Characterizations of the cross-ratio

Question

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$

What are the most prominent or most interesting theorems of the following form?

Theorem: The cross-ratio is the only function that satisfies this-particular-specified-condition.

Answer 1

Let $z_1, z_2, z_3, z_4 \in \Bbb P ^1 (\Bbb C) = \Bbb C \cup \{ \infty \}$, all of them distinct. The cross-ratio is essentially the only projective invariant of this system of points. More rigorously, let

$$X = \{ (z_1, z_2, z_3, z_4) \in \big( \Bbb P ^1 (\Bbb C) \big) ^4 \mid z_i \ne z_j \; \forall i \ne j \} ;$$

then $F : X \to \Bbb C$ is invariant under $z \mapsto \frac {az+b} {cz+d}$ $\forall a, b, c, d \in \Bbb C$ if and only if $\exists f :\Bbb C \to \Bbb C$ with

$$F(z_1, z_2, z_3, z_4) = f\left(\frac {(z_1 - z_3) (z_2 - z_4)} {(z_2 - z_3) (z_1 - z_4)}\right) \; \forall (z_1, z_2, z_3, z_4) \in X .$$

Let us see why.

First, one may find $a,b,c,d$ such that the resulting projective transformation sends $z_2, z_3, z_4$ to $1, 0, \infty$, respectively. Namely, the transformation $R$ given by

$$z \mapsto \frac {(z - z_3) (z_2 - z_4)} {(z - z_4) (z_2 - z_3)} .$$

Note that

$$R(z_1) = \frac {(z_1 - z_3) (z_2 - z_4)} {(z_2 - z_3) (z_1 - z_4)} ,$$

precisely the cross-ratio.

Now, if $F$ is invariant under all projective transformations, in particular it is invariant under $R$, so

$$F(z_1, z_2, z_3, z_4) = F\left(\frac {(z_1 - z_3) (z_2 - z_4)} {(z_2 - z_3) (z_1 - z_4)}, 1, 0, \infty\right) .$$

Call $f(z) = F(z, 1, 0, \infty)$ and the direct implication is done.

The converse is really easy: it is a straightforward computation to show that the cross-section is invariant under projective transformation, and therefore any function of it must also be so.

Therefore, the cross-ratio is essentially the only projective invariant of systems of $4$ distinct points, all others being functions of it.

Answer 2

The cross-ratio is the unique rational function of four complex numbers up to multiplication by a constant such that the cross-ratio of four distinct points is real if and only if the points are concyclic.