Cross-ratio

Definition

Let $z_1, z_2, z_3, z_4$ be four distinct complex numbers. Their cross-ratio is $$(z_1, z_2; z_3, z_4) = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)}.$$ Suppose that one of the four points is the point $\infty$ of the Riemann sphere and the other three are distinct complex numbers. Then the cross-ratio is the same formula without the two factors that contain $\infty$.

Remarks

Other books use other orders of the four points in the formula, so one must check the convention. For each Möbius transformation $f$, the cross-ratio $(f(z_1), f(z_2); f(z_3), f(z_4))$ is equal to $(z_1, z_2; z_3, z_4)$. With this convention, the map $z \mapsto (z, z_2; z_3, z_4)$ is the Möbius transformation that sends $z_2$, $z_3$, $z_4$ to $1$, $0$, $\infty$. The cross-ratio is a real number exactly when the four points are on one circle or on one line.

Explanations

L1 human

The cross-ratio is one number made from four points. Möbius transformations change distances and angles of figures in complicated ways, but they do not change this number. So the cross-ratio helps to find a Möbius map that sends given points to other given points.