cross ratioCrossRatio2005-07-14 13:09:552006-12-08 15:09:44Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{example}{Example}The \emph{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 $(a, b, c, d)$.
\section*{Examples}
\begin{example}
The cross ratio of $1$, $i$, $-1$, and $-i$ is
\[
\frac{1-(-1)}{1-(-i)}\cdot\frac{i-(-i)}{i-(-1)}
=\frac{4i}{(1+i)^2}=2.
\]
\end{example}
\begin{example}
The cross ratio of $1$, $2i$, $3$, and $4i$ is
\[
\frac{1-3}{1-4i}\cdot\frac{2i-4i}{2i-3}
=\frac{4i}{5+14i}
=\frac{56+20i}{221}.
\]
\end{example}
\section*{Properties}
\begin{enumerate}
\item
The cross ratio is invariant under M\"obius 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.
\item
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.
\item
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 M\"obius transformation which sends $b$ to $1$, $c$ to $0$, and $d$ to $\infty$.
\end{enumerate}
\begin{thebibliography}{1}
\bibitem{cite:A}
Ahlfors, L., \emph{Complex Analysis}. McGraw-Hill, 1966.
\bibitem{cite:B}
Beardon, A. F., \emph{The Geometry of Discrete Groups}. Springer-Verlag, 1983.
\bibitem{cite:H}
Henle, M., \emph{Modern Geometries: Non-Euclidean, Projective, and Discrete}. Prentice-Hall, 1997 [2001].
\end{thebibliography}