Distance in hyperbolic geometry

Question

In Euclidean geometry, we have that the distance between two points $p$ and $q$ in $\Re^n$ is $\sqrt{(p_1^2-q_1^2) + (p_2^2-q_2^2) + \ldots + (p_n^2-q_n^2) }$ (if we denote the points by $p = (p_1, p_2, \ldots, p_n)$ and $q = (q_1, q_2, \ldots, q_n)$). Is there any formula like this for distance between points in hyperbolic geometry? I know that for example in the Poincaré disc model we have a certain formula, another in the Klein model, and so on, but I was wondering if we can have some distance formula that exists independent of the model.

Answer 1

I would argue that your statement

I know that for example in the Poincaré disc model we have a certain formula, another in the Klein model, and so on

has a correspondence in Euclidean geometry: There is one formula for Cartesian coordinates (the one you quoted), another for polar coordinates, and so on. So in a certain sense, the models of hyperbolic geometry are just different coordinate systems for the common hyperbolic space they model. Yes, they do come with different visualizations, but that is because we have to embed them in Euclidean space to visualize them.

There is the concept called Cayley-Klein metric which works almost the same for different kinds of planar geometries, including Euclidean, hyperbolic, elliptic, pseudo-Euclidean and Galilean. Of course, you'd most likely associate the resulting representation of hyperbolic space with the Klein model. Furthermore, in that setup, Euclidean distances will have absolute value zero, and the only thing you can do is express the ratio between two Euclidean distances. Which makes sense, since Euclidean geometry, in contrast to hyperbolic, does not have any intrinsic reference length. However, there are formulations which encode this reference length in the description of the geometry, and then end up with the common formula for Euclidean distances. So Cayley-Klein metrics do form a kind of common language for distance measurements, but not in as simple a way as you might hope for.