cross ratio
The cross ratio^{} of the points $a$, $b$, $c$, and $d$ in $\u2102\cup \{\mathrm{\infty}\}$ is denoted by $[a,b,c,d]$ and is defined by
$$[a,b,c,d]=\frac{ac}{ad}\cdot \frac{bd}{bc}.$$ 
Some authors denote the cross ratio by $(a,b,c,d)$.
Examples
Example 1.
The cross ratio of $1$, $i$, $1$, and $i$ is
$$\frac{1(1)}{1(i)}\cdot \frac{i(i)}{i(1)}=\frac{4i}{{(1+i)}^{2}}=2.$$ 
Example 2.
The cross ratio of $1$, $2i$, $3$, and $4i$ is
$$\frac{13}{14i}\cdot \frac{2i4i}{2i3}=\frac{4i}{5+14i}=\frac{56+20i}{221}.$$ 
Properties

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 $T:\u2102\cup \{\mathrm{\infty}\}\to \u2102\cup \{\mathrm{\infty}\}$ defined by
$$T(z)=[z,b,c,d]$$ is the unique Möbius transformation which sends $b$ to $1$, $c$ to $0$, and $d$ to $\mathrm{\infty}$.
References
 1 Ahlfors, L., Complex Analysis. McGrawHill, 1966.
 2 Beardon, A. F., The Geometry^{} of Discrete Groups. SpringerVerlag, 1983.
 3 Henle, M., Modern Geometries: NonEuclidean, Projective, and Discrete. PrenticeHall, 1997 [2001].
Title  cross ratio 

Canonical name  CrossRatio 
Date of creation  20130322 15:23:31 
Last modified on  20130322 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  crossratio 
Related topic  MobiusTransformationCrossRatioPreservationTheorem 