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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 $(a, b, c, d)$
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 $$ \frac{1-3}{1-4i}\cdot\frac{2i-4i}{2i-3} =\frac{4i}{5+14i} =\frac{56+20i}{221}. $$
- 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.
- 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.
- The function $T:\mathbb{C}\cup\lbrace \infty \rbrace \to\mathbb{C}\cup\lbrace \infty\rbrace$ defined by $$ T(z) = [z, b, c, d\,] $$ is the unique Möbius transformation which sends $b$ to $1$ $c$ to $0$ and $d$ to $\infty$
- 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].
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