Imej:KleinBottle-Figure8-01.png
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KleinBottle-Figure8-01.png (75KB, MIME type: image/png)
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Figure-eight immersion of a Klein bottle into R3. Made with Mathematica. For a cut-away version see Image:KleinBottle-Figure8-02.png.
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[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 some fixed positive constant.
[edit] Mathematica source
Klein8[r_:2] =
Function[{u, v},
{
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Cos[u],
(r + Cos[u/2]Sin[v] - Sin[u/2]Sin[2v]) Sin[u],
Sin[u/2]Sin[v] + Cos[u/2]Sin[2v]
}
]
ParametricPlot3D[Evaluate[Klein8[][u, v]], {u, 0, 2Pi}, {v, 0, 2Pi},
PlotPoints -> 40, Boxed -> False, Axes -> False]
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