XV.’s time — White cloth with pale blue facings, trimmed handsomely and effectively with gold.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;URIEL&lt;/small&gt; B&lt;small&gt;ELL&lt;/small&gt;.

Princesse de Lamballe — Pink satin brocade, white petticoat, pink roses; hair ''poudré''.

M&lt;small&gt;ISS&lt;/small&gt; N. B&lt;small&gt;ATTYE&lt;/small&gt; (L&lt;small&gt;ONDON&lt;/small&gt;).

English dress of the period — Light blue satin dress[,] lace fichu, large black velvet hat with white ostrich feathers.

M&lt;small&gt;ISS&lt;/small&gt; &lt;small&gt;DE&lt;/small&gt; C&lt;small&gt;APELL&lt;/small&gt; B&lt;small&gt;ROOKE&lt;/small&gt;.

Lady of the time of Louis XV. — Pink figured silk, over white satin skirt, edged with gold gimp, Watteau back; hair powdered.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;EECH&lt;/small&gt;.

Garde du Corps du Roi — Crimson coat, white facings, and gold lace.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;EECH&lt;/small&gt;.

Madame de La Fayette — Old brocade, with lace and crimson roses and black velvet bows.

M&lt;small&gt;R&lt;/small&gt; F. C. H&lt;small&gt;UNTER&lt;/small&gt; B&lt;small&gt;LAIR&lt;/small&gt; (L&lt;small&gt;EAMINGTON&lt;/small&gt;).

Mousquetaire Uniform Louis XV., in white, scarlet, and gold.

C&lt;small&gt;APTAIN&lt;/small&gt; J&lt;small&gt;IM&lt;/small&gt; B&lt;small&gt;ARRY&lt;/small&gt; (L&lt;small&gt;ONDON&lt;/small&gt;).

Mousquetaire Louis XV. — Black knee-breeches, light blue coat and waistcoat, faced with white, and trimmed with gold lace.

M&lt;small&gt;R&lt;/small&gt; C&lt;small&gt;ARYL&lt;/small&gt; C&lt;small&gt;RAVEN&lt;/small&gt;.

Mousquetaire, Louis XVI. — White and gold.

T&lt;small&gt;HE&lt;/small&gt; E&lt;small&gt;ARL OF&lt;/small&gt; C&lt;small&gt;HESTERFIELD&lt;/small&gt;.

Court costume — Coat of pale blue corded silk, the cuffs, pocket flaps, and fronts all richly wrought with gold, while the buttons were old paste and amethysts. The knee-breeches were blue silk, and the blue silk stockings were clocked with gold, and Court shoes were worn, with diamond buckles, The waistcoat was of yellow satin, brocaded with pink rosebuds, and having old paste and amethyst buttons. A jabot of old lace was pinned with a diamond brooch, and the ruffles were of similar lace. A white wig was worn with a three-cornered gold-laced and white-plumed hat.

[6, Col. 3b–c]

C&lt;small&gt;OUNTESS&lt;/small&gt; C&lt;small&gt;AIRNS&lt;/small&gt;.

Duchess de Bouillon — Dress of light-hued satin, with relief of pink diamonds, and pink roses in the hair.

M&lt;small&gt;R&lt;/small&gt; C&lt;small&gt;HAPLIN&lt;/small&gt;.

Court suit of green silk velvet, with embroidered vest and white wig.

L&lt;small&gt;ORD&lt;/small&gt; C&lt;small&gt;HURCHILL&lt;/small&gt;.

Court costume — Blue brocade, with steel buttons and knot of ribbon, fringed with silver on one shoulder; white satin waistcoat and blue knee-breeches.

L&lt;small&gt;ORD&lt;/small&gt; C&lt;small&gt;LIFTON&lt;/small&gt;.

Officer of the Guards, Louis XV. — Coat of pale green cloth, turned back with crimson, and laced with gold.

M&lt;small&gt;ISS&lt;/small&gt; S&lt;small&gt;PENDER&lt;/small&gt; C&lt;small&gt;LAY&lt;/small&gt;.

Madme. Lamballe — A pretty pink and white brocade dress in the style of Louis XV., and with large hat, trimmed with ostrich plumes and roses. She carried a white wand surmounted by roses.

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ARLETON&lt;/small&gt;.

Watteau, Louis XV. — Blue silk brocaded dress, with little pink roses, and pink satin petticoat with Watteau pleat, hair arranged with pink wreath of roses and pink feathers.

M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;ARTWRIGHT&lt;/small&gt;.

White satin dress, trimmed with sable and point de gaze lace; musseline de soie fichu edged with lace, and caught up with clusters of pink roses.

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;HAMPION&lt;/small&gt; (N&lt;small&gt;ORFOLK&lt;/small&gt;).

Dress of old brocade; petticoat of Rose de Barri satin, trimmed with pearls and lace; lace fichu, large rose hat with plumes, and pearl ornaments.

M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;HAMBERLAYNE&lt;/small&gt; (S&lt;small&gt;TONEY&lt;/small&gt; T&lt;small&gt;HORPE&lt;/small&gt;).

Marquise Louis XV. — Pink satin petticoat, yellow flowered silk bodice, and train from the shoulders; pink satin ribbon and diamonds in the powdered hair.

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ARUTHERS&lt;/small&gt; (W&lt;small&gt;ARDINGTON&lt;/small&gt;, B&lt;small&gt;ANBURY&lt;/small&gt;).

Short-waisted dress of period Louis XV.; yellow brocade over yellow satin petticoat; old lace and roses.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;ERTRAM&lt;/small&gt; C&lt;small&gt;HAPLIN&lt;/small&gt;.

Period Louis XVI. — White satin coat and breeches, pink satin waistcoat.

M&lt;small&gt;R&lt;/small&gt; Q&lt;small&gt;UENTIN&lt;/small&gt; D&lt;small&gt;ICK&lt;/small&gt;.

Officer of the Household of Louis XV. — White cloth uniform, faced with blue, and braided with gold.

M&lt;small&gt;ISS&lt;/small&gt; D&lt;small&gt;RUMMOND&lt;/small&gt; (S&lt;small&gt;HERBOURNE&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;).

A blue brioche, brocaded with pink roses and leaves, and gaily trimmed with pink Banksia roses, petticoat of white satin, flounced with lovely Brussels lace, bodice [sic] of bioche silk, with white front trimmed with roses and old-fashioned gauze, necklace of roses and pearls, and wreath of roses.

HON. M&lt;small&gt;RS&lt;/small&gt; H&lt;small&gt;UBERT&lt;/small&gt; D&lt;small&gt;ORMER&lt;/small&gt; (L&lt;small&gt;ONDON&lt;/small&gt;).

Court dress Louis XVI. — Petticoat of pink satin, point lace flounce; overdress of dark red satin, in paniers, looped with red and pink roses, diamond and pearl ornaments.

M&lt;small&gt;R&lt;/small&gt; J. S. D&lt;small&gt;UGDALE&lt;/small&gt;, Q.C.

Recorder’s Court dress of the period — Black silk gown with lace ruffles, black silk stockings, buckle shoes, and full bottomed wig.

M&lt;small&gt;RS&lt;/small&gt; J. S. D&lt;small&gt;UGDALE&lt;/small&gt;.

A very handsome bright blue silk brocaded with white, and made ''à la'' Pompadour, with white satin front trimmed with dark fur, the bodice made with pearl trimmings, and a white muslin fichu tied at one side under a bunch of pink roses, hair dressed with blue feathers, wreath of pink roses, and a tiara of diamonds in front.

[6, Col. 3c–4a]

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ONSTANCE&lt;/small&gt; D&lt;small&gt;ORMER&lt;/small&gt; (H&lt;small&gt;ASTINGS&lt;/small&gt;.)

Marquise Louis XVI. — Gown of white silk brocaded with roses, Watteau back. Pearl ornaments.

L&lt;small&gt;ADY&lt;/small&gt; A&lt;small&gt;NGELA&lt;/small&gt; S&lt;small&gt;T&lt;/small&gt;. C&lt;small&gt;LAIR&lt;/small&gt; E&lt;small&gt;RSKINE&lt;/small&gt;.

Lady Mary Campbell — White muslin costume, with broad blue silk sash.

S&lt;small&gt;IR&lt;/small&gt; A&lt;small&gt;RCHIBALD&lt;/small&gt; E&lt;small&gt;DMONSTONE&lt;/small&gt;.

Mousquetaire — White cloth uniform, faced with blue and showing a blue waistcoat, the whole having a large amount of silver military braiding.

M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;INDSAY&lt;/small&gt; E&lt;small&gt;RIC&lt;/small&gt;-S&lt;small&gt;MITH&lt;/small&gt; (E&lt;small&gt;LFINSWARD&lt;/small&gt;, H&lt;small&gt;AYWARD'S&lt;/small&gt; H&lt;small&gt;EATH&lt;/small&gt;. [sic no paren]

Pompadour dress, period Louis XVI. — Yellow brocade, and white satin petticoat.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;AIRFAX&lt;/small&gt;-L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt;).

A Marquise. — Rich white brocade dress, with blue and straw brocade saque, edged with Brussels lace, and Brussels lace flounce, Vandycked round, petticoat with pink roses, lace ruffles and fichu, and pink roses and diamonds in the hair completed the costume.

M&lt;small&gt;R&lt;/small&gt; F&lt;small&gt;AIRFAX&lt;/small&gt;-L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt;).

Colonel George Lucy — Red lilac-coloured cloth suit, Court dress of the period, edged with silver lace, and belonged to Colonel G. Lucy in 1744; silk stockings of the same colour, high-heeled shoes, with diamond buckles, and knee buckles, lace ruffles, and cravat.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;ASIL&lt;/small&gt; J. F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Mousquetaire — Uniform of dark green cloth, faced with tan, and trimmed with silver, old point d’Alencon ruffles, tan silk sash, and cross belt of tan and silver.

M&lt;small&gt;R&lt;/small&gt; B. F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Mousquetaire — White uniform, with orange velvet facings braided with gold, crossbelt of white and gold, a yellow sash, and the high black leather boots of the period.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;-F&lt;small&gt;ITZGERALD&lt;/small&gt;.

Dress of the real old brocade of Louis XV.’s reign.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. C&lt;small&gt;LEMENT&lt;/small&gt; F&lt;small&gt;INCH&lt;/small&gt;.

Gentleman of the period Louis XV. — Coat of blue watered silk, with silver trimming, satin breeches to match, white satin vest, and black hat decked with silver.

M&lt;small&gt;ISS&lt;/small&gt; B&lt;small&gt;EATRICE&lt;/small&gt; F&lt;small&gt;ITZHERBERT&lt;/small&gt;.

Dress of pale blue satin, lined with pink, with pink roses on the corsage, Louis XVI. period. She wore a beautiful pearl necklace, with large diamond pendant, and two diamond sprays, all of which were given by George IV. to Mrs Fitzberbert.

M&lt;small&gt;AJOR&lt;/small&gt; F&lt;small&gt;OSBERY&lt;/small&gt; (W&lt;small&gt;ARWICK&lt;/small&gt;).

Mousquetaire — Claret-coloured tunic, with salmon-colour cuffs, lace ruffles, &amp;c.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;OSBERY&lt;/small&gt; (W&lt;small&gt;ARWICK&lt;/small&gt;).

Marquise — Pink satin petticoat, covered with lace, grey and pink brocaded bodice and train, pink roses and ostrich plume in powdered hair.

L&lt;small&gt;ADY&lt;/small&gt; G&lt;small&gt;ERARD&lt;/small&gt;.

Duchess de Pognac — Dress of pale blue brocade, decked with small roses, with front of pink satin; fichu of muslin and lace, and stomacher of lace and roses.

[6, Col. 4a–b]

L&lt;small&gt;ADY&lt;/small&gt; E&lt;small&gt;VA&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Polichinelle, Louis XV. — White satin gown ornamented at the bottom with a trelliswork of silver, studded with small pink roses; corsage to correspond, and fastened across the stomacher by large diamond hooks and eyes.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. S&lt;small&gt;IDNEY&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Officer of the Regiment of Swiss Guards — Coat of pale blue cloth, nearly bordering on green — quite a turquoise shade. The revers were white, and turned back from a white waistcoat braided with gold. The braiding was continued down the white revers of the coat and on the skirts; white satin knee-breeches, silk stockings, Court shoes, white wig, and three-cornered hat, trimmed with gold braid and white ostrich feathers, completed one of the most effective of military attires. A sword was, of course, worn.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. L&lt;small&gt;OUIS&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Mousquetaire, Louis XV. — Claret-coloured coat, laced with gold over white; a white silk sash, sword-belt of red cloth with gold, white knee-breeches, Court shoes, silk stockings, and the wig and three-cornered hat of the time.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;OUIS&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Dress of the period — Petticoat of deep rose-petal pink satin, with a full flounce of white lace headed by trails of roses; over-dress of white satin, brocaded with a design of roses and lined with pale-green satin, pointed bodice showing a pink vest laced across, and ruffles and fichu of Mechlin lace to correspond with the flounce. The hair was powdered and dressed high, with an ornament of roses and diamonds at one side.

C&lt;small&gt;APTAIN THE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. A&lt;small&gt;LWYN&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Mousquetaire — Coat of scarlet cloth, cuffs and fronts turned back with white and laced with gold, and broad red silk sash, white knee-breeches, silk stockings, and Court shoes.

THE H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;LWYN&lt;/small&gt; G&lt;small&gt;REVILLE&lt;/small&gt;.

Dame de la Court Louis XV. — Over-dress of pink mirror velvet bordered with dark fur, opening over a front of cream satin, long pink velvet sleeves with roses and fichu of fine old lace on the corsage; hair dressed ''a la'' Princess Lambale.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;ARKELL&lt;/small&gt;.

Grande Dame — Blue shot-satin dress adorned with point, d’Alençon lace, veiled with silver tissue under white gauze and tied up by a wide blue chiffon sash caught at the arm-holes with diamond buttons; ornaments, enamelled medallions set in diamonds.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;ASKELL&lt;/small&gt;.

Gainsborough costume — White satin and blue chiffon, Louis XVI.; old diamond necklace.

M&lt;small&gt;ISS&lt;/small&gt; G&lt;small&gt;ASKELL&lt;/small&gt;.

Costume, Louis XV. — Blue and pink costume of that period, with a very large black velvet hat, trimmed with blue feathers.

M&lt;small&gt;ISS&lt;/small&gt; G&lt;small&gt;ORDON&lt;/small&gt;.

All in white, lined with blue satin, the front of the bodice made of fine muslin, caught up with small pink roses; and a little wreath of pink roses in the hair, and diamonds.

M&lt;small&gt;ARCHIONESS OF&lt;/small&gt; H&lt;small&gt;ERTFORD&lt;/small&gt;.

Lady time of Louis XVI. — Black velvet dress and train, white satin front covered with old point lace; long pointed bodice with lace fichu, long velvet sleeves lined with white satin, front of dress covered with diamonds. White full-dress wig, with lace lappets and diamonds.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;RNEST&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt; (G&lt;small&gt;ROVE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;.)

Marquise Louis XVI. — Overdress of light green satin brocaded with pink roses and faced with pale pink satin over white satin petticoat, with lace flounce beaded with pink roses. Pearl and diamond ornaments.

[6, Col. 4b–c]

M&lt;small&gt;R&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt; (G&lt;small&gt;ROVE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

English Court dress — Black velvet, point lace ruffles.

M&lt;small&gt;R&lt;/small&gt; H. T. H&lt;small&gt;ICKMAN&lt;/small&gt;.

Court dress, time of Louis XV. — Black velvet coat, knee-breeches, trimmed with white lace.

M&lt;small&gt;RS&lt;/small&gt; H&lt;small&gt;UTTON&lt;/small&gt;.

Court costume — Dress with paniers of pale-green brocade over a white satin petticoat having a flounce of lace headed by roses.

M&lt;small&gt;ISS&lt;/small&gt; H&lt;small&gt;ODGSON&lt;/small&gt;.

''A la'' Watteau — Sang de boeuf coloured satin petticoat, trimmed with old lace, caught up with roses; a corset and polonaise of rose figured satin, the latter trimmed with deep revers of green satin; white wig; ornaments, pearls and diamonds.

M. J&lt;small&gt;OOSTENS&lt;/small&gt; (B&lt;small&gt;ELGIAN&lt;/small&gt; L&lt;small&gt;EGATION&lt;/small&gt;, L&lt;small&gt;ONDON&lt;/small&gt;).

Courtier, Louis XVI. — White satin knee breeches, claret velvet coat and waistcoat, point lace ruffles.

M&lt;small&gt;RS&lt;/small&gt; H. J&lt;small&gt;OLIFFE&lt;/small&gt; (G&lt;small&gt;OLDICOTE&lt;/small&gt;).

Marquise of Louis XV. — Blue silk brocaded dress with pink roses, the petticoat of pink satin trimmed with white lace and pink roses, and the over-dress turned back with green satin edged with gold embroidery.

L&lt;small&gt;ORD&lt;/small&gt; K&lt;small&gt;ENYON&lt;/small&gt;.

Officier of the Regiment du Roi, Louis XVI. — Handsome dress of white cloth faced with pale-blue and laced with gold.

T&lt;small&gt;HE&lt;/small&gt; [[Social Victorians/People/Keppel|H&lt;small&gt;ON&lt;/small&gt;. G&lt;small&gt;EORGE&lt;/small&gt; K&lt;small&gt;EPPEL&lt;/small&gt;]] (2, Wilton Crescent, London).

Mousequetaire — White cloth, with an exquisitely jewelled Order around his neck.

T&lt;small&gt;HE&lt;/small&gt; [[Social Victorians/People/Keppel|H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;EORGE&lt;/small&gt; K&lt;small&gt;EPPEL&lt;/small&gt;]].

Lady, time Louis XVI. — Gown of shell pink satin, pointed bodice, with full paniers, of antique brocade of the real deep rose shade known as du Barri sewn with silver thread and bouquets of roses. Full petticoat, of dull creamy-tinted satin, with a deep band round it of silver tissue embroidered with garlands of small leafless roses. The sleeves had long ruffles of old lace. The hair was powdered and dressed elaborately and high, with three rose du Barri feathers in it and a little cap of lace. The shoes were of pink satin, with diamond buckles.

T&lt;small&gt;HE&lt;/small&gt; E&lt;small&gt;ARL OF&lt;/small&gt; L&lt;small&gt;ONSDALE&lt;/small&gt;.

M. de Capuisan, Keeper of the Royal Stud, Louis XV. — Coat and knee-breeches were of ruby velvet, richly wrought with gold and with rare and valuable paste buttons on the former, while the vest was of pearl-white satin edged with very beautiful embroidery, white silk stockings, Court shoes with diamond buckles, lace ruffles and jabot with diamond brooch, jewelled hilted Court sword, and white wig with three-cornered hat with gold lace and white plumes.

L&lt;small&gt;ORD&lt;/small&gt; L&lt;small&gt;OVAT&lt;/small&gt;.

Comte d’Artagnas — Military costume of the period in white, faced with blue and laced with gold; an embroidered pouch slung from his belt, embroidered in gold, silk stockings, Court shoes, white wig, sword, and three-cornered hat.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;UCY&lt;/small&gt; (C&lt;small&gt;HARLECOTE&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

Madenoiselle de Montmirail — White satin petticoat, with deep flounce of Brussels lace, caught up with pompom; pink roses; witite brocade saque, laced with pearls; lace ruffles and fichu; large black velvet hat and plumes.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;AKIN&lt;/small&gt;.

Watteau costume — White satin brocade with white satin petticoat, festooned with roses.

[6, Col. 4c–5a]

T&lt;small&gt;HE&lt;/small&gt; P&lt;small&gt;ORTUGUESE&lt;/small&gt; M&lt;small&gt;INISTER&lt;/small&gt; (D&lt;small&gt;ON&lt;/small&gt; L&lt;small&gt;OUIE DE&lt;/small&gt; L&lt;small&gt;OUVERAL&lt;/small&gt; [Soveral]).

Mousquetaire of the 2nd Company of the Royal Household, Louis XV. — Scarlet, laced with gold and relieved with white, high black Mousquetaire boots, a plastron [sic] embroidered with the Royal arms, white wig, three-cornered hat gold-laced and white-plumed, sword.

M&lt;small&gt;R&lt;/small&gt; R&lt;small&gt;ICHARD&lt;/small&gt; L&lt;small&gt;ANT&lt;/small&gt; (N&lt;small&gt;AILCOTE&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;, C&lt;small&gt;OVENTRY&lt;/small&gt;).

Present day Court dress, with Louis XV. white wig.

H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;HANDOS&lt;/small&gt; L&lt;small&gt;EIGH&lt;/small&gt;.

The Duchesse de Polignac, period Louis XVI. — Petticoat of pale pink brocade, with corsage and train of sapphire blue velvet and lace fichu.

M&lt;small&gt;ISS&lt;/small&gt; V&lt;small&gt;IOLET&lt;/small&gt; L&lt;small&gt;EIGH&lt;/small&gt;.

Mdlle. de Chévreuse, period Louis XV. — Petticoat of white satin, with lattice work of pink roses, corsage with paniers and Watteau plait of sxy blue satin, lined with pale pink satin; powdered hair, with small wreath of roses, pearls, and white plume.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;RNEST&lt;/small&gt; L&lt;small&gt;ITTLE&lt;/small&gt; (&lt;small&gt;OF&lt;/small&gt; N&lt;small&gt;EWBOLD&lt;/small&gt; P&lt;small&gt;ACEY&lt;/small&gt;).

Lady of the reign of Louis XV. — Train of brocade, in white and purple, over dress of satin, trimmed with old lace and pink roses.

M&lt;small&gt;R&lt;/small&gt; L&lt;small&gt;INDSAY&lt;/small&gt; (R&lt;small&gt;ED&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;, B&lt;small&gt;ARFORD&lt;/small&gt;).

Court dress late 18th century, composed of black velvet.

M&lt;small&gt;RS&lt;/small&gt; E&lt;small&gt;DWARD&lt;/small&gt; L&lt;small&gt;UCAS&lt;/small&gt; (15, L&lt;small&gt;ENNOX&lt;/small&gt; G&lt;small&gt;ARDENS&lt;/small&gt;, L&lt;small&gt;ONDON&lt;/small&gt;)

Lady of Court Louis XVI. — Pink brocade, with green satin petticoat.

M&lt;small&gt;RS&lt;/small&gt; M&lt;small&gt;ORTON&lt;/small&gt; P. L&lt;small&gt;UCAS&lt;/small&gt; (T&lt;small&gt;HE&lt;/small&gt; O&lt;small&gt;AKS&lt;/small&gt;).

Court lady, Louis XV. — Black velvet bodice and train; and white satin petticoat trimmed Brussels lace and roses.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;ORTON&lt;/small&gt; P. L&lt;small&gt;UCAS&lt;/small&gt;.

Gentleman of the period — Black velvet Court dress, trimmed with steel, white satin waistcoat, and knee- breeches.

M&lt;small&gt;ISS&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt;-K&lt;small&gt;AYE&lt;/small&gt;.

Period Louis XVI. — Blue silk brocade, white petticoat, pink roses.

M&lt;small&gt;R&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt; L&lt;small&gt;ISTER&lt;/small&gt;-K&lt;small&gt;AYE&lt;/small&gt;.

Period Louis XVI. — Plum-coloured velvet coat and breeches, brocaded satin waistcoat.

M&lt;small&gt;RS&lt;/small&gt; L&lt;small&gt;ESLIE AND&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; E. L&lt;small&gt;YON&lt;/small&gt;.

A la Watteau — Dresses of white and gold flowered brioche, with plain white satin fronts handsomely painted, the design being large, full blown pink roses and butterflies; hip panniers, and the bodices were of gold brioche, with white satin fronts trimmed with lace, large pearls, and Cairngorm jewels, neck ruffles edged with pearls and gold, and aigrettes and velvet bow of pink and gold.

C&lt;small&gt;APTAIN THE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. H&lt;small&gt;EDWORTH&lt;/small&gt; L&lt;small&gt;AMBTON&lt;/small&gt;.

Courtier of Louis XVI. — Coat of bronze satin, richly-embroidered knee-breeches, and richly-embroidered waistcoat of pearl white corded silk, lace ruffles and jabot, and all the details to correspond.

D&lt;small&gt;UKE&lt;/small&gt; &lt;small&gt;OF&lt;/small&gt; M&lt;small&gt;ANCHESTER&lt;/small&gt;.

Marquis de Grammont — White satin knee-breeches, white silk stockings, shoes with paste buckles, a coat of real old Louis XVI. brocade, with a design of feathers in gold on a cream-coloured background and of pink rose sprays, the fronts, pockets, and cuffs all richly wrought in gold, and with fine old paste buttons. He had also an exquisite real lace jabot, fastened with an antique diamond brooch, a white satin waistcoat finely emboidered, white wig, and black three-cornered hat.

S&lt;small&gt;IR&lt;/small&gt; C. M&lt;small&gt;ORDAUNT&lt;/small&gt;, B&lt;small&gt;ART&lt;/small&gt;.

Gentleman of the period — Coat and knee-breeches of black velvet, with waistcoat of black broché, all three trimmed with cut steel buttons; lace ruffles and necktie; white wig, three-cornered hat, gold-headed cane finished with red and green ribbons; black silk stockings, and shoes with silver buckles.

L&lt;small&gt;ADY&lt;/small&gt; M&lt;small&gt;ORDAUNT&lt;/small&gt; (W&lt;small&gt;ALTON&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Princess de Lamballe — Bodice and train of rich white brocade, trimmed with lace and wreaths of small pink roses, the train being edged with them all round; over pink satin petticoat, ornamented with bows of lace, lace flounce, and vandykes of pink roses. Hair powdered, over a cushion with curls in the neck, with wreath of pink roses, and loops of pink ribbon: White satin shoes, with pink heels and pink rosettes. Row of pearls round neck.

M&lt;small&gt;R&lt;/small&gt; B&lt;small&gt;ASIL&lt;/small&gt; M&lt;small&gt;ONTGOMERY&lt;/small&gt;.

Mousquetaire — Uniform of white cloth and Royal blue velvet, embroidered in gold.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;ONCRIEFFE&lt;/small&gt;.

Gentleman of the period — Coat of pale ''vieux'' rose brocade trimmed with silver, satin breeches to match, sea-green satin waistcoat lightly worked over in silver, ruffles and jabot of white lace.

M&lt;small&gt;R&lt;/small&gt; R&lt;small&gt;OBERT&lt;/small&gt; O. M&lt;small&gt;ILNE&lt;/small&gt;.

Chevau léger de la Garde de Roi, Louis XVI. — Scarlet cloth coat, faced with gold lace, knee-breeches, three-cornered hat with ostrich feathers, large Hessian boots.

M&lt;small&gt;R&lt;/small&gt; J. M&lt;small&gt;ONCRIEFFE&lt;/small&gt;.

Courtier Louis XVI. — Coat of yellow brocade, heliotrope knee-breeches, and gold laced heliotrope waistcoat. The coat had paste buttons, and the details as to wig, sword, ruffles, shoes, stockings, and three-cornered hat, were all correctly carried out.

M&lt;small&gt;R&lt;/small&gt; H. M&lt;small&gt;ORDAUNT&lt;/small&gt; (W&lt;small&gt;ALTON&lt;/small&gt;).

Courtier of period Louis XV. — Coat of mauve brocade, full skirted and embroidered with gold, the waistcoat of mauve satin embroidered in floral design, knee breeches of shot manve and gold, silk stockings, Court shoes, powdered hair, lace ruffles, jabot and sword made up a costume, correct in every particular, of one of Louis XVI.’s courtiers.

M&lt;small&gt;RS&lt;/small&gt; H&lt;small&gt;ERBERT&lt;/small&gt; M&lt;small&gt;OLESWORTH&lt;/small&gt; (D&lt;small&gt;EVONPORT&lt;/small&gt;, D&lt;small&gt;EVON&lt;/small&gt;).

Court dress of Louis XVI. — Yellow satin gown in paniers, trimmed with light blue velvet and pink roses. Diamond ornaments.

C&lt;small&gt;APTAIN&lt;/small&gt; H&lt;small&gt;ERBERT&lt;/small&gt; M&lt;small&gt;OLESWORTH&lt;/small&gt; (F&lt;small&gt;IELD&lt;/small&gt; A&lt;small&gt;RTILLERY&lt;/small&gt;, D&lt;small&gt;EVONPORT&lt;/small&gt;, D&lt;small&gt;EVON&lt;/small&gt;).

Mousquetaire, Louis XV. — Claret velvet costume, slashed with pink satin.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;URRAY&lt;/small&gt; (D&lt;small&gt;IDDINGTON&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Short eau-de-nil satin skirt, trimmed band of passementarie, bodice and paniers of pink and white broché, looped up with pink satin bows; trimmed ruching of mousselaine-de-soie over pink satin; white wig; pink roses; ornaments, pearls and diamonds.

L&lt;small&gt;ADY&lt;/small&gt; A&lt;small&gt;NN&lt;/small&gt; M&lt;small&gt;URRAY&lt;/small&gt;.

Madam de Pompadour (after La Tour). — Dress was of white and gold, brocaded with large sprays of coloured flowers trimmed with ruching of vieux rose satin and lace; white wig; lace cap, with lappets and bow; ornaments, pearls and emeralds.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;ILLER&lt;/small&gt;-M&lt;small&gt;UNDAY&lt;/small&gt; [sic].

Pink dress with fichu.

M&lt;small&gt;RS&lt;/small&gt; M&lt;small&gt;ILLER&lt;/small&gt;-M&lt;small&gt;UNDAY&lt;/small&gt; [sic] (Shipley Hall, Derby).

Court dress — Petticoat of yellow satin, draped with white silk muslin edged with little pink roses; over-bodice low and double-breasted, of blue velvet with miniature and paste buttons, and revers faced with pale-hued brocade and finished with lace; headdress of Marie Theresè, Queen the Sicilies, a silk handkerchief tied in a knot at one side, very like the portrait of Madame le Brun in the Louvre Gallery done by herself; also plumes and jewels.

M&lt;small&gt;RS&lt;/small&gt; J. M&lt;small&gt;ENZIES&lt;/small&gt;.

Duchess d’Angoulêne — Dress of pink satin in a loose full skirt, with a frill round the hem, a white muslin bodice, and a blue sash. Her bonnet, one of the period, was of blue velvet, with purple and blue ostrich plumes.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;ERESFORD&lt;/small&gt; M&lt;small&gt;ELVILLE&lt;/small&gt;.

Dress of green satin, lined with rose silk, and trimmed with roses and Venetian point lace.

L&lt;small&gt;ORD&lt;/small&gt; R&lt;small&gt;ICHARD&lt;/small&gt; N&lt;small&gt;EVILLE&lt;/small&gt;.

Duc de Lanzun — Military coat of violet cloth, faced with white and braided with silver, violet silk stockings and knee-breeches, a white satin gold-embroidered waistcoat, silver epaulets, a sword in sword-belt, and all details to correspond.

MR H. D&lt;small&gt;U&lt;/small&gt; C. N&lt;small&gt;ORRIS&lt;/small&gt;.

Marquis of France, temps Louis XVI. — Coat and breeches rose silk, large diamond buttons to waistcoat, and superb diamond buckles on shoes. Sword hilt jewelled in steel.

M&lt;small&gt;R&lt;/small&gt; J. N&lt;small&gt;ORRIS&lt;/small&gt;.

Marquis of France — Green brocade velvet coat and breeches, white satin waistcoat, diamond buckles to shoes.

M&lt;small&gt;RS&lt;/small&gt; O&lt;small&gt;SBORNE&lt;/small&gt;.

Madame de Pompadour. — White silk dress, flowered with dandelions, and made with a Watteau back faced with pink, the petticoat of Rose du Barri silk with |ace flounces and jewelled trimmings, bodice also trimmed with lace and jewels, and long puffed tulle sleeves, drawn with narrow black ribbons; feathers in hair; gold and jewelled girdle.

M&lt;small&gt;ISS&lt;/small&gt; O&lt;small&gt;SBORNE&lt;/small&gt;.

Madame de Pompadour — In garden dress, pink petticoat with bands of black velvet at the edge, and overdress of white, brocaded with small rosebuds, large black velvet bow on one side under large pink roses, bodice ''en suite'' with lace and roses, and black bow on the right shoulder, and large white chip hat, wreathed in roses, and with black velvet and roses under the brim.

S&lt;small&gt;IR&lt;/small&gt; A&lt;small&gt;LGERNON&lt;/small&gt; O&lt;small&gt;SBORN&lt;/small&gt;, C&lt;small&gt;HICKSANDS&lt;/small&gt; P&lt;small&gt;RIORY&lt;/small&gt;.

Civilian costume — Black velvet, with steel buttons, lace cravat, white wig and sword.

P&lt;small&gt;RINCESS&lt;/small&gt; H&lt;small&gt;ENRY OF&lt;/small&gt; P&lt;small&gt;LESS&lt;/small&gt;.

Adrienne Lecouvreur — A handsome gown of ivory satin, the underskirt full, and embroidered, nearly half a yard deep, all round with gold, turquoise, amethyst, topaz, and briiliant jewelling. On the hip, the upper skirt was caught back with turquoise blue satin similarly embroidered. There were scalloped embroidered basques. Hair high and poudré; ornaments, diamond and turquoise.

P&lt;small&gt;RINCE&lt;/small&gt; H&lt;small&gt;ENRY OF&lt;/small&gt; P&lt;small&gt;LESS&lt;/small&gt;.

Vicomte de Bragebone — Green uniform of an Officer of the Guard of Louis XVI., faced with scarlet and relieved with white, the whole elaborately braided with gold. [Col 5c–6a]

C&lt;small&gt;OUNT&lt;/small&gt; P&lt;small&gt;ALFFY&lt;/small&gt; (A&lt;small&gt;USTRAILIAN&lt;/small&gt; E&lt;small&gt;MBASSY&lt;/small&gt;).

Court dress—Black velvet, point lace ruffles.

M&lt;small&gt;ISS&lt;/small&gt; P&lt;small&gt;ERRY&lt;/small&gt; (B&lt;small&gt;ITHAM&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;).

Louis XV. period. — Blue brocade, white satin petticoat, with beautiful deep old BrusseIs lace flounce.

L&lt;small&gt;ADY&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt;.

Lady of 1787 — This costume, which was '''designed by Madame Eloffe, dressmaker to Marie Antoinette''', was composed of a white satin skirt, waistcoat and sleeves of blue satin; the back of the bodice, puffs on the sleeves, and loose train being of striped canary yellow and mauve brocade; a fichu of white satin lace, and frills of lace on the sleeves. A white wig with long falling curls, and a very high turban of lace on a blue bandeau, with feathers and flowers on the left side; ornaments, diamonds and emeralds.

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ONSTANCE&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt;.

Bergère of about 1771 — Petticoat of pink satin, with festoons of lace, hooped up with pink and yellow roses; bodice and puffed top skirt of white brocade and pink roses, and large puffed sleeves; Tuscany straw Gainsborough hat, with high pink and white feathers; hair ''poudre''.

M&lt;small&gt;R&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; P&lt;small&gt;ETRE&lt;/small&gt; (W&lt;small&gt;HITLEY&lt;/small&gt; A&lt;small&gt;BBEY&lt;/small&gt;, C&lt;small&gt;OVENTRY&lt;/small&gt;).

Court dress (present day), with wig.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt;. M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;RANK&lt;/small&gt; P&lt;small&gt;ARKER&lt;/small&gt;.

Costume from a picture by Sir Joshua Reynolds — Grey brocade, white muslin fichu and cap; powdered hair.

M&lt;small&gt;ISS&lt;/small&gt; P&lt;small&gt;RACH&lt;/small&gt;.

Louis XV., Dame de Cours — White satin bodoce [sic] and tunic, jewelled over blue petticoat, trimmed with lace and roses, coffieur of the period.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RTHUR&lt;/small&gt; P&lt;small&gt;AGET&lt;/small&gt;.

Duchess d’Orleans — A white satin dress bordered with sable and richly embroidered with steel. She wore diamond ornaments across the top of her bodice, down the front of the dress, and round her neck, while in her white hair were three black ostrich plumes. Her dress sleeves were bound with roses.

E&lt;small&gt;ARL OF&lt;/small&gt; R&lt;small&gt;OSSLYN&lt;/small&gt;.

Duc de Nemours — Uniform of a Colonel of Dragoons, period Louis XVI.; French grey and cherry colour, elaborately braided with gold.

L&lt;small&gt;ORD&lt;/small&gt; R&lt;small&gt;OYSTON&lt;/small&gt;.

Souis Brigadier of Mousquetaires, Louis XVI. — White coat with gold-laced cuffs and collar, a tabard of sapphire blue edged with gold and white embroidery, a diamond Maltese cross suspended on the breast, white knee-breeches and silk stockings, Court shoes, white wig, and three-cornered hat.

T&lt;small&gt;HE&lt;/small&gt; &lt;small&gt;DUCHESS OF&lt;/small&gt; S&lt;small&gt;UTHERLAND&lt;/small&gt;.

Marie Leczinski, the wife of Louis XV. — White satin dress wrought handsomely with silver, and a regal robe, or mantle, of ruby velvet embroidered with gold. The front of the long pointed bodice was fairly ablaze with jewels — rubies, emeralds, and diamonds — of the Sutherland family collection. Along rivière of very large brilliants was used to loop up the Court mantle at the shoulders, a pointed diadem was worn on the forehead, and a large pearl ornament with pendant pear-shaped pearls at one side. The coiffure was low, in Louis XV. style.

L&lt;small&gt;ADY&lt;/small&gt; J&lt;small&gt;ANE&lt;/small&gt; S&lt;small&gt;EYMOUR&lt;/small&gt;.

Lady, time of Louis XVI. — Biue quilted satia petticoat, white satin over-skirt and bodice trimmed with broad Valenciennes lace and pale-pink roses; hair powdered, with chaplet of roses and pearl ornaments.

L&lt;small&gt;ADY&lt;/small&gt; F&lt;small&gt;EODORA&lt;/small&gt; S&lt;small&gt;TURT&lt;/small&gt;.

Madame la Marquise de la Pompadour — White satin costume, with the over-dress full and plain, and opening on a similar under-dress. The sleeves were of satin to the elbow, and finished with ruffles of point d’Alençon lace, while lace was arranged round the top of the bodice. The hair was worn high and ''poudré'', with a black cockade, the badge of the Queen’s Household, at one side. She wore a black satin cap sewn with diamonds and having a diamond aigrette.

B&lt;small&gt;ARONESS&lt;/small&gt; S&lt;small&gt;CHIMMELPENNINCK VON DER&lt;/small&gt; O&lt;small&gt;YE&lt;/small&gt;

Pompadour costume—Overdress of dark green satin en train, Watteau back, over petticoat of white satin, brocaded with flowers; pearl and diamond ornaments.

M&lt;small&gt;R&lt;/small&gt; M&lt;small&gt;AURICE&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt; (H&lt;small&gt;AMMINWOOD&lt;/small&gt;, E&lt;small&gt;AST&lt;/small&gt; G&lt;small&gt;RINSTEAD&lt;/small&gt;).

Gentleman of the period of Louis XVI., copied from an old family picture.

M&lt;small&gt;RS&lt;/small&gt; G&lt;small&gt;UY&lt;/small&gt; S&lt;small&gt;COTT&lt;/small&gt; (H&lt;small&gt;OCKLEY&lt;/small&gt;, M&lt;small&gt;ARTON&lt;/small&gt;).

Lady of the time of Louis XV. — Pink satin skirt, with Brussels lace and roses, white brocade body and overskirt, white wig, with pink roses and feather.

M&lt;small&gt;RS&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt; (T&lt;small&gt;HE&lt;/small&gt; L&lt;small&gt;AWN&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;). 

Lady of the Court of Louis XV. — A handsome gown of Louis XV. period, made of yellow brocade, lined and turned back with pale mauve satin and gold embroidery, over a petticoat of moss-green satin, with flounces of Brussels lace, and trails of westeria; bodice of yellow brocade, with fischu [sic] of white chiffon, fastened with green velvet bow and diamonds.

M&lt;small&gt;ISS&lt;/small&gt; M&lt;small&gt;AY&lt;/small&gt; S&lt;small&gt;ANDERS&lt;/small&gt; (S&lt;small&gt;NITTERFIELD&lt;/small&gt;).

Court dress of the period of Louis XVI. — White silk gown trimmed with gold, pink satin petticoat with revers of old point lace; hair ''poudré''; muslin fichu and pink roses.

MR S. S&lt;small&gt;ANDERS&lt;/small&gt; (S&lt;small&gt;NITTERFIELD&lt;/small&gt;).

Mousquetaire Corps du Garde de Louis XVI. — White frock coat with silver lace facings, blue waistcoat and breeches, white silk stockings, white wig, three-cornered hat, and sword.

M&lt;small&gt;ISS&lt;/small&gt; C. S&lt;small&gt;TARKEY&lt;/small&gt; (N&lt;small&gt;OTTS&lt;/small&gt;).

Dress of white brocade, lined with pink silk; pink silk petticoat, trimmed with old lace and wreaths of roses; hair ''poudré'', with pink roses; pearl ornaments.

M&lt;small&gt;RS&lt;/small&gt; B&lt;small&gt;EAUCHAMP&lt;/small&gt; S&lt;small&gt;COTT&lt;/small&gt;.

Period Louis XVI. — White and mauve satin brocade, trimmed with lace and fur, white wig.

M&lt;small&gt;RS&lt;/small&gt; A&lt;small&gt;RTHUR&lt;/small&gt; S&lt;small&gt;OMERSET&lt;/small&gt;.

Shepherdess ''à la'' Watteau — Pompadour style, the dress and hat being trimmed with pink roses, and her crook tied with similar flowers.

M&lt;small&gt;R&lt;/small&gt; O&lt;small&gt;SWALD&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt;.

Gentleman of the Court, Louis XVI. — Dark-striped yellow coat, with needlework design in flowers in shaded silk, embroidered cream silk waistcoat, and pale-green satin breeches.

C&lt;small&gt;APTAIN&lt;/small&gt; S&lt;small&gt;OMERSET&lt;/small&gt;.

Mousquetaire — Gold cloth with broad gold-bued silk sash.

H&lt;small&gt;ON&lt;/small&gt;. B. W. H. S&lt;small&gt;TONOR&lt;/small&gt;.

Mousquetaire, Louis XVI. — Knickerbockers of a dark blue, with a doublet to match, having a cross emblazoned on the breast, and deep point lace collar, white satin coat skirts almost like a simulation of armour below his doublet, and a long military cloak of French grey cloth lined with scarlet, three-cornered hat, white wig, sword, silk stockings, and Court shoes.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;HURSBY&lt;/small&gt;-P&lt;small&gt;ELHAM&lt;/small&gt;.

White satin, with white roses and lace.

M&lt;small&gt;ISS&lt;/small&gt; V&lt;small&gt;IOLET&lt;/small&gt; L&lt;small&gt;OFTUS&lt;/small&gt; T&lt;small&gt;OTTENHAM&lt;/small&gt;.

Pink and white brocade over white satin, and pink roses in her powdered hair.

M&lt;small&gt;R&lt;/small&gt; T&lt;small&gt;OWER&lt;/small&gt;.

Officier Garde Suisse — Scarlet coat lined with white, blue facings, and three-cornered hat.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;OWER&lt;/small&gt;.

Duchesse de Polignac — White satin, with old lace and rose coloured plumes.

M&lt;small&gt;RS&lt;/small&gt; T&lt;small&gt;REE&lt;/small&gt;.

Court Dame, Louis XV — Skirt of pink satin, with a tunic of most lovely silver tinsel brocade, having alternate stripes encllosing bunches of roses and baskets of flowers. Were the usual Watteau back and hip panniers, and the petticoat was arranged with a twist of chiffon above a frill of most beautiful cream lace, over these being a garland of pink roses, caught up on either side with a cluster of pink and cream feathers repeating to the top; the sleeves were made of brocade to the elbow, with hanging cream lace over pink chiffon and feathers on the shoulders, and the dress was completed by pink slippers with pink velvet bows, a Louis XV. fan of great beauty, and a staff of pink and green with green ribbons, roses, and feathers to match the dress. 

L&lt;small&gt;ORD&lt;/small&gt; G&lt;small&gt;REY DE&lt;/small&gt; W&lt;small&gt;ILTON&lt;/small&gt;.

Gentleman, temps Louis XV. — Coat of dark petunia velvet embroidered with gold, a white satin waistcoat elaborately gold laced, white silk stockings, jabot ruffles, wig, sword, hat, and shoes ''en suite''.

T&lt;small&gt;HE&lt;/small&gt; H&lt;small&gt;ON&lt;/small&gt; D&lt;small&gt;UDLEY&lt;/small&gt; W&lt;small&gt;ARD&lt;/small&gt;.

Mousquetaire — Uniform of dark blue cloth, with scarlet facings and elaborately braided with gold.

L&lt;small&gt;ADY&lt;/small&gt; W&lt;small&gt;ALLER&lt;/small&gt;.

Comtesse d’Artois — White satin quilted petticoat with a pearl at each corner of the pattern; gown of grey brocade lined with white satin; white lisse stomacher crossed by grey velvet bows fastened with diamonds.

M&lt;small&gt;ISS&lt;/small&gt; W&lt;small&gt;ALLER&lt;/small&gt;.

Fille de la Comtesse d’Artois — Gown of pale blue satin flecked with pink roses tied with ribbon, paniers and wreaths of roses. 

M&lt;small&gt;ISS&lt;/small&gt; C&lt;small&gt;ORNWALLIS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt;.

Mademoiselle de la Court — Costume after a picture by Roslin depicting a girl about to decorate the statue of Love. She wore an underdress of pale pink satin with gown of white satin with demi-train, lined with pink. The body was decked with tulle, and long tulle streamers were pendant from the sleeves; head-dress was roses and violets, with pink and white ribbons.

M&lt;small&gt;R&lt;/small&gt; W&lt;small&gt;ILLIAMS&lt;/small&gt;.

Black velvet Court dress, with Louis XV. wig.

M&lt;small&gt;RS&lt;/small&gt; F&lt;small&gt;RANCIS&lt;/small&gt; W&lt;small&gt;ILLIAMS&lt;/small&gt; (W&lt;small&gt;ATCHBURY&lt;/small&gt;, W&lt;small&gt;ARWICK&lt;/small&gt;).

Lady of the Court, Louis XV. — White satin bodice and train, with white satin petticoat trimmed with lace and pink roses.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt; (A&lt;small&gt;LSCOT&lt;/small&gt; P&lt;small&gt;ARK&lt;/small&gt;).

Lady of the Court of Louis XV. — A handsome gown of Louis XV. period, made of white broché silk, with bouquets of pink roses, over petticoat of rich pink satin with deep flounce of Brussels lace, caught up with trails of pink roses; bodice of same broché, with white chiffon fichu fastened in front with pink velvet bow and diamonds. [6, Col. 6c – 7, Col. 1a]

M&lt;small&gt;RS&lt;/small&gt; C&lt;small&gt;ARTLAND&lt;/small&gt;.

Lady of the court, Louis XV. — Green brocade bodice and skirt; white satin petticoat trimmed with old lace and roses; white Leghorn hat with roses.

M&lt;small&gt;ISS&lt;/small&gt; F&lt;small&gt;LEETWOOD&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;.

Lady of the period — Black peau de soie silk, the Watteau and over-skirt lined with white, pointed panniers, skirt caught up with roses, while black roses wee worn on the powdered hair. A splendid stomacher of diamonds and emeralds was worn.

L&lt;small&gt;IEUTENANT&lt;/small&gt;-C&lt;small&gt;OLONEL&lt;/small&gt; M&lt;small&gt;ILDMAY&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, C.B. (S&lt;small&gt;COTS&lt;/small&gt; G&lt;small&gt;UARDS&lt;/small&gt;).

Guardsman, 1790 — Red tunic, gold braid, [9R?] on buttons, white gaiters coming above the knee, black garters, wig, and three-cornered hat. 

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;HEATLEY&lt;/small&gt; (B&lt;small&gt;ERKSWELL&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;).

Lady of Louis XV. Court — Bodice and train of pale green and pearl coloured striped brocade, with bunches of pink roses; petticoat and corsage of pale green satin, embroidered in pearls, and high collar of lace; diamond ornaments, a wig with pink roses and diamond stars.

M&lt;small&gt;RS&lt;/small&gt; W&lt;small&gt;ILLIE&lt;/small&gt; L&lt;small&gt;OW&lt;/small&gt; (W&lt;small&gt;ELLESBOURNE&lt;/small&gt; H&lt;small&gt;OUSE&lt;/small&gt;).

Duchess of Gainsborough — Dress copied exactly from the portrait of the Hon. Mrs Graham, in brocade, with pink satin petticoat, a big black hat trimmed with white plumes, with diamonds.

M&lt;small&gt;R&lt;/small&gt; W. M. L&lt;small&gt;OW&lt;/small&gt;.

David Garrick — Costume worn by the actor, Mr Richard Wyndham, when impersonating that character, of rich purple vevet coat, purple satin waistcoat and knee-breeches, steel buttons, purple silk stockings, diamond buckles, black three-cornered hat, and steel sword.&lt;ref&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt;&lt;/blockquote&gt;

==Anthology==

====Quote Intro====
The day of the ball, the ''Coventry Evening Telegraph'' published the following:&lt;blockquote&gt;GRAND BAL POUDRE AT WARWICK CASTLE.

Writing this morning our Warwick representative says: Warwick Castle will tonight be the scene of a memorable spectacle, the Earl and Countess of Warwick having invited about four hundred guests to a ''bal poudre'', in which the costumes were to be of the style of the Louis XIV. and XV. period. The event has been looked forward to with considerable interest by the ''élite'' of the fashionable world, on account of the prominent position occupied by the Countess in society. Great preparations were made the Castle, the greater portion of which has been most lavishly decorated in the light and airy French style of the period. The dancing will take place in the Cedar drawing-room, the adjoining rooms having been set apart as retiring rooms. Supper will be served in the Great Hall, where the whole of the guests will be able to sit down together. The decorations have been carried out under the personal supervision of the hostess, who has received the valuable assistance and advice of Mr. Caryll Craven. The dance music will be supplied by Worm's famous &quot;White Viennese&quot; Band, while Johnson's (Manchester) Band will discourse in the supper room. The hostess will be dressed as &quot;Mary Antoinette,&quot; Queen of Louis XVI. Her costume will be of rose-coloured brocade with a gold pattern, and a sky-blue velvet train embroidered with gold fleur-de lys. Lady Warwick's relative, the Duchess of Sutherland, will appear as the wife of Louis XV. in a costume of white and silver with crimson velvet train and silver fleur-de-lys. Lord Warwick will be in the dress of a military officer of the period, while Prince Francis of Teck has signified his intention of appearing in the uniform of &quot;the Royals&quot; (of the period). Owing to the demise of Lord Randolph Churchill, the Duke of Marlborough will not be present.

The house party at the Castle included the Austro-Hungarian Ambassador, the Portuguese Minister, Prince Francis of Teck, Prince and Princess Henry of Pless and Miss Cornwallis West, Duchess of Sutherland and Lady Angela St. Clair Erskine, Duke of Manchester, Earl and Countess of Rosslyn, Earl of Lonsdale, Earl of Burford, Earl of Chesterfield, Countess Cairns, Lord Clifden, Lord Kenyon, Lady Gerard, Lord Grey de Wilton, Lord Royston, Lord Lovat, Lady Norreys, Lady Eva Greville, Lord Richard Neville, Hon H. and Lady Fedora Sturt, Hon. H. Stonor, Captain the Hon. Hedworth Lambton, Mr. F. Menzies and Miss Muriel Wilson, Miss Naylor, Mr. Arthur Paget, Mr. Cyril Foley, Mr. C. de Murietta, and Mr. Layoock.

The following accepted invitations to the ball, and most of them brought parties with them, the guests numbering in all about four hundred:— The Earl and Countess of Aylesford, Mr. and Mrs. Aubrey Cartwright, Mr. and Mrs. J. Stratford Dugdale, Mr. and Mrs. Chamberlayne, Sir C. and Lady Mordaunt, Mr. and Mrs. Smythe, Lord and Lady Hertford, Lady and Miss Waller, Mr. J. and Mr. J. P. Arkwright, Mr. and Mrs. M. Lucas, Mr. and Mrs. J. B. Dugdale (18), Mr., Lady Anne, and Miss Murray, Captain and Mrs. Brinkley, Mr. and Mrs. Guy Scott, Mr., Mrs., and Miss Irwin, Mr. and Miss Perry, Major and Mrs. Fosbery, Mr. Lindsay, Mr. R. Paget, Sir A. and Lady Hodgson, Mrs. Beauchamp Scott, Major and Mrs. Norris, Mr. and Mrs. Tree, Mr., Mrs. and Miss Granville, Mr. and Mrs. Joliffe, Captain and Mrs. Osborne, Mr. and Mrs. E. Little, Mr., Mrs., and Miss Lakin, Officers 6th Reg. District, Mr. Batchelor, Hon. Mrs. and Miss Chandos Leigh, Colonel and Mrs. Paulet, Mr. F. Hunter Blair, Mr. J. Alston, Mr. and Mrs. Hutton, Captain and Mrs. Cowan, Mr. and Mrs. Williams, Mr. and the Misses Allfrey, Mr. and Mrs. Wilson Fitzgerald, Mr. and Mrs. W. Allfrey, Mrs. and Miss Drummond, Mr. and Mrs. Shaw, Mr. R. and Mr. J. Lant, Mr. and Mrs. Sanders and party, Captain Lafone, Sir F. and Lady Peel, Captain and Mrs. Keighly-Peach, Miss Nicol and party, Mr. and Mrs. Fred Shaw, Mr. and Mrs. Cove Jones, Mr. and Lady G. Petre, Mr. R. Barnes, Mr. and Mrs. Wheatley, Mr., Mrs., and Miss Ramsden, Mr. and Mrs. W. M. Low, Mrs. Basil Montgomery, Mr. and Mrs. Thursby-Pelham, Mr. and Mrs. H. Chamberlain, Mr. Francis, Mr. and Mrs. Graham, Mr. and Mrs. West, Mr. and Mrs. Sam Smith, Officers 17th Lancers.&lt;ref&gt;&quot;Grand Bal Poudre at Warwick Castle.&quot; ''Coventry Evening Telegraph'' 01 February 1895, Friday: 3 [of 4], Col. 4a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000337/18950201/021/0003.&lt;/ref&gt;&lt;/blockquote&gt;The report from the ''Morning Post'' the next day:&lt;blockquote&gt;The Countess of Warwick's Bal Poudré at Warwick Castle last night was attended by a company of nearly 400 guests, and was a brilliant success. The magnificent suite of apartments was superbly decorated with choice flowers, while the many treasures of antiquity and historic interest which the Castle contains were displayed in the various rooms. The choice of costume was restricted to the period covering the reigns of Louis XV. and Louis XVI., with powdered hair or white wigs, but gentlemen were given the option of appearing in English Court dress with Louis XV. wigs. The Countess of Warwick, who represented Marie Antoinette, wore a dress of rose-coloured material brocaded with gold, with a train of sky-blue velvet, embroidered with fleur-de-lis. The Earl of Warwick was attired in a Maison du Roi costume of rich velvet, with gold and diamond buttons. Prince Francis of Teck wore the uniform of the period of his own regiment, the Royals. The Duchess of Sutherland, as the wife of Louis XV., was in a costume of white and silver, with a crimson velvet train embroidered with silver fleur-de-lis. Prince Henry of Pless wore a blue military dress of the period with red facings, while the Earl of Rosslyn donned the uniform of a Colonel of the reign of Louis XVI. The Hon. H. Sturt represented the Church of the period as an Abbé, and Mr. W. Low the stage as David Garrick. Amongst the other guests were the Austro-Hungarian Ambassador, the Portuguese Minister, Princess Henry of Pless, Lady Angela St. Clair Erskine, the Duke of Manchester, the Earl and Countess of Rosslyn, the Earl of Lonsdale, the Earl of Burford, the Earl of Chesterfield, Countess Cairns, Lord Clifden, Lord Kenyon, Lady Gerard, Lord Grey de Wilton, Lord Royston, Lord Lovat, Lady Norreys, Lady Eva Greville, Lord Richard Nevill, Lady Feodorowna Sturt, the Hon. S. Greville, the Hon. H. Stonor, Captain the Hon. Hedworth Lambton, Mrs. Menzies, Miss Muriel Wilson [sic no comma] Miss Naylor, Mr. Arthur Paget, Mr. Cyril Foley, Mr. C. de Murrieta, Mr. Caryl Craven, Mr. Kennard, and Mr. Laycock. The Countess of Aylesford brought a large party from Packington Hall. Herr Würm's White Viennese Band occupied the orchestra. Dancing commenced at nine o'clock, and at midnight the entire company sat down to supper in the large banqueting hall. The assembly was undoubtedly one of the most brilliant which has ever been gathered together within the walls of the historic Castle.&lt;ref&gt;&quot;Arrangements for This Day.&quot; ''Morning Post'' 02 February 1895, Saturday: 5 [of 10], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18950202/052/0005.&lt;/ref&gt;&lt;/blockquote&gt;

== Notes and Questions ==
# 

==References==</text>
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    <title>African Arthropods/''Ormyrus'' wasps on a ''Euclea'' stem gall</title>
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      <comment>New resource with &quot;==''Ormyrus'' wasps on a ''Euclea'' stem gall== [w:Ormyrus|''Ormyrus''] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenopte...&quot;</comment>
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      <text bytes="1251" sha1="m1euxd06p2vak1f65qgvpzdh7fuiq5i" xml:space="preserve">==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[w:Ormyrus|''Ormyrus''] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
 These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
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      <text bytes="1252" sha1="muvdyiyzcks3pl2esgea0y4n78vvo5p" xml:space="preserve">==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
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      <comment>New resource with &quot;==''Ormyrus'' wasps on a ''Euclea'' stem gall== [[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenop...&quot;</comment>
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      <text bytes="1252" sha1="muvdyiyzcks3pl2esgea0y4n78vvo5p" xml:space="preserve">==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>muvdyiyzcks3pl2esgea0y4n78vvo5p</sha1>
    </revision>
    <revision>
      <id>2689229</id>
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      <timestamp>2024-11-28T19:09:07Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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      <comment>intro</comment>
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      <text bytes="2655" sha1="fpojce3fak2tww1gpeb3mcrx54layjb" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=vanNoort2024&gt;van Noort, 2024. Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org, accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including &quot;Tanaostigmodes tambotis&quot;, which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>fpojce3fak2tww1gpeb3mcrx54layjb</sha1>
    </revision>
    <revision>
      <id>2689230</id>
      <parentid>2689229</parentid>
      <timestamp>2024-11-28T19:10:17Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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      <text bytes="2654" sha1="s8m18moxd2idddn77fppz7bichdkd0j" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). www.waspweb.org, accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including &quot;Tanaostigmodes tambotis&quot;, which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>s8m18moxd2idddn77fppz7bichdkd0j</sha1>
    </revision>
    <revision>
      <id>2689231</id>
      <parentid>2689230</parentid>
      <timestamp>2024-11-28T19:11:11Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2689231</origin>
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      <text bytes="2679" sha1="2ds6pgmfnyku3wqybnog0p55rycfz9b" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). [http://www.waspweb.org www.waspweb.org], accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including &quot;Tanaostigmodes tambotis&quot;, which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>2ds6pgmfnyku3wqybnog0p55rycfz9b</sha1>
    </revision>
    <revision>
      <id>2689232</id>
      <parentid>2689231</parentid>
      <timestamp>2024-11-28T19:13:12Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>refs</comment>
      <origin>2689232</origin>
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      <format>text/x-wiki</format>
      <text bytes="2723" sha1="497ukd5bmhe0dycytg16fibvmaxcwy3" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants). [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm], accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including &quot;Tanaostigmodes tambotis&quot;, which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>497ukd5bmhe0dycytg16fibvmaxcwy3</sha1>
    </revision>
    <revision>
      <id>2689233</id>
      <parentid>2689232</parentid>
      <timestamp>2024-11-28T19:14:49Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>link</comment>
      <origin>2689233</origin>
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      <format>text/x-wiki</format>
      <text bytes="2722" sha1="ql9l5p5er1l74hx4x1zgjii29cf97uk" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including &quot;Tanaostigmodes tambotis&quot;, which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>ql9l5p5er1l74hx4x1zgjii29cf97uk</sha1>
    </revision>
    <revision>
      <id>2689234</id>
      <parentid>2689233</parentid>
      <timestamp>2024-11-28T19:17:31Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <minor />
      <comment>italics</comment>
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      <format>text/x-wiki</format>
      <text bytes="2724" sha1="1xbkyl7fu8j46rfnst3tlp2v83as6nz" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* Epichrysomallidae includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* Melanosomellidae has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* Ormyridae is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* Tanaostigmatidae contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>1xbkyl7fu8j46rfnst3tlp2v83as6nz</sha1>
    </revision>
    <revision>
      <id>2689235</id>
      <parentid>2689234</parentid>
      <timestamp>2024-11-28T19:19:54Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2689235</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2810" sha1="1u1pbxwokc9xw06ahsdxrufgnph2rfi" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects.
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>1u1pbxwokc9xw06ahsdxrufgnph2rfi</sha1>
    </revision>
    <revision>
      <id>2689237</id>
      <parentid>2689235</parentid>
      <timestamp>2024-11-28T20:10:31Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="2810" sha1="707lxp1wrfbrn1c8du0e0xj1glola1u" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''Ficus'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''Asparagus'' species, ''Halleriaphagus phagolucida'', which forms galls on ''Halleria'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''Spirostachys africana''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>707lxp1wrfbrn1c8du0e0xj1glola1u</sha1>
    </revision>
    <revision>
      <id>2689238</id>
      <parentid>2689237</parentid>
      <timestamp>2024-11-28T20:13:15Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>links</comment>
      <origin>2689238</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2881" sha1="3nn3xty1gc7hbt8jr0ugarn6vkal2cd" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>3nn3xty1gc7hbt8jr0ugarn6vkal2cd</sha1>
    </revision>
    <revision>
      <id>2689239</id>
      <parentid>2689238</parentid>
      <timestamp>2024-11-28T20:14:46Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2689239</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2889" sha1="dcm0qoiyfdo9omaxja8kdd5c23z2ijg" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus (genus)|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>dcm0qoiyfdo9omaxja8kdd5c23z2ijg</sha1>
    </revision>
    <revision>
      <id>2689241</id>
      <parentid>2689239</parentid>
      <timestamp>2024-11-28T20:16:23Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2689241</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2897" sha1="1xbb6ghbtodaqhoyfkwhegq3an9fbee" xml:space="preserve">Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus (genus)|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria (plant)|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>1xbb6ghbtodaqhoyfkwhegq3an9fbee</sha1>
    </revision>
    <revision>
      <id>2689260</id>
      <parentid>2689241</parentid>
      <timestamp>2024-11-29T08:04:22Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>Intro and Cynipidae</comment>
      <origin>2689260</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3480" sha1="ejfcb52znc6ryv6vrjs08m96tvwgmwf" xml:space="preserve">In the northern hemisphere, many plant galls are induced by wasps of the family [[Cynipidae]], but this family is rarely encountered in the Afrotropics, where it comprises only four described species, all found in South Africa.&lt;ref name=WaspwebAfroCynipidae&gt;van Noort, 2024. [https://www.waspweb.org/Cynipoidea/Cynipidae/Classification/Classification_Afrotropical_Cynipidae.htm Classification and checklist of Afrotropical cynipid wasps], accessed on 29 November 2024.&lt;/ref&gt; In the Afrotropics, most gall-associated wasps are chacidoids.

=Chalcidoidea associated with plant galls=

Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus (genus)|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria (plant)|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>ejfcb52znc6ryv6vrjs08m96tvwgmwf</sha1>
    </revision>
    <revision>
      <id>2689261</id>
      <parentid>2689260</parentid>
      <timestamp>2024-11-29T08:05:21Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>link</comment>
      <origin>2689261</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3492" sha1="5j0ylnsrkn767ds3dbu3tjhr9ufmina" xml:space="preserve">In the northern hemisphere, many plant galls are induced by wasps of the family [[w:Cynipidae|Cynipidae]], but this family is rarely encountered in the Afrotropics, where it comprises only four described species, all found in South Africa.&lt;ref name=WaspwebAfroCynipidae&gt;van Noort, 2024. [https://www.waspweb.org/Cynipoidea/Cynipidae/Classification/Classification_Afrotropical_Cynipidae.htm Classification and checklist of Afrotropical cynipid wasps], accessed on 29 November 2024.&lt;/ref&gt; In the Afrotropics, most gall-associated wasps are chacidoids.

=Chalcidoidea associated with plant galls=

Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus (genus)|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria (plant)|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.


==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
      <sha1>5j0ylnsrkn767ds3dbu3tjhr9ufmina</sha1>
    </revision>
    <revision>
      <id>2689262</id>
      <parentid>2689261</parentid>
      <timestamp>2024-11-29T08:05:58Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Chalcidoidea associated with plant galls */</comment>
      <origin>2689262</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3491" sha1="fexs6p42iufat07br0927iquxp4nlpz" xml:space="preserve">In the northern hemisphere, many plant galls are induced by wasps of the family [[w:Cynipidae|Cynipidae]], but this family is rarely encountered in the Afrotropics, where it comprises only four described species, all found in South Africa.&lt;ref name=WaspwebAfroCynipidae&gt;van Noort, 2024. [https://www.waspweb.org/Cynipoidea/Cynipidae/Classification/Classification_Afrotropical_Cynipidae.htm Classification and checklist of Afrotropical cynipid wasps], accessed on 29 November 2024.&lt;/ref&gt; In the Afrotropics, most gall-associated wasps are chacidoids.

=Chalcidoidea associated with plant galls=

Most chalcidoid wasps in the families Cynipencyrtidae, Epichrysomallidae, Melanosomellidae, Ormyridae and Tanaostigmatidae are associated with plant galls; their larvae and pupae develop within galls. Together, these families comprise the ‘Gall Clade’ within the Chalcidoidea.&lt;ref name=vanNoort2024/&gt; 

In the Afrotropics four of these families are represented:&lt;ref name=waspweb2024&gt;van Noort, 2024. [https://www.waspweb.org/Classification/Classification_Afrotropical_Hymenoptera.htm Classification of Afrotropical Hymenoptera (Wasps, Bees, Ants)], accessed on 28 November 2024.&lt;/ref&gt;
* [[w:Epichrysomallidae|Epichrysomallidae]] includes about 16 described Afrotropical species in seven genera; they are exclusively associated with figs (genus ''[[w:Ficus|Ficus]]'');
* [[w:Melanosomellidae|Melanosomellidae]] has four known species in the Afrotropics, two of which were introduced from Australia for the biological control of invasive Australian wattle trees;
* [[w:Ormyridae|Ormyridae]] is represented by three species of ''Asparagobius'' which form galls on ''[[w:Asparagus (genus)|Asparagus]]'' species, ''Halleriaphagus phagolucida'', which forms galls on ''[[w:Halleria (plant)|Halleria]]'' species, two species of ''Ouma'' (biology unknown), and 14 described species of ''Ormyrus'', which are hyperparasitoids, attacking other gall-forming insects;
* [[w:Tanaostigmatidae|Tanaostigmatidae]] contains five African species including ''Tanaostigmodes tambotis'', which is phytophagous, forming galls on the stems of tamboti trees (''[[w:Spirostachys africana|Spirostachys africana]]''); the biologies of the other species are unknown.

==''Ormyrus'' wasps on a ''Euclea'' stem gall==
[[w:Ormyrus|''Ormyrus'']] is a genus of parasitoid wasps with a worldwide distribution. Most species are thought to be hyperparasitoids, i.e. their larvae feed on the larvae of gall-forming insects; these galls are found on a wide variety of plants.&lt;ref name=vanNoort2024&gt;van Noort, S., Mitroiu, M.D., Burks, R., Gibson, G., Hanson, P., Heraty, J., Janšta, P., Cruaud, A. and Rasplus, J.Y., 2024. Redefining Ormyridae (Hymenoptera, Chalcidoidea) with establishment of subfamilies and description of new genera. Systematic Entomology, 49(3), pp.447-494. https://doi.org/10.3897/zookeys.644.10035&lt;/ref&gt;
These phtographs show the activity of ''Ormyrus'' wasps on a stem gall on a magic gwarrie (''[[w:Euclea divinorum|Euclea divinorum]]'') shrub. In this case the gall-forming insect, and host of the ''Ormyrus'' parasite, is not known.
&lt;gallery mode=packed heights=250&gt;
Ormyrus males 2024 09 28 iN 245187710 01.jpg|Male Ormyrus wasps on a stem gall, waiting for a female to emerge
Ormyrus female 2024 10 01 iN 249534371 02.jpg|Female ''Ormyrus'' wasp ovipositing into a stem gall
Ormyrus_males_2024_09_28_iN_245187710_02.jpg|Stem gall on a ''Euclea divinorum'' shrub
&lt;/gallery&gt;

==References==
{{reflist}}</text>
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In the following lectures, we will enhance our methods by considering equivalence relations for algebraic structures and the formation of residue classes. For different algebraic structures
{{
Extra/Bracket
|text=like groups, rings, vector spaces|
|Ipm=|Epm=,
}}
these constructions follow the same scheme; therefore, we describe first this construction for groups.

{{Subtitle|Groups}}

For an element
{{
Relationchain
| g
|\in| G
||
||
||
|pm=
}} 
of a 
{{
Extra/Bracket
|text=multiplicatively written|
|Ipm=|Epm=
}}
group {{mat|term= G |pm=}} and
{{
Relationchain
| n
|\in| \N
||
||
||
|pm=,
}}
we write
{{
Relationchain/display
| g^n
|| g \cdots g
||
||
||
|pm=
}}
{{
Extra/Bracket
|text={{mat|term= n |pm=}} times|
|Ipm=|Epm=
}}
and
{{
Relationchain/display
| g^{n}
|| {{mabr| g^{-1} |}}^{-n}
||
||
||
|pm=
}}
for
{{
Relationchain
| n
|\in| \Z_-
||
||
||
|pm=.
}} 
Due to the exponent rules, see
{{
Exerciselink
|Preword=||Exercisename=
Group theory/Exponent rules/Fact/Proof/Exercise
|Nr=
|pm=,
}}
this fits together well. For permutations and invertible matrices, we have encountered the order of an element several times already.

{{
inputdefinition
|Group theory/Order of an element/Definition||
}}

{{
inputdefinition
|Group theory/Cyclic group/Definition||
}}

This means that there exists an element
{{
Relationchain
| g
|\in| G
||
||
||
|pm=
}}
{{
Extra/Bracket
|text=a {{Keyword|generator|pm=}}|
|Ipm=|Epm=
}}
such that every element in {{mat|term= G |pm=}} can be written as {{mathl|term= g^n |pm=}} with some 
{{
Relationchain
| n
|\in| \Z
||
||
||
|pm=.
}}
The group {{mathl|term= (\Z,+,0) |pm=}} is cyclic, we can take
{{
Mathcor|term1=
1
|or|term2=
-1
|pm=
}}
as a generator. Also all subgroups of {{mat|term=\Z|pm=}} are cyclic themselves, as the following theorem shows.

{{
inputfactproofexercise
|Subgroups of Z/One generator/Fact|Theorem||
}}

{{
inputexample
|Z modulo n/Representatives/Associativity and group/Cyclic/Example||
}}

{{Subtitle|Group homomorphisms}}

We have mentioned group homomorphisms already in the 18th lecture in the context of the sign of a permutation. {{:Group homomorphism/Introduction/Section}}

{{Subtitle|Group isomorphisms}}

{{:Group isomorpisms/Automorphism/Introduction/Section}}

{{Subtitle|The kernel of a group homomorphism}}

{{:Group homomorphism/Kernel/Injectivity criterion/Introduction/Section|extra1=As for linear mappings, we have again the {{Keyword|kernel criterion for injectivity|pm=.}} }} 

{{Subtitle|The image of a group homomorphism}}

{{
inputfactproof
|Group homomorphism/Image/Subgroup/Fact|Lemma||
}}

{{
inputexample
|Group homomorphism/R to C/e^it/Kernel and image/Example||
}}</text>
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In the following lectures, we will enhance our methods by considering equivalence relations for algebraic structures and the formation of residue classes. For different algebraic structures
{{
Extra/Bracket
|text=like groups, rings, vector spaces|
|Ipm=|Epm=,
}}
these constructions follow the same scheme; therefore, we describe first this construction for groups.

{{Subtitle|Groups}}

For an element
{{
Relationchain
| g
|\in| G
||
||
||
|pm=
}} 
of a 
{{
Extra/Bracket
|text=multiplicatively written|
|Ipm=|Epm=
}}
group {{mat|term= G |pm=}} and
{{
Relationchain
| n
|\in| \N
||
||
||
|pm=,
}}
we write
{{
Relationchain/display
| g^n
|| g \cdots g
||
||
||
|pm=
}}
{{
Extra/Bracket
|text={{mat|term= n |pm=}} times|
|Ipm=|Epm=
}}
and
{{
Relationchain/display
| g^{n}
|| {{mabr| g^{-1} |}}^{-n}
||
||
||
|pm=
}}
for
{{
Relationchain
| n
|\in| \Z_-
||
||
||
|pm=.
}} 
Due to the exponent rules, see
{{
Exerciselink
|Exercisename=
Group theory/Exponent rules/Fact/Proof/Exercise
|Nr=
|pm=,
}}
this fits together well. For permutations and invertible matrices, we have encountered the order of an element several times already.

{{
inputdefinition
|Group theory/Order of an element/Definition||
}}

{{
inputdefinition
|Group theory/Cyclic group/Definition||
}}

This means that there exists an element
{{
Relationchain
| g
|\in| G
||
||
||
|pm=
}}
{{
Extra/Bracket
|text=a {{Keyword|generator|pm=}}|
|Ipm=|Epm=
}}
such that every element in {{mat|term= G |pm=}} can be written as {{mathl|term= g^n |pm=}} with some 
{{
Relationchain
| n
|\in| \Z
||
||
||
|pm=.
}}
The group {{mathl|term= (\Z,+,0) |pm=}} is cyclic, we can take
{{
Mathcor|term1=
1
|or|term2=
-1
|pm=
}}
as a generator. Also all subgroups of {{mat|term=\Z|pm=}} are cyclic themselves, as the following theorem shows.

{{
inputfactproofexercise
|Subgroups of Z/One generator/Fact|Theorem||
}}

{{
inputexample
|Z modulo n/Representatives/Associativity and group/Cyclic/Example||
}}

{{Subtitle|Group homomorphisms}}

We have mentioned group homomorphisms already in the 18th lecture in the context of the sign of a permutation. {{:Group homomorphism/Introduction/Section}}

{{Subtitle|Group isomorphisms}}

{{:Group isomorphisms/Automorphism/Introduction/Section}}

{{Subtitle|The kernel of a group homomorphism}}

{{:Group homomorphism/Kernel/Injectivity criterion/Introduction/Section|extra1=As for linear mappings, we have again the {{Keyword|kernel criterion for injectivity|pm=.}} }} 

{{Subtitle|The image of a group homomorphism}}

{{
inputfactproof
|Group homomorphism/Image/Subgroup/Fact|Lemma||
}}

{{
inputexample
|Group homomorphism/R to C/e^it/Kernel and image/Example||
}}</text>
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{{
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|pm=
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{{mat|term= G |pm=}} is called {{Word of definition|cyclic|pm=}} if it is generated by one element.
|Textform=Definition
|Category=   
|Word of definition=Cyclic group
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Mathematical text/Fact
|Text=
{{
Factstructure
|Situation=
|Condition=
The
{{
Definitionlink
|subgroups|
|pm=
}}
of {{mat|\Z |pm=}} are precisely 
|Segue=
|Conclusion=
the subsets of the form
{{
Relationchain/display
| \Z d 
||  {{Setcond| kd|k \in \Z}}
||
||
||
|pm=
}}
with a uniquely determined nonnegative number {{mat|term= d |pm=.}}
|Extra=
}}
|Textform=Fact 
|Category=
|Factname=Theorem about the subgroups of {{mat|term= \Z|pm=}} 
}}</text>
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Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
A subset of the form {{mat|term=  \Z d |pm=}} is a subgroup due to the distributive law. Let now
{{
Relationchain
|H
| \subseteq| \Z
||
||
||
|pm=
}}
be a subgroup. In case
{{
Relationchain
|H
|| 0
||
||
||
|pm=,
}}
we can take
{{
Relationchain
|d
|| 0
||
||
||
|pm=.
}}
Hence, we may assume that {{mat|term= H |pm=}} contains beside {{mat|term= 0 |pm=}} another element {{mat|term= x |pm=.}} If {{mat|term= x |pm=}} id negative, then subgroup {{mat|term= H |pm=}} must contain its negative {{mat|term= -x|pm=,}} , which is positive. This means that {{mat|term= H |pm=}} contains also a positive number. Let {{mat|term= d |pm=}} denote the smallest positive number in {{mat|term= H |pm=.}} We claim
{{
Relationchain
|H
||\Z d
||
||
||
|pm=.
}}
Here, the inclusion
{{
Relationchain 
| \Z d 
|\subseteq |H
||
||
||
|pm=
}}
is clear, as all
{{
Extra/Bracket
|text=positive and negative|
|Ipm=|Epm=
}}
multiples of {{mat|term= d |pm=}} must belong to the subgroup. To show the inverse inclusion, let
{{
Relationchain
|h
|\in|H
||
||
||
|pm=
}}
be arbitrary. Due to 
{{
Factlink
|Preword=the|division with remainder|Factname=
Division with remainder/Z/Fact
|Nr=
|pm=,
}}
we have
{{
 Math/display|term=
h=qd+r \text{ with } 0 \leq r &lt; d
|pm=.
}}
Because of
{{
mathcor|term1=
 h \in H
|and|term2=
qd \in H
|pm=,
}} 
also
{{
Relationchain
| r
|| h-qd 
|\in| H
||
||
||
|pm=
}}
holds. Because of the choice of {{mat|term= d |pm=}} and 
{{
Relationchain
|r
|&lt;|d
||
||
||
|pm=,
}}
we must have
{{
Relationchain
|r
|| 0
||
||
||
|pm=.
}}
This means
{{
Relationchain
|h
||qd
||
||
||
|pm=.
}}
Therefore
{{
Relationchain
|h
|\in|\Z d
||
||
||
|pm=,
}}
thus
{{
Relationchain
|H 
|\subseteq |\Z d
||
||
||
||
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
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Mathematical text/Example
|Text=
Let
{{
Relationchain
| n
|\in| \N_+
||
||
||
|pm=,
}} 
and consider on
{{
Relationchain/display
| {{op:Zmod|n}} 
|| \{0,1 {{commadots}} n-1\}
||
||
||
|pm=
}}
the 
{{
Definitionlink
|binary operation|
|Context=|
|pm=
}}
{{
Relationchain/display
|  a + b 
| {{defeq|}} | (a+b) \mod n
|| \begin{cases} a+b, \text{ if } a+b &lt;n\, , \\ a+b-n,  \text{ if } a+b \geq n \, . \end{cases} 
|| 
|| 
|pm=
}}
With this operation, we have a
{{
Definitionlink
|group|
|Context=|
|pm=
}}
due to
{{
Exerciselink
|Exercisename=
Representatives/Associativity and group/Exercise
|Nr=
|pm=.
}}
Because we can write every element as a certain sum of {{mat|term= 1 |pm=}} with itself, this is a 
{{
Definitionlink
|cyclic group|
|Context=|
|pm=.
}}
|Textform=Example 
|Category=
}}</text>
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We consider the analytic mapping
{{
Mapping/display
|name=
|\R| \Complex
|t| e^{ {{imaginary unit|}} t}{{=}}\cos t +  {{imaginary unit|}} \sin t
|pm=.
}}
Due to the exponential law
{{
Extra/Bracket
|text=or the addition theorems for the trigonometric functions|
|Ipm=|Epm=,
}} 
we have
{{
Relationchain 
| e^{  {{imaginary unit|}} (t+s)}
|| e^{ {{imaginary unit|}} t} e^{ {{imaginary unit|}} s}
||
||
||
|pm=.
}}
Therefore, this is a
{{
Definitionlink
|group homomorphism|
|pm=
}}
from the additive group {{mathl|term= (\R,+,0) |pm=}} into the multiplicative group {{mathl|term= ({{op:Unit group|\Complex}}, \cdot, 1) |pm=.}} We determine the 
{{
Definitionlink
|kernel|
|Context=group
|pm=
}}
and the image of this mapping. To determine the kernel, we must identify those real numbers {{mat|term= t |pm=}} fulfilling
{{
Mathcor/display|term1=
\cos t = 1
|and|term2=
\sin t = 0
|pm=.
}}
Because of the periodicity of the trigonometric functions, this is the case if and  only if {{mat|term= t |pm=}} is an integer  multiple of {{mat|term= 2 \pi|pm=.}} Hence, the kernel is the 
{{
Definitionlink
|subgroup|
|pm=
}}
{{mathl|term= 2 \pi \Z|pm=.}} For a point in the image, we have
{{
Relationchain
| {{op:Modulus|e^{ {{imaginary unit|}} t} }}
|| \sin^2 t + \cos^2 t
|| 1
|pm=;
}}
therefore, the image point belongs to the complex unit circle. The trigonometric functions run through the complete unit circle, so that the image group is the complex unit circle with its complex multiplication.
|Textform=Example 
|Category=
}}</text>
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Mathematical text/Fact
|Text=
{{
Factstructure
|Situation=
Let
{{
Mathcor|term1=
G
|and|term2=
H
|pm=
}}
denote
{{
Definitionlink
|groups|
|pm=,
}}
and let
{{
Mapping
|name=\varphi
|G|H
||
|pm=
}}
be a
{{
Definitionlink
|group homomorphism|
|pm=.
}}
|Condition=
|Segue=
|Conclusion=
Then the 
{{
Definitionlink
|image|
|pm=
}}
of {{mat|term= \varphi |pm=}} is a 
{{
Definitionlink
|subgroup|
|pm=
}}
of {{mat|term= H |pm=.}}
|Extra=
}}
|Textform=Fact 
|Category=
}}</text>
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  <page>
    <title>Group homomorphism/Image/Subgroup/Fact/Proof</title>
    <ns>0</ns>
    <id>316756</id>
    <revision>
      <id>2689221</id>
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        <username>Bocardodarapti</username>
        <id>289675</id>
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      <comment>New resource with &quot;{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Let {{ Relationchain |B | {{defeq|}} |  {{op:image| \varphi |}}  || || || |pm=. }} We have {{ Relationchain  | e_H ||\varphi(e_G) |\in | B || || |pm=. }} Let {{ Relationchain | h_1,h_2 |\in| B || || || |pm=. }}  Then there exist {{ Relationchain | g_1,g_2 |\in| G || || || |pm= }}  such that {{ Mathcor|term1= \varphi(g_1)=h_1 |and|term2= \varphi(g_2)=h_2 |pm=. }} Therefore, {{ Relationchain...&quot;</comment>
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Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
Let
{{
Relationchain
|B
| {{defeq|}} |  {{op:image| \varphi |}} 
||
||
||
|pm=.
}}
We have
{{
Relationchain 
| e_H
||\varphi(e_G)
|\in | B
||
||
|pm=.
}}
Let
{{
Relationchain
| h_1,h_2
|\in| B
||
||
||
|pm=.
}} 
Then there exist
{{
Relationchain
| g_1,g_2
|\in| G
||
||
||
|pm=
}} 
such that
{{
Mathcor|term1=
\varphi(g_1)=h_1
|and|term2=
\varphi(g_2)=h_2
|pm=.
}}
Therefore,
{{
Relationchain
| h_1 \cdot h_2
|| \varphi(g_1) \cdot  \varphi(g_2)
|| \varphi(g_1 \cdot g_2)
|\in| B
|pm=.
}}
Similarly, for
{{
Relationchain
| h
|\in| B
||
||
||
|pm=
}} 
there exists a
{{
Mathcor|term1=
g \in G
|fulfilling|term2=
\varphi(g)=h
|pm=.
}}
Hence,
{{
Relationchain
| h^{-1}
|| (\varphi(g))^{-1}
|| \varphi(g^{-1})
|\in| B
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
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Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
Let
{{
Relationchain
|B
| {{defeq|}} |  {{op:Image| \varphi |}} 
||
||
||
|pm=.
}}
We have
{{
Relationchain 
| e_H
||\varphi(e_G)
|\in | B
||
||
|pm=.
}}
Let
{{
Relationchain
| h_1,h_2
|\in| B
||
||
||
|pm=.
}} 
Then there exist
{{
Relationchain
| g_1,g_2
|\in| G
||
||
||
|pm=
}} 
such that
{{
Mathcor|term1=
\varphi(g_1)=h_1
|and|term2=
\varphi(g_2)=h_2
|pm=.
}}
Therefore,
{{
Relationchain
| h_1 \cdot h_2
|| \varphi(g_1) \cdot  \varphi(g_2)
|| \varphi(g_1 \cdot g_2)
|\in| B
|pm=.
}}
Similarly, for
{{
Relationchain
| h
|\in| B
||
||
||
|pm=
}} 
there exists a
{{
Mathcor|term1=
g \in G
|fulfilling|term2=
\varphi(g)=h
|pm=.
}}
Hence,
{{
Relationchain
| h^{-1}
|| (\varphi(g))^{-1}
|| \varphi(g^{-1})
|\in| B
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
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  <page>
    <title>Group homomorphism/Introduction/Section</title>
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    <id>316757</id>
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      <id>2689223</id>
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        <id>289675</id>
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      <comment>New resource with &quot;{{ Mathematical section{{{opt|}}} |Content= {{ inputdefinition |Group homomorphism/Definition|| }}  The set of all group homomorphisms from {{mat|term= G |pm=}} to {{mat|term= H |pm=}} is denoted by {{ Math/display|term=   {{op:Homomorphisms|G|H|}}  |pm=. }} {{ Definitionlink |Premath= |Linear mappings| |Context=| |pm= }} between vector spaces are in particular group homomorphisms. The following two lemmas follow directly from the definition. {{ inputfactproof |Group hom...&quot;</comment>
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Mathematical section{{{opt|}}}
|Content=
{{
inputdefinition
|Group homomorphism/Definition||
}}

The set of all group homomorphisms from {{mat|term= G |pm=}} to {{mat|term= H |pm=}} is denoted by
{{
Math/display|term=
  {{op:Homomorphisms|G|H|}} 
|pm=.
}}
{{
Definitionlink
|Premath=
|Linear mappings|
|Context=|
|pm=
}}
between vector spaces are in particular group homomorphisms. The following two lemmas follow directly from the definition.
{{
inputfactproof
|Group homomorphism/Inverse to inverse/Fact|Lemma||
}}

{{
inputfactprooftrivial
|Group homomorphism/Categorial properties/Fact|Lemma||
}}

{{
inputexample
|Group homomorphism/Z to Z/Example||
}}

{{
inputexample
|Group homomorphism/Z to Z mod d/Directly/Example||
}}

{{
inputexample
|Group homomorphism/Determinant/Example||
}}

{{
inputexample
|Group homomorphism/Sign/Example||
}}

{{
inputfactproof
|Group homomorphism/Z to group/Fact|Lemma||
}}

This lemma can be stated quickly by saying
{{
Relationchain
| G
| \cong | {{op:Homomorphisms| \Z | G |}} 
||
||
||
|pm=.
}}
It is more difficult to characterize the group homomorphisms from a group {{mat|term= G |pm=}} to {{mat|term= \Z |pm=.}} The group homomorphisms from {{mat|term= \Z |pm=}} to {{mat|term= \Z |pm=}} are just the multiplications with a fixed integer number {{mat|term= a |pm=,}} that is,
{{
Mapping/display
|name=
| \Z | \Z
| x | ax
|pm=.
}}
|Textform=Section
|Category=
|}}</text>
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    <revision>
      <id>2689226</id>
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        <username>Bocardodarapti</username>
        <id>289675</id>
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Mathematical section{{{opt|}}}
|Content=
{{
inputdefinition
|Group homomorphism/Definition||
}}

The set of all group homomorphisms from {{mat|term= G |pm=}} to {{mat|term= H |pm=}} is denoted by
{{
Math/display|term=
  {{op:Homomorphisms|G|H|K=}} 
|pm=.
}}
{{
Definitionlink
|Premath=
|Linear mappings|
|Context=|
|pm=
}}
between vector spaces are in particular group homomorphisms. The following two lemmas follow directly from the definition.
{{
inputfactproof
|Group homomorphism/Inverse to inverse/Fact|Lemma||
}}

{{
inputfactprooftrivial
|Group homomorphism/Categorial properties/Fact|Lemma||
}}

{{
inputexample
|Group homomorphism/Z to Z/Example||
}}

{{
inputexample
|Group homomorphism/Z to Z mod d/Directly/Example||
}}

{{
inputexample
|Group homomorphism/Determinant/Example||
}}

{{
inputexample
|Group homomorphism/Sign/Example||
}}

{{
inputfactproof
|Group homomorphism/Z to group/Fact|Lemma||
}}

This lemma can be stated quickly by saying
{{
Relationchain
| G
| \cong | {{op:Homomorphisms| \Z | G |K=}} 
||
||
||
|pm=.
}}
It is more difficult to characterize the group homomorphisms from a group {{mat|term= G |pm=}} to {{mat|term= \Z |pm=.}} The group homomorphisms from {{mat|term= \Z |pm=}} to {{mat|term= \Z |pm=}} are just the multiplications with a fixed integer number {{mat|term= a |pm=,}} that is,
{{
Mapping/display
|name=
| \Z | \Z
| x | ax
|pm=.
}}
|Textform=Section
|Category=
|}}</text>
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  </page>
  <page>
    <title>Template:Inputfactprooftrivial</title>
    <ns>10</ns>
    <id>316758</id>
    <revision>
      <id>2689224</id>
      <timestamp>2024-11-28T17:53:18Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{#ifeq: {{SUBPAGENAME}}|latex|&lt;div class=&quot;latex&quot;&gt;&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;\inputfactproof&lt;br /&gt;{{{{1}}}{{{fv|}}}}&lt;br /&gt;{{{{2}}}}&lt;br/&gt;{{#if:{{{3|}}}|{&lt;nowiki/&gt;{{{3|}}}&lt;nowiki/&gt;}|{} }}&lt;br/&gt;{{{:{{{1}}}|extra1={{{extra1|}}}||extra2={{{extra2|}}}||extra3={{{extra3|}}}}} }&lt;br /&gt;{This is trivial.} }&lt;/div&gt; |&lt;br clear=&quot;left&quot;/&gt;&lt;div class=&quot;factproof&quot;&gt; &lt;h2&gt; [[{{{1}}}|{{{2}}}]]{{#switch: {{SUBPAGENAME}}|refcontrol={{#ifexist:{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|{{:...&quot;</comment>
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      <text bytes="1130" sha1="c71gk9469ny34vdtqdqfvj9zh18guxc" xml:space="preserve">{{#ifeq: {{SUBPAGENAME}}|latex|&lt;div class=&quot;latex&quot;&gt;&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;\inputfactproof&lt;br /&gt;{{{{1}}}{{{fv|}}}}&lt;br /&gt;{{{{2}}}}&lt;br/&gt;{{#if:{{{3|}}}|{&lt;nowiki/&gt;{{{3|}}}&lt;nowiki/&gt;}|{} }}&lt;br/&gt;{{{:{{{1}}}|extra1={{{extra1|}}}||extra2={{{extra2|}}}||extra3={{{extra3|}}}}}
}&lt;br /&gt;{This is trivial.}
}&lt;/div&gt;
|&lt;br clear=&quot;left&quot;/&gt;&lt;div class=&quot;factproof&quot;&gt;
&lt;h2&gt; [[{{{1}}}|{{{2}}}]]{{#switch: {{SUBPAGENAME}}|refcontrol={{#ifexist:{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|{{:{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber}} [[{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|change]]|[[{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|Create referencenumber]]|}}|#default=}}&lt;/h2&gt;
&lt;div class=&quot;content&quot; style=&quot;{{{STYLE|font-style:italic;}}}&quot;&gt;{{:{{{1}}}|extra1={{{extra1|}}}||extra2={{{extra2|}}}||extra3={{{extra3|}}}}}&lt;/div&gt;
&lt;h3&gt; [[{{{1}}}/Proof|Proof]]  &lt;span class=&quot;noprint&quot;&gt;[{{fullurl:{{{1|}}}/Proof|action=edit}} &amp;nbsp;]&lt;/span&gt; &lt;/h3&gt; 
&lt;div class=&quot;content&quot;&gt;
{{qed|STYLE=text-align:justify;|TEXT=This is trivial.}}
&lt;/div&gt;&lt;/div&gt;
}}|}}
&lt;noinclude&gt;{{Semantic template|}}&lt;/noinclude&gt;</text>
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    <revision>
      <id>2689225</id>
      <parentid>2689224</parentid>
      <timestamp>2024-11-28T17:54:37Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
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      <text bytes="1031" sha1="t1l5qdwjf5yil3d2jpo3fypbn9jb4h7" xml:space="preserve">{{#ifeq: {{SUBPAGENAME}}|latex|&lt;div class=&quot;latex&quot;&gt;&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;\inputfactproof&lt;br /&gt;{{{{1}}}{{{fv|}}}}&lt;br /&gt;{{{{2}}}}&lt;br/&gt;{{#if:{{{3|}}}|{&lt;nowiki/&gt;{{{3|}}}&lt;nowiki/&gt;}|{} }}&lt;br/&gt;{{{:{{{1}}}|extra1={{{extra1|}}}||extra2={{{extra2|}}}||extra3={{{extra3|}}}}}
}&lt;br /&gt;{This is trivial.}
}&lt;/div&gt;
|&lt;br clear=&quot;left&quot;/&gt;&lt;div class=&quot;factproof&quot;&gt;
&lt;h2&gt; [[{{{1}}}|{{{2}}}]]{{#switch: {{SUBPAGENAME}}|refcontrol={{#ifexist:{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|{{:{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber}} [[{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|change]]|[[{{#titleparts:{{FULLPAGENAME}}|1|}}/{{{1|}}}/Factreferencenumber|Create referencenumber]]|}}|#default=}}&lt;/h2&gt;
&lt;div class=&quot;content&quot; style=&quot;{{{STYLE|font-style:italic;}}}&quot;&gt;{{:{{{1}}}|extra1={{{extra1|}}}||extra2={{{extra2|}}}||extra3={{{extra3|}}}}}&lt;/div&gt;
&lt;h3&gt; Proof &lt;/h3&gt; 
&lt;div class=&quot;content&quot;&gt;
{{qed|STYLE=text-align:justify;|TEXT=This is trivial.}}
&lt;/div&gt;&lt;/div&gt;
}}
&lt;noinclude&gt;{{Semantic template|}}&lt;/noinclude&gt;</text>
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  <page>
    <title>Group isomorphisms/Automorphism/Introduction/Section</title>
    <ns>0</ns>
    <id>316759</id>
    <revision>
      <id>2689227</id>
      <timestamp>2024-11-28T18:58:36Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical section{{{opt|}}} |Content= {{ inputdefinition |Group isomorphism/Definition|| }}  Bijective  {{ Definitionlink |Premath= |linear mappings| |Context=| |pm= }} are in particular group isomorphisms.  {{ inputdefinition |Groups/Isomorphic/Definition|| }}  {{ inputfactproof |Bijective group homomorphism/Inverse mapping/Homomorphism/Fact|Lemma|| }}  {{ inputexample |Group isomorphism/Real exponential function/Example|| }} Isomorphic groups are equal with respe...&quot;</comment>
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Mathematical section{{{opt|}}}
|Content=
{{
inputdefinition
|Group isomorphism/Definition||
}}

Bijective 
{{
Definitionlink
|Premath=
|linear mappings|
|Context=|
|pm=
}}
are in particular group isomorphisms.

{{
inputdefinition
|Groups/Isomorphic/Definition||
}}

{{
inputfactproof
|Bijective group homomorphism/Inverse mapping/Homomorphism/Fact|Lemma||
}}

{{
inputexample
|Group isomorphism/Real exponential function/Example||
}}
Isomorphic groups are equal with respect to their group-theoretic properties. An isomorphism of a group to itself is called {{Keyword|automorphism|pm=.}} The set of all automorphisms on {{mat|term=G|pm=}} form, with the composition of mappings, a group, which is denoted by {{mathl|term=  {{op:Aut|G|}} |pm=}} and which is called the {{Keyword|automorphism group|pm=}} of {{mat|term=G|pm=.}} Important examples of automorphisms are the so-called inner automorphisms.

{{
inputdefinition
|Group theory/Inner automorphism/Definition||
}}
The mapping {{mat|term=\kappa_g|pm=}} is also called the {{Keyword|conjugation|pm=}} with {{mat|term=g|pm=.}} If {{mat|term=G|pm=}} is a commutative Gruppe, then, because of
{{
Relationchain
| gxg^{-1}
|| xgg^{-1}
|| x
||
||
|pm=,
}}
the identity is the only inner automorphism. Therefore, this concept is only interesting for non-commutative groups.

{{
inputfactproof
|Inner automorphism/Automorphism/Fact|Lemma||
}}

{{
inputexample
|Matrices/Inner automorphisms/Base change/Example||
}}
|Textform=Section
|Category=
|}}</text>
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    <title>Template:Op:Aut</title>
    <ns>10</ns>
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      <timestamp>2024-11-28T18:59:12Z</timestamp>
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        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;&lt;includeonly&gt;{{#switch: {{#titleparts:{{FULLPAGENAME}}|1|-1}} |latex |#default= \operatorname{Aut}{{#if:{{{2|}}}|_{{{2|}}}|}} \, {{{1|}}} }}&lt;/includeonly&gt;&lt;noinclude&gt;{{Semantic template|}}&lt;/noinclude&gt;&quot;</comment>
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|latex
|#default= \operatorname{Aut}{{#if:{{{2|}}}|_{{{2|}}}|}} \, {{{1|}}}
}}&lt;/includeonly&gt;&lt;noinclude&gt;{{Semantic template|}}&lt;/noinclude&gt;</text>
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  <page>
    <title>Group homomorphism/Kernel/Injectivity criterion/Introduction/Section</title>
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    <id>316761</id>
    <revision>
      <id>2689236</id>
      <timestamp>2024-11-28T19:24:00Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical section{{{opt|}}} |Content= {{ inputdefinition |Group homomorphism/Kernel/Definition|| }}  {{ inputfactproof |Group homomorphism/Kernel/Subgroup/Fact|Lemma|| }}  {{ inputimage |Group homomorphism|svg| 300px {{!}}  {{!}}  |epsname=Group_homomorphism |User=Cronholm 144 |Domain= |License=CC-by-Sa 2.5 }} {{{extra1|}}} {{ inputfactproof |Group homomorphism/Injectivity and kernel/Fact|Lemma|| }} |Textform=Section |Category= |}}&quot;</comment>
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Mathematical section{{{opt|}}}
|Content=
{{
inputdefinition
|Group homomorphism/Kernel/Definition||
}}

{{
inputfactproof
|Group homomorphism/Kernel/Subgroup/Fact|Lemma||
}}

{{
inputimage
|Group homomorphism|svg| 300px {{!}}  {{!}} 
|epsname=Group_homomorphism
|User=Cronholm 144
|Domain=
|License=CC-by-Sa 2.5
}}
{{{extra1|}}}
{{
inputfactproof
|Group homomorphism/Injectivity and kernel/Fact|Lemma||
}}
|Textform=Section
|Category=
|}}</text>
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  <page>
    <title>Media literacy for the Arab World per Ahmed Al-Rawi</title>
    <ns>0</ns>
    <id>316762</id>
    <revision>
      <id>2689240</id>
      <timestamp>2024-11-28T20:16:21Z</timestamp>
      <contributor>
        <username>DavidMCEddy</username>
        <id>218607</id>
      </contributor>
      <comment>create</comment>
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      <text bytes="5891" sha1="527ksozqhclmdltxv0ll4ras5gjpnun" xml:space="preserve">:''This is a discussion of an interview 2024-11-21 with Simon Fraser University professor Ahmed Al-Rawi&lt;ref name=AlRawiSFU&gt;&lt;!--SFU homepage of Ahmed Al-Rawi--&gt;{{cite Q|Q131349551}}&lt;/ref&gt; about his research into how to understand and counter the rise in political polarization and violence worldwide. A 29:00 mm:ss podcast excerpted from the companion video will be posted here after it is released to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt; 

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs&lt;/ref&gt; and treating others with respect.&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] is different: Contributors there are asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;'' 

::[Video of the interview coming soon.] 
&lt;!--[[File: ... .webm|thumb|Legal concerns of Wikimedia Europe]]--&gt;
&lt;!--[[File: ... .ogg|thumb|29:00 mm:ss extract from interview recorded 2024-10-25 regarding the legal concerns of Wikimedia Europe.]]--&gt;

[[w:Simon Fraser University|Simon Fraser University]] professor Ahmed Al-Rawi&lt;ref name=AlRawiSFU/&gt; discusses the media literacy laboratory he co-founded at the Lebanese American University in Beiruit&lt;ref&gt;&lt;!-- Author archives: Ahmed Al-Rawi, LLRX--&gt;{{cite Q|Q131349668}}&lt;/ref&gt; and his research into how to understand and counter the rise in political polarization and violence worldwide. He is interviewed by Spencer Graves.

Al-Rawi is the author or co-author of a dozen books in the last dozen years plus co-editor of three others and author of dozens of articles.&lt;ref name=AlRawiCV&gt;&lt;!--Curriculum Vitae: Ahmed Al-Rawi--&gt;{{cite Q|Q131349693}}&lt;/ref&gt; Most of his publications describe the increase in political polarization and violence worldwide in recent decades and what might be done to counter it. His research has focused primarily on the Arab World and on Canada. At Simon Fraser and elsewhere he has taught classes on media, communications, democracy and power.

Al-Rawi is currently an Associate Professor of News, Social Media &amp; Public Communication in the School of Communication, Faculty of Communication, Art &amp; Technology at Simon Fraser University in [[w:Vancouver|Vancouver]], British Columbia, Canada and a scientist with the [[w:International Panel on the Information Environment|International Panel on the Information Environment]]&lt;ref&gt;&lt;!--Ahmed Al-Rawi, IPIE--&gt;{{cite Q|Q131349735}}&lt;/ref&gt; He has previously taught at other universities in Canada as well as in the [[w:Netherlands|Netherlands]] and in [[w:Oman|Oman]]. Twenty years ago he worked as a freelance radio journalist for the Pacifica Radio Network and before that as a translator for Iraq National Television, [[w:Baghdad|Baghdad]], Iraq.

== The threat == 

Internet company executives have knowingly increased political polarization and violence including the [[w:Rohingya genocide|Rohingya genocide]] in [[w:Myanmar|Myanmar]], because doing otherwise might have reduced their profits. Documentation of this is summarized in [[:Category:Media reform to improve democracy]].

==Discussion == 

:''[Interested readers are invite to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Ahmed Al-Rawi (2025, forthcoming) Mediated Racism and democracy in Canada: Interrogating the news industry, political systems, and public discourses, Routledge--&gt;{{cite Q|Q131349901|date=2025b}}
* &lt;!-- Ahmed Al-Rawi (2025, forthcoming) Disruptive Information in Canada, Bloomsbury--&gt;{{cite Q|Q131349920|date=2025a}}
* &lt;!-- Ahmed Al-Rawi et al. (2025) The Canadian Far-Right and Conspiracy Theories, Routledge--&gt;{{cite Q|Q131349937}}
* &lt;!-- Ahmed Al-Rawi (2024) The Iraqi Spring: Social Media and Political Activism, Amsterdam U. Pr.--&gt;{{cite Q|Q131350073}}
* &lt;!-- Ahmed Al-Rawi (2024) Online hate on social media, Palgrave Macmillan--&gt;{{cite Q|Q131350104}}
* &lt;!-- Ahmed Al-Rawi (2024) ISIS' propaganda machine : global mediated terrorism, Routledge--&gt;{{cite Q|Q131350154}}
* &lt;!-- Ahmed Al-Rawi (2023) Supernatural Creatures in Arabic Literary Tradition, Routledge--&gt;{{cite Q|Q131350208}}
* &lt;!-- Ahmed Al-Rawi (2021) Cyberwars in the Middle East, Routledge--&gt;{{cite Q|Q131350317}}
* &lt;!-- Ahmed Al-Rawi (2020) News 2.0: Journalists, Audiences and News on Social Media, Wiley-Blackwell--&gt;{{cite Q|Q131350446}}
* &lt;!-- Ahmed Al-Rawi (2020) Women's Activism and New Media in the Arab World, SUNY Pr.--&gt;{{cite Q|Q131350555}}
* &lt;!-- Ahmed Al-Rawi (2017) Islam on YouTube : Online Debates, Protests, and Extremism, Palgrave Macmillan--&gt;{{cite Q|Q131350577}}
* &lt;!-- Ahmed Al-Rawi (2012) Media Practice in Iraq, Springer--&gt;{{cite Q|Q131350656}}

[[Category:Politics]]
[[Category:Freedom and abundance]]
[[Category:Media reform to improve democracy]]</text>
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      <text bytes="991" sha1="4bzjq55qfhdvfeiklk250jkpofgt49n" xml:space="preserve">== Blocked ==

&lt;div class=&quot;user-block&quot; style=&quot;background:#ffe0e0; border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px&quot;&gt;
[[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: &lt;code&gt;&lt;nowiki&gt;{{unblock|your reason here ~~~~}}&lt;/nowiki&gt;&lt;/code&gt;
&lt;hr&gt;
If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.&lt;/div&gt;

—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:21, 28 November 2024 (UTC)</text>
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|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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Mathematical text/Fact
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{{
Factstructure
|Situation=
Let 
{{
Mathcor|term1=
G
|and|term2=
H
|pm=
}}
denote
{{
Definitionlink
|groups|
|pm=,
}}
and let
{{
Mapping
|name=\varphi
|G|H
||
|pm=
}}
be a
{{
Definitionlink
|group homomorphism|
|pm=.
}}
|Condition=
|Segue=
|Conclusion=
Then
{{
Relationchain
| \varphi (e_G)
|| e_H
||
||
||
|pm=
}}
and
{{
Relationchain 
| (\varphi(g))^{-1} 
|| \varphi  {{mabr| g^{-1} |}} 
||
||
||
|pm=
}}
for every
{{
Relationchain
| g
|\in| G 
||
||
||
|pm=.
}}
|Extra=
}}
|Textform=Fact 
|Category=
|Request=Inverse element under group homomorphism
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Proofstructure
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|Notation=
|Proof=
{{
Part of proof
|Goal=|Strategy=
|Proof=
To prove the first statement, consider
{{
Relationchain/display
| \varphi(e_G)
|| \varphi(e_G e_G)
|| \varphi(e_G) \varphi(e_G)
||
||
|pm=.
}}
Multiplication with {{mathl|term=  \varphi(e_G)^{-1} |pm=}} yields
{{
Relationchain
| e_H
|| \varphi(e_G)
||
||
||
|pm=.
}}
|Closure of part=
}} 
{{
Part of proof
|Goal=|Strategy=
|Proof=
To prove the second claim, we use
{{
Relationchain/display
| \varphi {{mabr| g^{-1} |}} \varphi(g)
|| \varphi {{mabr| g^{-1} g |}} 
|| \varphi(e_G)
|| e_H
|pm=.
}}
This means that {{mathl|term=  \varphi {{mabr| g^{-1} |}} |pm=}} has the property that characterizes the inverse element of {{mathl|term=  \varphi(g) |pm=.}} Since the inverse element in a group is, due to
{{
Factlink
|Factname=
Group/Inverse element/Unique/Fact
|Nr=
|pm=,
}}
uniquely determined, we must have
{{
Relationchain
| \varphi {{mabr| g^{-1} |}} 
|| (\varphi(g))^{-1}
||
||
||
|pm=.
}} 
|Closure of part=
}}
|Closure=
}}
|Textform=Proof 
|Category=See
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Let {{mathl|term= F,G,H |pm=}} denote
{{
Definitionlink
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|pm=.
}}
|Segue=Then the following properties hold.
|Conclusion=
{{
Enumeration4
|The identity
{{
Mapping/display
|name= {{op:identity||}} 
| G|G
||
|pm=
}}
is a
{{
Definitionlink
|group homomorphism|
|pm=.
}}
|If
{{
Mathcor|term1=
\varphi:F \rightarrow G
|and|term2=
\psi: G \rightarrow H
|pm=
}}
are group homomorphisms, then the composition
{{
Mapping 
|name= \psi \circ \varphi
| F|H
||
|pm=
}}
is a group homomorphism.
|For a
{{
Definitionlink
|subgroup|
|Context=|
|pm=
}}
{{
Relationchain
|F
|\subseteq|G
||
||
||
|pm=,
}}
the inclusion {{mathl|term= F \hookrightarrow G |pm=}} is a group homomorphism.
|Let {{mat|term= \{e\} |pm=}} be the
{{
Definitionlink
|trivial group|
|pm=.
}}
Then the mapping {{mathl|term= \{e\} \rightarrow G |pm=}} that sends {{mat|term= e |pm=}} to {{mat|term= e_G|pm=}} is a group homomorphism. Moreover, the
{{
Extra/Bracket
|text=constant|
|Ipm=|Epm=
}}
mapping {{mathl|term= G \rightarrow \{e\} |pm=}} is a group homomorphism.
}}
|Extra=
}}
|Textform=Fact 
|Category=
|Factname=
}}</text>
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Let
{{
Relationchain
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||
||
||
|pm=
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be fixed. The mapping
{{
Mapping/display
|name=
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|n|dn
|pm=,
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is a
{{
Definitionlink
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|Context=|
|pm=.
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This follows immediately from the distributive law. For
{{
Relationchain
|d
|\geq|1
||
||
||
|pm=,
}}
this mapping is
{{
Definitionlink
|injective|
|Context=|
|pm=,
}}
and the image is the 
{{
Definitionlink
|subgroup|
|Context=|
|pm=
}}
{{
Relationchain
| \Z d
|\subseteq| \Z
||
||
||
|pm=.
}}
For
{{
Relationchain
|d
|| 0
||
||
||
|pm=,
}}
we have the zero mapping. For
{{
Relationchain
|d
||1
||
||
||
|pm=,
}}
the mapping is the 
{{
Definitionlink
|identity|
|Context=|
|pm=.
}}
For
{{
Relationchain
|d
|\geq|2
||
||
||
|pm=,
}}
the mapping is not 
{{
Definitionlink
|surjective|
|Context=|
|pm=.
}}
|Textform=Example 
|Category=
}}</text>
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Relationchain
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||
||
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{{
Relationchain/display
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||
||
||
|pm=,
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together with the addition described in
{{
Exerciselink
|Exercisename=
Z modulo n/Representatives/Associativity and group/Exercise
|Nr=
|pm=,
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which makes it a group. The mapping
{{
Mapping/display
|name= \varphi
| \Z | {{op:Zmod|d|}} 
|| 
|pm=
}}
that sends an integer number {{mat|term= n |pm=}} to its remainder after division by {{mat|term= d |pm=}}  is a 
{{
Definitionlink
|group homomorphism|
|Context=|
|pm=.
}}
For, if
{{
Mathcor|term1=
m=ad+r
|and|term2=
n=bd+s
|pm=
}}
are given with
{{
Relationchain
|0
|\leq| r,s
|&lt;|d
||
||
|pm=,
}}
then
{{
Relationchain/display
|m+n
||(a+b)d +r+s
||
||
||
|pm=.
}}
Here, it may happen that
{{
Relationchain
|r+s
|\geq|d
||
||
||
|pm=.
}}
In this case,
{{
Relationchain/display
| \varphi(m+n)
|| r+s-d
||
||
||
|pm=,
}}
and this coincides with the addition of
{{
Mathcor|term1=
r
|and|term2=
s
|pm=
}}
in {{mathl|term=  {{op:Zmod|d|}} |pm=.}} This mapping is surjective, but not injective.
|Category=
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For a
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Definitionlink
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|Context=|
|pm=
}}
{{mat|term= K |pm=}} and
{{
Relationchain
| n
|\in| \N_+
||
||
||
|pm=,
}}
the
{{
Definitionlink
|determinant|
|Context=|
|pm=
}}
{{
Mapping/display
|name=  {{op:Determinant||}} 
|  {{op:GLG|n|K}}|K^\times
| M |  {{op:Determinant|M|}}
|pm=,
}}
is a
{{
Definitionlink
|group homomorphism|
|Context=|
|pm=.
}}
This follows from
{{
Factlink
|Preword=the|multiplication theorem for the determinant|Factname=
Determinant/Multiplication theorem/Fact
|Nr=
|pm=
}}
and
{{
Factlink
|Factname=
Determinant/Zero, linear dependent and rank property/Fact
|Nr=
|pm=.
}}
|Textform=Example 
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Mathematical text/Example
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The assignment
{{
Mapping/display
|name=
|S_n | \{1,-1\}
|\pi| {{op:Sign|\pi}}
|pm=,
}}
where {{mat|term= S_n|pm=}} denotes the 
{{
Definitionlink
|permutation group|
|Context=|
|pm=
}}
for {{mat|term= n |pm=}} elements, is a
{{
Definitionlink
|group homomorphism|
|pm=,
}}
due to
{{
Factlink
|Factname=
Permutation/Sign/Group homomorphism/Fact
|Nr=
|pm=.
}}
|Textform=Example 
|Category=
}}</text>
      <sha1>t6ajut3ey280b9vkrguv81ptni93v92</sha1>
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  </page>
  <page>
    <title>Group homomorphism/Z to group/Fact</title>
    <ns>0</ns>
    <id>316772</id>
    <revision>
      <id>2689266</id>
      <timestamp>2024-11-29T08:28:48Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Fact |Text= {{ Factstructure |Situation= |Condition= Let {{mat|term= G |pm=}} denote a  {{ Definitionlink |group| |pm=. }} |Segue= |Conclusion= Then there is a correspondence between group elements {{ Relationchain | g |\in| G  |pm= }} and {{ Definitionlink |group homomorphisms| |pm= }} {{mat|term= \varphi|pm=}} from {{mat|term= \Z|pm=}} to {{mat|term= G |pm=,}} given by {{  Math/display|term= g \longmapsto  (   n \mapsto g^n )  \text{ and } \varphi...&quot;</comment>
      <origin>2689266</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="591" sha1="ai0sqnbbafj1696qkg06653pn0j9c90" xml:space="preserve">{{
Mathematical text/Fact
|Text=
{{
Factstructure
|Situation=
|Condition=
Let {{mat|term= G |pm=}} denote a 
{{
Definitionlink
|group|
|pm=.
}}
|Segue=
|Conclusion=
Then there is a correspondence between group elements
{{
Relationchain
| g
|\in| G 
|pm=
}}
and
{{
Definitionlink
|group homomorphisms|
|pm=
}}
{{mat|term= \varphi|pm=}} from {{mat|term= \Z|pm=}} to {{mat|term= G |pm=,}} given by
{{
 Math/display|term=
g \longmapsto  (   n \mapsto g^n )  \text{ and } \varphi \longmapsto \varphi(1)
|pm=.
}}
|Extra=
}}
|Textform=Fact 
|Category=
|Request=Homomorphisms from Z to a group G 
}}</text>
      <sha1>ai0sqnbbafj1696qkg06653pn0j9c90</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Z to group/Fact/Proof</title>
    <ns>0</ns>
    <id>316773</id>
    <revision>
      <id>2689269</id>
      <timestamp>2024-11-29T08:35:44Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Let {{ Relationchain | g |\in| G  |pm= }} be fixed. That the mapping {{ Mapping/display |name= \varphi_g | \Z | G | n | g^n |pm=, }} is a   {{ Definitionlink |group homomorphism| |Context=| |pm=, }} is just a reformulation of {{ Factlink |Preword=the|exponential laws|Factname= Group theory/Exponential laws/Fact |Nr= |pm=. }} Because of {{ Relationchain | \varphi_g(1) || g^{1} || g || || |pm...&quot;</comment>
      <origin>2689269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="990" sha1="e9ck3ctws43is1m9ebcx6in65wuovne" xml:space="preserve">{{
Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
Let
{{
Relationchain
| g
|\in| G 
|pm=
}}
be fixed. That the mapping
{{
Mapping/display
|name= \varphi_g
| \Z | G
| n | g^n
|pm=,
}}
is a  
{{
Definitionlink
|group homomorphism|
|Context=|
|pm=,
}}
is just a reformulation of
{{
Factlink
|Preword=the|exponential laws|Factname=
Group theory/Exponential laws/Fact
|Nr=
|pm=.
}}
Because of
{{
Relationchain
| \varphi_g(1)
|| g^{1}
|| g
||
||
|pm=,
}}
we obtain from the power mapping {{mat|term= \varphi_g |pm=}} the group element back. Moreover, a group homomorphism
{{
Mapping
|name= \varphi
| \Z | G
||
|pm=
}} 
is uniquely determined by {{mathl|term= \varphi(1) |pm=,}} as
{{
Relationchain
| \varphi(n)
|| (\varphi(1))^{n}
||
||
||
|pm=
}}
for {{mat|term= n |pm=}} positive, and
{{
Relationchain 
| \varphi(n)
|| {{mabr| (\varphi(1))^{-1} |}}^{-n}
||
||
||
|pm=
}}
for {{mat|term= n |pm=}} negative must hold.
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
      <sha1>e9ck3ctws43is1m9ebcx6in65wuovne</sha1>
    </revision>
  </page>
  <page>
    <title>Linear algebra (Osnabrück 2024-2025)/Permutation/Sign/Group homomorphism/Fact/Factreferencenumber</title>
    <ns>0</ns>
    <id>316774</id>
    <revision>
      <id>2689270</id>
      <timestamp>2024-11-29T08:36:46Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{Number in course{{{opt|}}}|Theorem|18|13|}}&quot;</comment>
      <origin>2689270</origin>
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      <format>text/x-wiki</format>
      <text bytes="45" sha1="37oi0wen69csgppj4mmli4mnahtl6lb" xml:space="preserve">{{Number in course{{{opt|}}}|Theorem|18|13|}}</text>
      <sha1>37oi0wen69csgppj4mmli4mnahtl6lb</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Kernel/Definition</title>
    <ns>0</ns>
    <id>316775</id>
    <revision>
      <id>2689273</id>
      <timestamp>2024-11-29T08:46:33Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Definition |Text= {{:Group homomorphism/Situation|pm=.}} Then the  {{ Definitionlink |Premath= |preimage| |Context=| |pm= }} of the neutral element is called the {{Word of definition|kernel|pm=}} of {{mat|term=\varphi|pm=,}} denoted by {{ Relationchain/display | {{op:Kern| \varphi}} || \varphi^{-1}(e_H) || {{Setcond|g \in G|\varphi(g){{=}}e_H }} || || |pm=. }} |Textform=Definition |Category=    |Word of definition=Kernel (group homomorphism) }}&quot;</comment>
      <origin>2689273</origin>
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      <format>text/x-wiki</format>
      <text bytes="469" sha1="avs8xteeccklvw76b1t8bo8i7daan86" xml:space="preserve">{{
Mathematical text/Definition
|Text=
{{:Group homomorphism/Situation|pm=.}} Then the 
{{
Definitionlink
|Premath=
|preimage|
|Context=|
|pm=
}}
of the neutral element is called the {{Word of definition|kernel|pm=}} of {{mat|term=\varphi|pm=,}} denoted by
{{
Relationchain/display
| {{op:Kern| \varphi}}
|| \varphi^{-1}(e_H)
|| {{Setcond|g \in G|\varphi(g){{=}}e_H }}
||
||
|pm=.
}}
|Textform=Definition
|Category=   
|Word of definition=Kernel (group homomorphism)
}}</text>
      <sha1>avs8xteeccklvw76b1t8bo8i7daan86</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Situation</title>
    <ns>0</ns>
    <id>316776</id>
    <revision>
      <id>2689274</id>
      <timestamp>2024-11-29T08:49:21Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Situation{{{opt|}}} |Text= Let {{ mathcor|term1= {{{G|G}}} |and|term2= {{{H|H}}} }} be {{ Definitionlink |groups| |pm=, }} and let {{ Mapping/display |name={{{\varphi|\varphi}}} |{{{G|G}}}|{{{H|H}}} || }} be a {{{extra1|}}} {{ Definitionlink |group homomorphism| |pm={{{pn|}}} }}  |Textform=Situation |}}&quot;</comment>
      <origin>2689274</origin>
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      <format>text/x-wiki</format>
      <text bytes="325" sha1="mjlxbzb7zy63px7vdo967d3318jph10" xml:space="preserve">{{
Mathematical text/Situation{{{opt|}}}
|Text=
Let
{{
mathcor|term1=
{{{G|G}}}
|and|term2=
{{{H|H}}}
}}
be
{{
Definitionlink
|groups|
|pm=,
}}
and let
{{
Mapping/display
|name={{{\varphi|\varphi}}}
|{{{G|G}}}|{{{H|H}}}
||
}}
be a {{{extra1|}}}
{{
Definitionlink
|group homomorphism|
|pm={{{pn|}}}
}} 
|Textform=Situation
|}}</text>
      <sha1>mjlxbzb7zy63px7vdo967d3318jph10</sha1>
    </revision>
    <revision>
      <id>2689275</id>
      <parentid>2689274</parentid>
      <timestamp>2024-11-29T08:49:50Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <origin>2689275</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="325" sha1="6vexshwi5cx1vatuesehhel4zm7wwka" xml:space="preserve">{{
Mathematical text/Situation{{{opt|}}}
|Text=
Let
{{
mathcor|term1=
{{{G|G}}}
|and|term2=
{{{H|H}}}
}}
be
{{
Definitionlink
|groups|
|pm=,
}}
and let
{{
Mapping/display
|name={{{\varphi|\varphi}}}
|{{{G|G}}}|{{{H|H}}}
||
}}
be a {{{extra1|}}}
{{
Definitionlink
|group homomorphism|
|pm={{{pm|}}}
}} 
|Textform=Situation
|}}</text>
      <sha1>6vexshwi5cx1vatuesehhel4zm7wwka</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Kernel/Subgroup/Fact</title>
    <ns>0</ns>
    <id>316777</id>
    <revision>
      <id>2689276</id>
      <timestamp>2024-11-29T08:53:10Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Fact |Text= {{ Factstructure |Situation= {{:Group homomorphism/Situation|pm=.}}  |Condition= |Segue= |Conclusion= Then the {{ Definitionlink |kernel| |Context=group |pm= }} of {{mat|term= \varphi |pm=}} is a {{ Definitionlink |subgroup| |pm= }} of {{mat|term= G |pm=.}} |Extra= }} |Textform=Fact  |Category= |Factname= |Request=Kernel of a group homomorphism }}&quot;</comment>
      <origin>2689276</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="382" sha1="e5tuq826rcvh6sfv5qzkcruqlxk7p9a" xml:space="preserve">{{
Mathematical text/Fact
|Text=
{{
Factstructure
|Situation=
{{:Group homomorphism/Situation|pm=.}} 
|Condition=
|Segue=
|Conclusion=
Then the
{{
Definitionlink
|kernel|
|Context=group
|pm=
}}
of {{mat|term= \varphi |pm=}} is a
{{
Definitionlink
|subgroup|
|pm=
}}
of {{mat|term= G |pm=.}}
|Extra=
}}
|Textform=Fact 
|Category=
|Factname=
|Request=Kernel of a group homomorphism
}}</text>
      <sha1>e5tuq826rcvh6sfv5qzkcruqlxk7p9a</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Kernel/Subgroup/Fact/Proof</title>
    <ns>0</ns>
    <id>316778</id>
    <revision>
      <id>2689277</id>
      <timestamp>2024-11-29T08:56:27Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= Because of {{ Relationchain | \varphi(e_G) || e_H || || || |pm=, }} we have {{ Relationchain | e_G |\in|  {{op:Kern|\varphi|}}  || || || |pm=. }} Let {{ Relationchain | g,g' |\in|  {{op:Kern|\varphi|}}  || || || |pm=. }}  Then {{ Relationchain/display | \varphi(g g') || \varphi(g) \varphi(g') || e_H e_H || e_H  || |pm=; }} therefore, also {{ Relationchain | g g' |\in| {{op:Kern|\varphi|}}...&quot;</comment>
      <origin>2689277</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="997" sha1="dii16sj1sfugs3hk40caxq6fuqvkk5z" xml:space="preserve">{{
Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
Because of
{{
Relationchain
| \varphi(e_G)
|| e_H
||
||
||
|pm=,
}}
we have
{{
Relationchain
| e_G
|\in|  {{op:Kern|\varphi|}} 
||
||
||
|pm=.
}}
Let
{{
Relationchain
| g,g'
|\in|  {{op:Kern|\varphi|}} 
||
||
||
|pm=.
}} 
Then
{{
Relationchain/display
| \varphi(g g')
|| \varphi(g) \varphi(g')
|| e_H e_H
|| e_H 
||
|pm=;
}}
therefore, also
{{
Relationchain
| g g'
|\in| {{op:Kern|\varphi|}} 
||
||
||
|pm=.
}}
Hence, the kernel is a submonoid. Now, let
{{
Relationchain
| g
|\in|  {{op:Kern|\varphi|}} 
||
||
||
|pm=,
}} 
and consider the inverse element {{mat|term=  g^{-1} |pm=.}} Due to
{{
Factlink
|Factname=
Group homomorphism/Inverse to inverse/Fact
|Nr=
|pm=,
}}
we have
{{
Relationchain/display
| \varphi {{mabr| g^{-1} |}} 
|| (\varphi (g))^{-1} 
|| e_H^{-1}
|| e_H
||
|pm=;
}}
Hence,
{{
Relationchain
| g^{-1}  
|\in| {{op:Kern|\varphi|}} 
||
||
||
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
      <sha1>dii16sj1sfugs3hk40caxq6fuqvkk5z</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Injectivity and kernel/Fact</title>
    <ns>0</ns>
    <id>316779</id>
    <revision>
      <id>2689278</id>
      <timestamp>2024-11-29T09:31:07Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Fact |Text= {{ Factstructure |Situation= Let  {{ Mathcor|term1= G |and|term2= H |pm= }} be {{ Definitionlink |groups| |pm=. }} |Condition= |Segue= |Conclusion= A {{ Definitionlink |group homomorphism| |pm= }} {{ Mapping |name= \varphi |G|H || |pm= }} is  {{ Definitionlink |injective| |pm= }} if and only if the {{ Definitionlink |kernel| |context=group |pm= }} of {{mat|term= \varphi|pm=}} is trivial. |Extra= }} |Textform=Fact  |Category= |Factname=Ker...&quot;</comment>
      <origin>2689278</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="507" sha1="05iupnj7asazsopzf1up021zy5dvoby" xml:space="preserve">{{
Mathematical text/Fact
|Text=
{{
Factstructure
|Situation=
Let 
{{
Mathcor|term1=
G
|and|term2=
H
|pm=
}}
be
{{
Definitionlink
|groups|
|pm=.
}}
|Condition=
|Segue=
|Conclusion=
A
{{
Definitionlink
|group homomorphism|
|pm=
}}
{{
Mapping
|name= \varphi
|G|H
||
|pm=
}}
is 
{{
Definitionlink
|injective|
|pm=
}}
if and only if the
{{
Definitionlink
|kernel|
|context=group
|pm=
}}
of {{mat|term= \varphi|pm=}} is trivial.
|Extra=
}}
|Textform=Fact 
|Category=
|Factname=Kernel criterion for injectivity
}}</text>
      <sha1>05iupnj7asazsopzf1up021zy5dvoby</sha1>
    </revision>
  </page>
  <page>
    <title>Group homomorphism/Injectivity and kernel/Fact/Proof</title>
    <ns>0</ns>
    <id>316780</id>
    <revision>
      <id>2689279</id>
      <timestamp>2024-11-29T09:38:22Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Proof |Text= {{ Proofstructure |Strategy= |Notation= |Proof= If {{mat|term= \varphi|pm=}} is injective, then every element {{ Relationchain | h |\in| H  || || || |pm= }} is hit by at most one element from {{mat|term= G |pm=.}} As {{mat|term= e_G |pm=}} is sent to {{mat|term= e_H |pm=,}} no further element can be sent to {{mat|term= e_H |pm=.}} Therefore, {{ Relationchain |  {{op:Kern|\varphi|}}  || \{e_G\} || || || |pm=. }} Now assume that this holds...&quot;</comment>
      <origin>2689279</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1037" sha1="5fkzm5kfspc5j7n1n6azvakaztw1x8k" xml:space="preserve">{{
Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
If {{mat|term= \varphi|pm=}} is injective, then every element
{{
Relationchain
| h
|\in| H 
||
||
||
|pm=
}}
is hit by at most one element from {{mat|term= G |pm=.}} As {{mat|term= e_G |pm=}} is sent to {{mat|term= e_H |pm=,}} no further element can be sent to {{mat|term= e_H |pm=.}} Therefore,
{{
Relationchain
|  {{op:Kern|\varphi|}} 
|| \{e_G\}
||
||
||
|pm=.
}}
Now assume that this holds. Let
{{
Relationchain
|  g, \tilde{g} 
|\in| G
||
||
||
|pm=
}}
be elements mapping to
{{
Relationchain
| h
|\in| H 
||
||
||
|pm=.
}}
Then
{{
Relationchain/display
| \varphi {{mabr| g \tilde{g}^{-1} |}} 
|| \varphi(g) \varphi (\tilde{g})^{-1}
|| h h^{-1}
|| e_H 
||
|pm=;
}}
hence,
{{
Relationchain
|  g \tilde{g}^{-1}
|\in| {{op:Kern| \varphi |}} 
||
||
||
|pm=,
}}
and so
{{
Relationchain
| g \tilde{g}^{-1}
|| e_G
||
||
||
|pm=
}}
by the condition. Therefore, 
{{
Relationchain
|g
||\tilde{g}
||
||
||
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
      <sha1>5fkzm5kfspc5j7n1n6azvakaztw1x8k</sha1>
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