The Cross-Ratio

Complex Analysis · Axiom Academy

One number, built from four points, that no Möbius transformation can change — the fundamental invariant of the complex plane. Given four distinct points z_1, z_2, z_3, z_4 in the extended plane , the cross-ratio is the single complex number you get by combining their differences in a fixed pattern: numerator pairs in green , denominator pairs in red It is built only from differences z_i - z_j . The animation links each of the four factors to the segment it measures, then assembles them into the ratio. When one of the points is , take the limit — every factor containing that point cancels in pairs. For instance: 2. Why Möbius Maps Can't Touch It Here is the property that makes the cross-ratio matter. If f is any Möbius transformation, then applying it to all four points leaves the cross-ratio unchanged: Watch the four points stream to their images under . The whole configuration warps — yet the cross-ratio readout never budges. Why it's true. Every Möbius map is a composition of three simple moves, and each one survives: The cross-ratio uses only differences z_i - z_j , and the +b cancels in every difference. Unchanged. Each factor picks up a copy of a — two on top, two on bottom — so they cancel in the ratio. Unchanged. A direct calculation: , and all the z_i z_j factors cancel. Unchanged. Every Möbius map is built from these three. If each building block preserves the cross-ratio, the composition does too.

This is the written version of the interactive lesson above. See the full Complex Analysis course.