= Cross-ratio normalization of four projective points
{title2=$(P_0P_1:P_2P_3)=T(P_3),\quad(T(P_0),T(P_1),T(P_2))=(\infty,0,1)$}
For four distinct <projective points> on the complex <projective line>, use the convention
$$
(P_0P_1:P_2P_3)=\frac{[p_0,p_2][p_1,p_3]}{[p_0,p_3][p_1,p_2]},
$$
where $[p_i,p_j]$ is the <determinant> of chosen representative vectors. Rescaling any representative cancels, and applying an <invertible matrix> multiplies every bracket by its determinant, which also cancels. Thus the expression is a projective invariant. The <projective general linear group action on the projective line> has a unique element $T$ taking the first three points to $\infty,0,1$. In those coordinates the displayed expression is exactly $T(P_3)$. Consequently the <cross-ratio> completely determines the <orbit> of the ordered quadruple under the <projective linear group>. Its value lies in $\mathbb C\setminus\{0,1\}$.
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