Describe the stereographic projection map from the sphere $S^{2}$ to the extended complex plane $\mathbf{C}_{\infty}$, positioned equatorially. Prove that $w, z \in \mathbf{C}_{\infty}$ correspond to antipodal points on $S^{2}$ if and only if $w=-1 / \bar{z}$. State, without proof, a result which relates the rotations of $S^{2}$ to a certain group of Möbius transformations on $\mathbf{C}_{\infty}$.

Show that any circle in the complex plane corresponds, under stereographic projection, to a circle on $S^{2}$. Let $C$ denote any circle in the complex plane other than the unit circle; show that $C$ corresponds to a great circle on $S^{2}$ if and only if its intersection with the unit circle consists of two points, one of which is the negative of the other.

[You may assume the result that a Möbius transformation on the complex plane sends circles and straight lines to circles and straight lines.]