Definition:Cross-Ratio

Definition

Points on a Line

Let $A$, $B$, $C$ and $D$ be points on a straight line.

The cross-ratio of $A$, $B$, $C$ and $D$, in that specific order, is defined as:

$\dfrac {AC / CB} {AD / DB}$

or equivalently:

$\dfrac {AC \cdot DB} {AD \cdot CB}$

It can be denoted:

$\set {A, B; C, D}$

Lines through Origin

Let $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$ be lines through the origin $O$ whose equations embedded in the Cartesian plane are as follows:

\(\ds \LL_1: \ \ \)

\(\ds y\)

\(=\)

\(\ds \lambda x\)

\(\ds \LL_2: \ \ \)

\(\ds y\)

\(=\)

\(\ds \mu x\)

\(\ds \LL_3: \ \ \)

\(\ds y\)

\(=\)

\(\ds \lambda' x\)

\(\ds \LL_4: \ \ \)

\(\ds y\)

\(=\)

\(\ds \mu' x\)

The cross-ratio of $\LL_1$, $\LL_2$, $\LL_3$ and $\LL_4$, in that specific order, is defined and denoted:

$\tuple {\lambda \mu, \lambda', \mu'} := \dfrac {\paren {\lambda - \lambda'} \paren {\mu - \mu'} } {\paren {\lambda - \mu'} \paren {\mu - \lambda'} }$

Complex Analysis

Let $z_1, z_2, z_3, z_4$ be distinct complex numbers.

The cross-ratio of $z_1, z_2, z_3, z_4$ is defined and denoted:

$\paren {z_1, z_3; z_2, z_4} = \dfrac {\paren {z_1 - z_2} \paren {z_3 - z_4} } {\paren {z_1 - z_4} \paren {z_3 - z_2} }$

Also known as

Some sources do not hyphenate cross-ratio, leaving it as: cross ratio.