Annotated Definitions

The Cross-ratio

The cross-ratio of four points is $[z_1, z_2, z_3, z_4] := \frac{z_1 - z_3}{z_1 - z_4} \frac{z_2 - z_4}{z_2 - z_3}$.

By taking $z_1, z_2, z_3, z_4 \in \mathbb{R}$ we define the cross-ratio on (ordered) quadruples of points on the real line, and thereby, by interpreting the differences as signed distances, on ordered quadruples of collinear points in Euclidean space. We can extend this definition to the projective line or projective space by allowing $z_i = \infty$.

By taking $z_1, z_2, z_3, z_4 \in \mathbb{C} \cup \{\infty\}$, we define the cross-ratio on quadruples of points in the Riemann sphere $\hat{\mathbb{C}}$. (More generally, we can define the cross-ratio on quadruples of points in any field.)

The cross-ratio is interesting because it is the essential projective invariant (in 2 dimensions); indeed, this might be taken as the more “natural” or “universal” definition, or what Tim Gowers refers to when he says that [about tensor products in particular, but also mathematical objects in general] “exactly how they are defined is not important: what matters is the properties they have.”

More precisely, the cross-ratio is invariant under projective transformations, i.e. (in 2 dimensions) under $\mathrm{Aut}(\hat{\mathbb{C}}) \cong \mathrm{PSL}(2, \mathbb{C}) := \mathrm{SL}(2, \mathbb{C}) / \pm I$. It is essentially the only projective invariant of ordered quadruples of points, in the sense of a universal property (any projective invariant can be bijectively transformed into the cross-ratio), since $\mathrm{PSL}(2, \mathbb{C})$ acts simply transitively on ordered triples of points in $\hat{\mathbb{C}}$. [I feel like I don’t quite understand the details of the argument here yet.]

Furthermore, quadruples of points are the natural choice of objects to  determine (2-8dimensional) projective invariants on: the Euclidean distance between 2 points is invariant under translations or rotations and the ratio of the distances between 3 points is invariant under Euclidean similarities, but $\mathrm{PSL}(2, \mathbb{C})$ acts transitively on ordered triples of points in $\hat{\mathbb{C}}$.

This property of invariance of cross-ratio allows us to define hyperbolic distance in terms of the cross-ratio: more precisely, taking the Poincaré half-plane model $\mathbb{H}$ of the hyperbolic plane, we have that $d_{\mathbb{H}}(x,y) = \log [x', y, x, y']$ where $x', y'$ are the endpoints on $\partial\mathbb{H}$ of the geodesic between $x$ and $y$ (we may replace $\mathbb{H}$ with the Poincaré disc model $\mathbb{D}$ throughout.)

To prove this we conformally map our points to the imaginary axis and treat only this special case: note that $\mathrm{Aut} (\mathbb{H}) \subset \mathrm{Aut}(\widehat{\mathbb{C}})$ and also that $\mathrm{Aut} (\mathbb{H})$ preserves the cross-ratio. Now WLOG let $x, y$ lie on the imaginary axis with $x = i$ and $y = ai$ (where $a \in \mathbb{R}$.) Then $d_{\mathbb{H}}(x,y) = a$ and $[x',y,x,y'] = \frac{0-ai}{0-i} \frac{\infty - i}{\infty-ai} = a$, as desired.