I have 4 points $P_0=[1:2], P_1=[3:4], P_2=[5:6], P_3=[7,8]$ in $\mathbb KP^1$ and would like to evaluate the cross-ratio.
It is given by the following:
$\pi:\mathbb KP^1\rightarrow G$ is the unique projective map with $\pi([1:0])=P_0, \pi([0:1])=P_1, \pi([1:1])=P_2$ and $\pi([x_0:x_1])=P_3$
The cross-ratio is euqal to $\frac{x_1}{x_0}$
My problem is, how to evaluate $x_0$ and $x_1$, may you could help me with that.
Pay attention to the fact that we have
$$P_2\,,\,P_3\in l: [1:2]+t[2:2]=P_0+t(P_1-P_0)$$
as a projective map, $\,\pi\,$ must map collinear points to collinear points...can you take it from here?