18.9: Complex cross-ratio

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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.