cross ratio

The cross ratioMathworldPlanetmath of the points a, b, c, and d in {} is denoted by [a,b,c,d] and is defined by


Some authors denote the cross ratio by (a,b,c,d).


Example 1.

The cross ratio of 1, i, -1, and -i is

Example 2.

The cross ratio of 1, 2i, 3, and 4i is



  1. 1.

    The cross ratio is invariant under Möbius transformations and projective transformations. This fact can be used to determine distancesMathworldPlanetmath between objects in a photograph when the distance between certain reference points is known.

  2. 2.

    The cross ratio [a,b,c,d] is real if and only if a, b, c, and d lie on a single circle on the Riemann sphere.

  3. 3.

    The function T:{}{} defined by


    is the unique Möbius transformation which sends b to 1, c to 0, and d to .


  • 1 Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.
  • 2 Beardon, A. F., The GeometryMathworldPlanetmath of Discrete Groups. Springer-Verlag, 1983.
  • 3 Henle, M., Modern Geometries: Non-Euclidean, Projective, and Discrete. Prentice-Hall, 1997 [2001].
Title cross ratio
Canonical name CrossRatio
Date of creation 2013-03-22 15:23:31
Last modified on 2013-03-22 15:23:31
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Definition
Classification msc 51N25
Classification msc 30C20
Classification msc 30F40
Synonym cross-ratio
Related topic MobiusTransformationCrossRatioPreservationTheorem