$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 18.9: Complex cross-ratio

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Let $$u$$, $$v$$, $$w$$, and $$z$$ be four distinct complex numbers. Recall that the complex number

$$\dfrac{(u-w) \cdot (v-z)}{(v-w) \cdot (u-z)}$$

is called the complex cross-ratio of $$u$$, $$v$$, $$w$$, and $$z$$; it is denoted by $$(u,v;w,z)$$.

If one of the numbers $$u$$, $$v$$, $$w$$, $$z$$ is $$\infty$$, then the complex cross-ratio has to be defined by taking the appropriate limit; in other words, we assume that $$\dfrac{\infty}{\infty}=1$$. For example,

$$(u, v; w, \infty)=\dfrac{(u-w)}{(v-w)}.$$

Assume that $$U$$, $$V$$, $$W$$, and $$Z$$ are the points with complex coordinates $$u$$, $$v$$, $$w$$, and $$z$$ respectively. Note that

\begin{aligned} \dfrac{UW\cdot VZ}{VW\cdot UZ}&=|(u,v;w,z)|, \\ \measuredangle WUZ +\measuredangle ZVW&=\arg\frac{u-w}{u-z}+\arg\frac{v-z}{v-w}\equiv \\ &\equiv \arg(u,v;w,z).\end{aligned}

These equations make it possible to reformulate Theorem 10.2.1 using the complex coordinates the following way:

Theorem $$\PageIndex{1}$$

Let $$UWVZ$$ and $$U'W'V'Z'$$ be two quadrangles such that the points $$U'$$, $$W'$$, $$V'$$, and $$Z'$$ are inverses of $$U$$, $$W$$, $$V$$, and $$Z$$ respectively. Assume $$u$$, $$w$$, $$v$$, $$z$$, $$u'$$, $$w'$$, $$v'$$, and $$z'$$ are the complex coordinates of $$U$$, $$W$$, $$V$$, $$Z$$, $$U'$$, $$W'$$, $$V'$$, and $$Z'$$ respectively.

Then

$$(u',v';w',z')=\overline{(u,v;w,z)}.$$

The following exercise is a generalization of the theorem above. It has a short solution using Proposition 18.8.1.

Exercise $$\PageIndex{1}$$

Show that complex cross-ratios are invariant under fractional linear transformations.

That is, if a fractional linear transformation maps four distinct complex numbers $$u, v, w, z$$ to complex numbers $$u', v', w', z'$$ respectively, then

$$(u',v';w',z') = (u,v;w,z).$$

Hint

Check the statement for each elementary transformation. Then apply Proposition 18.8.1.