Image:KleinBottle-Figure8-01.png

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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
Source

Own drawing, Mathematica 5.1
Date

05/08/06
Author

Fropuff, Inductiveload
Permission
(Reusing this image)

The original image was released into the public domain by Fropuff:

Public domain This image 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.

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The derived, redrawn, edited image was released into the public domain by Inductiveload:

Public domain This image 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.

العربية | Български | Deutsch | Ελληνικά | English | Español | Français | Italiano | 日本語 | 한국어 | Polski | Русский | Српски / Srpski | Svenska | தமிழ் | Українська | ‪中文(繁體)‬ | ‪中文(简体)‬  +/-
Other versions For a cut-away version see Image:KleinBottle-Figure8-02.png.

[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.

x = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \cos u
y = \left(r + \cos\frac{u}{2}\sin v - \sin\frac{u}{2}\sin 2v\right) \sin u
z = \sin\frac{u}{2}\sin v + \cos\frac{u}{2}\sin 2v

[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 -> 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.

Historique du fichier

Cliquer sur une date et une heure pour voir le fichier tel qu’il était à ce moment-là

Date et heureDimensionsUtilisateurCommentaire
actuel16 février 2007 à 08:09560×400 (149 Kio)Inductiveload
5 mars 2005 à 00:55300×226 (75 Kio)Dbenbenn (losslessly compressed with pngcrush, 18% smaller)
3 mars 2005 à 19:45300×226 (91 Kio)Fropuff (Figure-eight immersion of a Klein bottle. {{PD}})

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