The cross-ratio normalizes three marked points to $1$, $0$, and $\infty$.Suggest a slogan
Let \(z_2,z_3,z_4\in\CC\) be distinct. The cross-ratio map is the Möbius transformation \[ R(z) \coloneqq (z, z_2, z_3, z_4) \coloneqq {z - z_3\over z-z_4}\cdot{z_2 - z_4 \over z_2 - z_3}. \] It satisfies \(R(z_2)=1\), \(R(z_3)=0\), and \(R(z_4)=\infty\).
What is the cross-ratio map, and where does it send $z_2, z_3, z_4$?