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Summary
Description |
Figure-eight immersion of a Klein bottle into R3. Made with Mathematica. English: The "figure 8" immersion of the Klein bottle.
Italiano: L'immersione a "figura 8" della bottiglia di Klein.
Русский: Реализация бутылки Клейна в виде восьмерки
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Date |
05/08/06 |
Source |
Own drawing, Mathematica 5.1 |
Author |
Fropuff, Inductiveload |
Permission ( Reusing this file) |
The original image was released into the public domain by Fropuff:
Public domainPublic domainfalsefalse |
![Public domain](../0/36.png) |
This work has been released into the public domain by its author, Fropuff. This applies worldwide. In some countries this may not be legally possible; if so: Fropuff grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse
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The derived, redrawn, edited image was released into the public domain by Inductiveload:
Public domainPublic domainfalsefalse |
![Public domain](../0/36.png) |
This work has been released into the public domain by its author, Inductiveload. This applies worldwide. In some countries this may not be legally possible; if so: Inductiveload grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse
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![Gnome-x-office-drawing.svg](../14/1498.png) |
File:KleinBottle-Figure8-01.svg is a vector version of this file. It should be used in place of this raster image when superior.
File:KleinBottle-Figure8-01.png ![Gnome-go-next.svg](../14/1499.png) File:KleinBottle-Figure8-01.svg
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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 some fixed positive constant.
![x = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u](20822.png)
![y = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u](20823.png)
![z = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v](20824.png)
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 -> 60, Boxed -> False, Axes -> False, ImageSize -> 800]
This image was then antialised with Chris Hill's code, made transparent around the surface and had stray pixels removed in an image editor.
File usage
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