Cross-ratio invariance under Möbius transformations

Statement

Let $T$ be a Möbius transformation, and let $z_1,z_2,z_3,z_4$ be four distinct points of the Riemann sphere. Then

[Tz_1,Tz_2;Tz_3,Tz_4]=[z_1,z_2;z_3,z_4].

Proof

Substitution proves the identity for

T(z)=\frac{az+b}{cz+d};

the factors of $ad-bc$ and the denominators cancel. Equivalently, the cross-ratio is the projective coordinate that sends an ordered triple to $(1,0,\infty)$ , so it is unchanged by changing homogeneous coordinates.

Consequence

The cross-ratio is the basic invariant of ordered quadruples under the Möbius group. In fact, equality of cross-ratios is sufficient for two ordered quadruples of distinct points to lie in the same Möbius orbit, by the sharp three-transitivity of the Möbius group.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapter 3, §§1–2.