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# 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. 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 $\infty$.

# 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].

## Mathematics Subject Classification

51N25*no label found*30C20

*no label found*30F40

*no label found*

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