Cross-ratio

The cross-ratio of four distinct points $z,z_1,z_2,z_3\in \hat{\mathbb{C}}$ is

$[z,z_1,z_2,z_3]=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}.$

Convention warning. Different textbooks define the cross-ratio with different orderings of the four points — some swap $z_1$ and $z_3$, some negate, some take the reciprocal. All six permutations are related by Möbius transformations of the cross-ratio itself, so the “invariant under Möbius transformations” content holds in every convention, but the specific algebraic formula differs. Always check the convention before cross-referencing. This note uses the convention in which $[z_1,z_1,z_2,z_3]=0$ and $[z_3,z_1,z_2,z_3]=\infty$ — the first reference point maps to $0$, the third to $\infty$.

The cross-ratio is the fundamental invariant of Möbius transformations: if $f$ is a Möbius transformation, then

$[f(z),f(z_1),f(z_2),f(z_3)]=[z,z_1,z_2,z_3].$

Three points to three points

The cross-ratio is the tool that solves the three-points-to-three-points problem for Möbius transformations.

Given $z_1\mapsto w_1$, $z_2\mapsto w_2$, $z_3\mapsto w_3$, the unique Möbius transformation sending $z\mapsto w$ satisfies

$[w,w_1,w_2,w_3]=[z,z_1,z_2,z_3].$

Solve this equation for $w$ as a function of $z$ — the result is the Möbius transformation.

Worked example

Find the Möbius transformation sending $0\mapsto 1$, $1\mapsto i$, $-1\mapsto -i$.

Direct approach with $f(z)=(az+b)/(cz+d)$: $f(0)=b/d=1$, so $b=d$. $f(1)=(a+b)/(c+d)=i$; $f(-1)=(-a+b)/(-c+d)=-i$. Solving: $a=i$, $c=-i$. Taking $b=1$: $f(z)=(iz+1)/(-iz+1)$.

Verify: $f(0)=1$, $f(1)=(i+1)/(1-i)=(1+i{)}^2/((1-i)(1+i))=2i/2=i$. ✓ $f(-1)=(1-i)/(1+i)=-i$. ✓

Cross-ratio with $\infty$

If one of the four points is $\infty$, the formula uses the convention that ratios involving $\infty$ simplify limit-style. For example,

$[z,z_1,z_2,\infty ]={\lim }_{z_3\to \infty }[z,z_1,z_2,z_3]=\frac{(z-z_1)}{(z_2-z_1)}$

(the $(z_2-z_3)/(z-z_3)\to 1$ as $z_3\to \infty$).

So the cross-ratio extends naturally to the Riemann sphere.

Real cross-ratio ↔ concyclic

A surprising and beautiful fact: four points lie on a circle (or a line) if and only if their cross-ratio is real.

This is closely related to the fact that Möbius transformations preserve “circles and lines” — see Möbius transformation.

In context

The cross-ratio is a tool of projective geometry going back to ancient Greek mathematics. In complex analysis, it parameterizes Möbius transformations and witnesses their action on four-point configurations.

In transmission-line theory and the Smith chart, cross-ratio-style invariants help track impedance transformations as a wave propagates along a line.