Preserves straight lines and cross ratios

Given four collinear points $ {\bf A}_1$ , $ {\bf A}_2$ ,$ {\bf A}_3$ and $ {\bf A}_4$ , their cross ratio is defined as

$\displaystyle \frac{\overline{{\bf A}_1 {\bf A}_3}\;\; \overline{{\bf A}_2 {\bf A}_4}}
{\overline{{\bf A}_1 {\bf A}_4}\;\; \overline{{\bf A}_2 {\bf A}_3}}

if $ {\bf A}_4$ is a point at infinity the the cross ratio is given as

$\displaystyle \frac{\overline{{\bf A}_1 {\bf A}_3}}
{\overline{{\bf A}_2 {\bf A}_3}}

The cross ratio is independent of the choice of the projective coordinate system.

Subhashis Banerjee 2008-01-20