Image:KleinBottle-01.png

From Wikipedia, the free encyclopedia

Image
File history
File links

No higher resolution available.
KleinBottle-01.png‎ (240 × 300 pixels, file size: 64 KB, MIME type: image/png)

This is a file from the Wikimedia Commons. The description on its description page there is shown below.Commons is a freely licensed media file repository. You can help.

1 Summary
2 See also
3 Licensing
4 Parameterization
5 Mathematica source

Summary

Standard immersion of a Klein bottle into R³. Made with Mathematica.

Licensing

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.

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 $0 \leq u < \pi$ :

$x = 6 \cos u(1 + \sin u) + 4r\left(1 - \frac{\cos u}{2}\right) \cos u \cos v$

$y = 16 \sin u + 4r\left(1 - \frac{\cos u}{2}\right) \sin u \cos v$

$z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v$

For $\pi \leq u < 2\pi$ :

$x = 6 \cos u(1 + \sin u) - 4r\left(1 - \frac{\cos u}{2}\right) \cos v$

$y = 16 \sin u\,$

$z = 4r\left(1 - \frac{\cos u}{2}\right) \sin v$

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.

	Date/Time	Dimensions	User	Comment
current	23:39, 12 December 2006	240×300 (64 KB)	Mahahahaneapneap	(pngcrushed)
	10:21, 15 September 2006	240×300 (77 KB)	Dark knight	(Added transparency, unfortunately dimension rose)
	02:23, 4 March 2005	240×300 (57 KB)	Dbenbenn	(losslessly compressed with pngcrush, 20% smaller)
	17:44, 3 March 2005	240×300 (71 KB)	Fropuff	(Standard immersion of a Klein bottle. {{PD}})

File links

The following pages on Schools Wikipedia link to this image (list may be incomplete):

Retrieved from "http://en.wikipedia.org/wiki/Image:KleinBottle-01.png"