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Summary
Standard immersion of a Klein bottle into R3. Made with Mathematica.
See also
Image:Klein bottle.svg
Licensing
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I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide.
In case this is not legally possible: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.
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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 :
For :
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}]
File history
Click on a date/time to view the file as it appeared at that time.
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Date/Time |
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User |
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current |
23:39, 12 December 2006 |
240×300 (64 KB) |
Mahahahaneapneap |
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10:21, 15 September 2006 |
240×300 (77 KB) |
Dark knight |
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02:23, 4 March 2005 |
240×300 (57 KB) |
Dbenbenn |
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17:44, 3 March 2005 |
240×300 (71 KB) |
Fropuff |
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File links
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