On 1/11/08, Henry Baker <hbaker1@pipeline.com> wrote:

Very cool animation of Moebius transformations:

http://youtube.com/watch?v=JX3VmDgiFnY

I trawled through the (mostly unrewarding) comments attached to the video --- the amount of repetition in these postings is almost as noteworthy as their dismally juvenile style and general lack of content. However amongst the dross turned up the following enquiry, contributed independently by TheHeadlessIndian and reg2800, who wonder about representing loxodromic transformations using this technique. [For the uninitiated, these are the continuous "motions" which result from iterating a general infinitesimal Moebius transformation, each particle pursuing a sigmoid path spiralling between a pair of fixed points.] Thinking about this I promptly tumbled into an elephant trap. [I have spent a lot of time sitting in these over Xmas, one way and another...] The spin groups are isomorphic, so their Cartan subgroups --- essentially the elementary motions --- must correspond; but if the projection sphere pursues a helical path in space [say, along an axis lying in the plane] I cannot understand how any projected path can possibly have any fixed points! In several special cases which are easy to think about --- e.g. spiral dilations, elliptic transformations (non-intersecting coaxial circular paths) --- the correspondence does work; although I can't seem to manage hyperbolic (intersecting circular paths). Can anybody sort me out here (along with TheHeadlessIndian and reg2800)? Fred Lunnon