cross ratio

The cross ratio of the points $a$ , $b$ , $c$ , and $d$ in $\mathbb{C}\cup\{\infty\}$ is denoted by $[a,b,c,d\,]$ and is defined by

[a,b,c,d\,]=\frac{a-c}{a-d}\cdot\frac{b-d}{b-c}.

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

Examples

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

\frac{1-(-1)}{1-(-i)}\cdot\frac{i-(-i)}{i-(-1)}=\frac{4i}{(1+i)^{2}}=2.

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

\frac{1-3}{1-4i}\cdot\frac{2i-4i}{2i-3}=\frac{4i}{5+14i}=\frac{56+20i}{221}.

1.

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

The function $T:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\}$ defined by

$T(z)=[z,b,c,d\,]$

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

1 Ahlfors, L., Complex Analysis. McGraw-Hill, 1966.
2 Beardon, A. F., The Geometry 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