= Solution
Use the <cross-ratio> convention compatible with the permutation identities in this paper. Represent a <projective point> $P_i\in\mathbb{CP}^1$ by a nonzero vector $p_i\in\mathbb C^2$, and write $[p_i,p_j]=\det(p_i,p_j)$. Define
$$
\boxed{(P_0P_1:P_2P_3)=\frac{[p_0,p_2][p_1,p_3]}{[p_0,p_3][p_1,p_2]}.}
$$
The denominator is nonzero because the four <projective points> are distinct. Rescaling any one representative multiplies numerator and denominator by the same nonzero <scalar>, so the definition is independent of the chosen <homogeneous coordinates>. In finite affine coordinates it reads
$$
(z_0z_1:z_2z_3)=\frac{(z_0-z_2)(z_1-z_3)}{(z_0-z_3)(z_1-z_2)},
$$
with the <point at infinity of the complex projective line> handled by the homogeneous formula.
A <projective transformation> is represented by an <invertible matrix> $A\in\operatorname{GL}_2(\mathbb C)$. Since $[Ap_i,Ap_j]=\det(A)[p_i,p_j]$, the two factors of $\det(A)$ in the numerator cancel the two in the denominator. Therefore \b[every <projective transformation> preserves this <cross-ratio>]. This also shows that the formula descends from matrices to the <projective linear group>, whose representatives differ by a nonzero common <scalar>.
Back to article page
