nLab cross ratio

Any triple of distinct points in a projective line over a field $K$ may be transformed by a projective transformation (an element of PGL(2,K), in the case of $K = \mathbf{C}$ (the complex numbers) known as a Möbius transformation) to another given triple of distinct points (i.e., the group $PGL(2,K)$ 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 $0,1,\infty$. The unique element in $PGL_2(K)$ that takes $(A, B, C)$ to $(\infty, 0, 1)$ may be described as the fractional linear transformation

$$p_{A, B, C}: z \mapsto \frac{z - B}{z - A} \cdot \frac{C - A}{C - B}$$

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.,

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

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

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

$$\lambda, \frac{1}\lambda, 1-\lambda, \frac{1}{1-\lambda}, \frac{\lambda - 1}\lambda, \frac\lambda{\lambda-1}$$

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

• F. Labourie, What is a Cross Ratio, Notices of the AMS, 55 (2008), no. 10, pp.1234–1235 pdf
• wikipedia cross-ratio, Möbius transformation

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