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[edit] Summary
Standard immersion of a Klein bottle into R3. Made with Mathematica.
[edit] See also
Image:Klein bottle.svg
[edit] Licensing
[edit] Parameterization
This immersion of the Klein bottle into R3 is given by the following parameterization. Here the parameters u and v run from 0 to 2π and r is a fixed positive constant.
For
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For
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[edit] Mathematica source
KleinBottle[r_:1] =
Function[{u, v},
UnitStep[Sin[u]]
{
6 Cos[u](1 + Sin[u]) + 4r(1 - Cos[u]/2) Cos[u]Cos[v],
16 Sin[u] + 4r(1 - Cos[u]/2) Sin[u]Cos[v],
4r(1 - Cos[u]/2) Sin[v]
}
+ (1 - UnitStep[Sin[u]])
{
6 Cos[u](1 + Sin[u]) - 4r(1 - Cos[u]/2) Cos[v],
16 Sin[u],
4r(1 - Cos[u]/2) Sin[v]
}
]
ParametricPlot3D[Evaluate[KleinBottle[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
PlotPoints -> {50, 19}, Boxed -> False, Axes -> False,
ViewPoint -> {0.454, -2.439, -2.301}]
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