Cross ratios, inversions and stereographic projections
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 correspond to points P, Q on , and d denotes the angular distance from P to Q on , show that is the cross ratio of the points , 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, 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 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