the cross-ratio

Mobius transformations distort almost everything — lengths, areas, which point is a center, even whether a curve is a circle or a line. So a natural and valuable question is: is there any quantity, built from points, that a Mobius map leaves completely alone? There is exactly one fundamental such invariant, and it is built from four points. It is called the cross-ratio, and it is the conserved currency of Mobius geometry.

Given four distinct points z_1, z_2, z_3, z_4, their cross-ratio is the number (z_1, z_2; z_3, z_4) = ((z_1 - z_3)(z_2 - z_4)) / ((z_1 - z_4)(z_2 - z_3)). (Conventions for the ordering differ between textbooks, so always check which one your source uses.) The headline theorem is invariance: if w = f(z) is any Mobius transformation and w_k = f(z_k), then the cross-ratio of the four images equals the cross-ratio of the four originals. The map can scatter the four points all over the sphere, but this particular combination of their differences comes out unchanged. The reason is that the cross-ratio is exactly engineered to cancel out the rotation, scaling, translation, and inversion that any Mobius map is built from.

Two payoffs make this central. First, four points lie on a common line-or-circle if and only if their cross-ratio is a real number — a clean algebraic test for concyclicity that needs no picture. Second, invariance is the engine behind 'three points determine a Mobius map' and the explicit formulas for the standard mappings: to build the map sending z_1, z_2, z_3 to chosen targets, you set the cross-ratio of (z, z_1; z_2, z_3) equal to the cross-ratio of (w, w_1; w_2, w_3) and solve for w. The cross-ratio is the workhorse formula; its invariance is why the workhorse pulls.