Cross-ratio - Transformational Approach

Transformational Approach

The cross-ratio is invariant under the projective transformations of the line. In the case of a complex projective line, or the Riemann sphere, these transformation are known as Möbius transformations. A general Möbius transformation has the form

These transformations form a group acting on the Riemann sphere, the Möbius group.

The projective invariance of the cross-ratio means that

The cross-ratio is real if and only if the four points are either collinear or concyclic, reflecting the fact that every Möbius transformation maps generalized circles to generalized circles.

The action of the Möbius group is simply transitive on the set of triples of distinct points of the Riemann sphere: given any ordered triple of distinct points, (z₂,z₃,z₄), there is a unique Möbius transformation f(z) that maps it to the triple (1,0,∞). This transformation can be conveniently described using the cross-ratio: since (z,z₂,z₃,z₄) must equal (f(z),1;0,∞) which in turn equals f(z), we obtain

An alternative explanation for the invariance of the cross-ratio is based on the fact that the group of projective transformations of a line is generated by the translations, the homotheties, and the multiplicative inversion. The differences z_j - z_k are invariant under the translations

where a is a constant in the ground field F. Furthermore, the division ratios are invariant under a homothety

for a non-zero constant b in F. Therefore, the cross-ratio is invariant under the affine transformations.

In order to obtain a well-defined inversion mapping

the affine line needs to be augmented by the point at infinity, denoted ∞, forming the projective line P1(F). Each affine mapping f: F → F can be uniquely extended to a mapping of P1(F) into itself that fixes the point at infinity. The map T swaps 0 and ∞. The projective group is generated by T and the affine mappings extended to P1(F). In the case F = C, the complex plane, this results in the Möbius group. Since the cross-ratio is also invariant under T, it is invariant under any projective mapping of P1(F) into itself.