Distance in hyperbolic geometry

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.

• It is possible to impose a single coordinate system on the hyperbolic plane, for example. Changing that to polar coordinates around a single origin does give distances by one of the hyperbolic Laws of Cosines; there are two of those. But for higher dimension, not so much. The closest thing is Weierstrass coordinates, and I would say they do not do what you want. en.wikipedia.org/wiki/Hyperboloid_model – Will Jagy Sep 19 '13 at 1:23
• Why is it like that? I thought that hyperbolic geometry was "just as right" as Euclidean geometry (for a lack of better words)? – Sid Sep 19 '13 at 1:29
• It is. Euclidean geometry was studied without a coordinate system for over 1900 years. A coordinate system is not a particularly natural construct. What is your background at this point? – Will Jagy Sep 19 '13 at 1:56
• Third year math student taking a course in differential geometry. I realize that my questions are not really bright. I just think it's weird how hard it is to find distances between two points in hyperbolic geometry compared to Euclidean geometry. – Sid Sep 19 '13 at 2:15
• Alright. So you have the background to read this pdf: zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf where I never used a coordinate system or any of the models you know. Show it to your instructor as well, I'm quite proud of it. – Will Jagy Sep 19 '13 at 2:21