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Math Help - Cross ratios, inversions and stereographic projections

  1. #1
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    Cross ratios, inversions and stereographic projections

    Hi all,

    I wasn't sure whether to put this in here or not, it's regarding geometry, stereographic projections and mobius maps and so on but this seemed to be the most appropriate place. Anyway:

    If u,v \in \mathbb{C} correspond to points P, Q on S^2, and d denotes the angular distance from P to Q on S^2, show that -\tan^2(\frac{d}{2}) is the cross ratio of the points u, v, \frac{-1}{u^*}, \frac{-1}{v^*}, taken in an appropriate order (which you should specify). (The star denotes complex conjugation - I'm not sure how to do the 'bar' in latex!)

    Now I'm useless at geometry, but if I recall correctly, \frac{-1}{u^*} would correspond to the stereographic projection of the point (-P), right? And likewise with v - other than that however, I really can't see a smart way to do this. I certainly don't want to try all 6 permutations of the 4 points and see what pops up on the cross ratio, but at the same time I can't see intuitively where the -\tan^2(\frac{d}{2}) could have come from in order to try and work out how to take the cross ratio to get the desired result. Please help!

    Many thanks in advance, Mathmos6
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  2. #2
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    Anyone?
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