| course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
15 |
|
2.II.12A |
2007 |
(i) The spherical circle with centre $P \in S^{2}$ and radius $r, 0<r<\pi$, is the set of all points on the unit sphere $S^{2}$ at spherical distance $r$ from $P$. Find the circumference of a spherical circle with spherical radius $r$. Compare, for small $r$, with the formula for a Euclidean circle and comment on the result.
(ii) The cross ratio of four distinct points $z_{i}$ in $\mathbf{C}$ is
$$\frac{\left(z_{4}-z_{1}\right)\left(z_{2}-z_{3}\right)}{\left(z_{4}-z_{3}\right)\left(z_{2}-z_{1}\right)} .$$
Show that the cross-ratio is a real number if and only if $z_{1}, z_{2}, z_{3}, z_{4}$ lie on a circle or a line.
[You may assume that Möbius transformations preserve the cross-ratio.]