In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.
The cross-ratio is preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of points, which underlies its importance for projective geometry. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
Cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.
Read more about Cross-ratio: Definition, Terminology and History, Projective Geometry, Definition in Homogeneous Coordinates, Role in Non-Euclidean Geometry, Six Cross-ratios, Transformational Approach, Differential-geometric Point of View, Higher-dimensional Generalizations