course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
9 |
|
1.II.13B |
2001 |
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.]