# Random points on a non-zero curvature 2D space

For some computational project, I'm interested in the pairwise distance matrix between random points on a unit square of $\mathbb{R}^2$.

I now want to extend this case to non-zero curvature 2D spaces, but I don't see what is the proper way to spread random points on such spaces. Does one define the random distribution on $[0,1]^2$ and maps it to the space through an appropriate coordinate transform, or is there a way to do it directly ?

How would you expect the distance matrix to change with curvature ?

@Aaron Mazel-Gee : That's what I was more or less thinking also, but I'm lacking the mathematical link there. For example, on a Euclidean space, the infinitesimal area unit is dx x dy so a uniform distribution on both x and y is ok. On a sphere, this small area can be expressed as rdrd$\theta$, so a uniform distribution on $\theta$ and a uniform distribution over $r^2$ might be ok. And the same goes for a hyperbolic space ? –  AlexPof Jul 11 '11 at 18:51
@Aaron Mazel-Gee : sorry, it got mixed up with something else. So for a unit sphere : $sin \phi d\phi d\theta$, so I have to choose a uniform distribution over $\theta$ and $cos \phi$. If I'm guessing correctly, on the hyperbolic half-plane it would be over x and $1/y$, and other models of hyperbolic space give rather complicated formulas. –  AlexPof Jul 11 '11 at 19:19