crossratios

Cross ratio is a projective invariant associated with an ordered quadruple of distinct points on a projective line. It is a fundamental tool in projective geometry because it remains unchanged under projective (Möbius) transformations, which are the automorphisms of the projective line.

For four distinct points a, b, c, d on the real or complex line (or more generally on P^1 over a field), the cross ratio is defined in an affine chart by

(a,b;c,d) = (a − c)(b − d) / ((a − d)(b − c)).

A coordinate-free viewpoint uses a projective transformation sending a, b, c to 0, 1, ∞; the image of d under this transformation is the cross ratio.

Properties include that the cross ratio is invariant under any projective transformation. It depends on the quadruple up to the natural action of permuting the four points, with the six possible permutations yielding related values via standard fractional-linear transformations. The cross ratio is real when all four points are real and in a suitable order, and the value −1 corresponds to a harmonic quadruple, a configuration with special symmetry.

Generalizations and uses: the concept extends to quadruples on any projective line over a field and can be used to parameterize the moduli of four points up to projective equivalence, via the model 0, 1, ∞, t where t is the cross ratio. Applications appear in geometry, computer vision, and complex analysis, where cross ratios serve as robust invariants under projective changes of coordinates.