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

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

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

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