# Thread: Cross ratios, inversions and stereographic projections

1. ## 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$\displaystyle \in \mathbb{C}$ correspond to points P, Q on $\displaystyle S^2$, and d denotes the angular distance from P to Q on $\displaystyle S^2$, show that $\displaystyle -\tan^2(\frac{d}{2})$ is the cross ratio of the points $\displaystyle u, v, \frac{-1}{u^*}, \frac{-1}{v^*}$, 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, $\displaystyle \frac{-1}{u^*}$ 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 $\displaystyle -\tan^2(\frac{d}{2})$ could have come from in order to try and work out how to take the cross ratio to get the desired result. Please help!