Cross-ratio

Given four points $A, B, C, D$ on a projective line (over a field where subtraction/division makes sense in an affine chart), the cross-ratio $(A, B; C, D)$ is a quantity invariant under projectivities.

Definition via normalization

Assume $A, B, C$ are distinct. By the Fundamental theorem of projective geometry there is a unique projectivity sending

A \mapsto 0, B \mapsto 1, C \mapsto \infty .

Then the image of $D$ under this unique projectivity is defined to be the cross-ratio $(A, B; C, D)$ .

Coordinate formula

In an affine coordinate where the points are represented by scalars (still denoted $A, B, C, D$ ), one has

(A, B; C, D) = \frac{(C - A) (D - B)}{(C - B) (D - A)} .

Invariance

For any projectivity $f$ of the projective line,

(A, B; C, D) = (f (A), f (B); f (C), f (D)) .