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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 ?

Thank you for your answers !

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Most likely you're going to want to have this depend on the area of your surface: if two regions have the same area, then they have equal probability of containing a random point. – Aaron Mazel-Gee Jul 11 '11 at 18:47
@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
That's the idea, but I think you're using polar coordinates where you should be using spherical coordinates...? – Aaron Mazel-Gee Jul 11 '11 at 19:07
@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
Yes, I think that sounds about right. I don't think the Poicare disk model would be so bad, either. – Aaron Mazel-Gee Jul 11 '11 at 19:32

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