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

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

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

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

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