Knowlpedia

Definition

Cross-ratio

A Möbius-invariant coordinate of an ordered quadruple on the projective line.

Definition

For an ordered quadruple of distinct points z1,z2,z3,z4∈C^z_1,z_2,z_3,z_4\in\widehat{\mathbb C}, this knowl uses the cross-ratio

[z1,z2;z3,z4]=(z1−z3)(z2−z4)(z1−z4)(z2−z3).[z_1,z_2;z_3,z_4] =\frac{(z_1-z_3)(z_2-z_4)} {(z_1-z_4)(z_2-z_3)}.

Cases involving ∞\infty are defined by taking limits.

Examples

For instance, taking the fourth point to infinity gives

[z1,z2;z3,∞]=z1−z3z2−z3.[z_1,z_2;z_3,\infty]=\frac{z_1-z_3}{z_2-z_3}.
Projective meaning

There is a unique sending z2,z3,z4z_2,z_3,z_4 to 1,0,∞1,0,\infty. Its value at z1z_1 is the displayed cross-ratio. This makes projective invariance immediate.

The precise invariance statement is the . Conversely, a bijection of the sphere preserving all complex cross-ratios is .

Ordering convention

The order matters. Permuting four distinct points changes a value λ\lambda among

λ,1λ,1−λ,11−λ,λλ−1,λ−1λ.\lambda,\quad \frac1\lambda,\quad 1-\lambda,\quad \frac1{1-\lambda},\quad\frac{\lambda}{\lambda-1},\quad \frac{\lambda-1}{\lambda}.

Other books choose a permutation of the four arguments as their defining convention, so formulas should always be checked against the stated ordering.

Real and circular criterion

Four distinct points lie on one exactly when their cross-ratio is real. This connects the invariant to the circle-preserving geometry of Möbius transformations.

References
  1. Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapter 3.