# 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

###### 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

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

## Properties

1. 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. 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 $T:\mathbb{C}\cup\{\infty\}\to\mathbb{C}\cup\{\infty\}$ defined by

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

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

## References

• 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 CrossRatio 2013-03-22 15:23:31 2013-03-22 15:23:31 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 51N25 msc 30C20 msc 30F40 cross-ratio MobiusTransformationCrossRatioPreservationTheorem