Geometry | Part IB, 2001

Describe geometrically the stereographic projection map ϕ\phi from the unit sphere S2S^{2} to the extended complex plane C=C\mathbb{C}_{\infty}=\mathbb{C} \cup \infty, and find a formula for ϕ\phi. Show that any rotation of S2S^{2} about the zz-axis corresponds to a Möbius transformation of C\mathbb{C}_{\infty}. You are given that the rotation of S2S^{2} defined by the matrix

(001010100)\left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{array}\right)

corresponds under ϕ\phi to a Möbius transformation of C\mathbb{C}_{\infty}; deduce that any rotation of S2S^{2} about the xx-axis also corresponds to a Möbius transformation.

Suppose now that u,vCu, v \in \mathbb{C} correspond under ϕ\phi to distinct points P,QS2P, Q \in S^{2}, and let dd denote the angular distance from PP to QQ on S2S^{2}. Show that tan2(d/2)-\tan ^{2}(d / 2) is the cross-ratio of the points u,v,1/uˉ,1/vˉu, v,-1 / \bar{u},-1 / \bar{v}, taken in some order (which you should specify). [You may assume that the cross-ratio is invariant under Möbius transformations.]

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