cross ratio

Any triple of distinct points in a projective line over a field KK may be transformed by a projective transformation (an element of PGL(2,K), in the case of K=CK = \mathbf{C} (the complex numbers) known as a Möbius transformation) to another given triple of distinct points (i.e., the group PGL(2,K)PGL(2,K) is 3-transitive on the projective line). Therefore no projective invariant can be attached to a triple of points and if three points are the starting parameters for some problem in projective geometry we may and typically do give them fixed values of 0,1,0,1,\infty. The unique element in PGL 2(K)PGL_2(K) that takes (A,B,C)(A, B, C) to (,0,1)(\infty, 0, 1) may be described as the fractional linear transformation

p A,B,C:zzBzACACBp_{A, B, C}: z \mapsto \frac{z - B}{z - A} \cdot \frac{C - A}{C - B}

The simplest projective invariant can be attached to a quadruple of points and it is the cross ratio. It can be expressed in various ways, e.g.,

(A,B,C,D)ACADBDBC (A,B,C,D) \coloneqq \frac{A - C}{A - D}\frac{B - D}{B - C}

which is the value p A,B,C(D)p_{A, B, C}(D) using the fractional linear transformation above.

If the cross ratio for (A,B,C,D)(A,B,C,D) is λ\lambda and the order of the four points is changed than the cross ratio takes one of the following values, depending on the permutation:

λ,1λ,1λ,11λ,λ1λ,λλ1 \lambda, \frac{1}\lambda, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda - 1}\lambda, \frac\lambda{\lambda-1}

An elementary introduction can be found in

  • Lucienne Félix, The modern aspect of mathematics, (Rus. transl.: Элементарная математика в современном изложении, 1967)

and a quick overviews in

There is a noncommutative version

  • Vladimir Retakh, Noncommutative cross-ratios, Journal of Geometry and Physics 82 (2014) 13-17 arxiv/1401.5770 doi
category: geometry

Last revised on March 27, 2018 at 07:54:30. See the history of this page for a list of all contributions to it.