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What is this video trying to tell us? http://www.youtube.com/watch?v=JX3VmDgiFnY

The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic projection is wrong (since for example some fractional linear transformations have only one fixed point, which is impossible for the rotation).

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    $\begingroup$ You should read ima.umn.edu/~arnold/papers/moebius.pdf if you haven't already. From the first Google hit for the name of the video: ima.umn.edu/~arnold/moebius. Also linked from the Youtube page. $\endgroup$
    – Jonas Meyer
    Commented May 26, 2010 at 6:15
  • $\begingroup$ The video shows for instance the translations, which have one fixed point. $\endgroup$
    – Jonas Meyer
    Commented May 26, 2010 at 6:28
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    $\begingroup$ Indeed, as far as I can tell the video makes no such claim as stated in the question. I am voting to close. $\endgroup$
    – Harald Hanche-Olsen
    Commented May 26, 2010 at 12:29
  • $\begingroup$ Mobius transformations are beautiful. In particular loxodromic elements (generated in the video by lifting the sphere while spinning about the north-south axis) give rise to wonderful patterns both visually and inside of geometric topology. $\endgroup$
    – Sam Nead
    Commented May 26, 2010 at 13:20

1 Answer 1

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Any Möbius transformations is a composite of a rotation of S2 (3 degrees of freedom), along with a translation and dilation of ℝ2 (3 degrees of freedom), adding up to the six dimensions of the Lie group PSL(2,ℂ) = group of Möbius transformations.

In the video, the translations are depicted by letting the sphere move left and right on the surface of the plane, while the dilations are depicted by lifting the sphere in the third dimension.

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  • $\begingroup$ Thanks. I am glad that there is some mathematical content in this video. $\endgroup$
    – Evgeny Shinder
    Commented May 26, 2010 at 16:18

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