cross ratio

The cross ratio of the points $a$, $b$, $c$, and $d$ in $ℂ\cup \{\mathrm{\infty }\}$ 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 $T:ℂ\cup \{\mathrm{\infty }\}\to ℂ\cup \{\mathrm{\infty }\}$ defined by

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

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