hyperbolic distance between two random points

[Steven Finch]

Hello!

Let D denote the disk of radius R, centered at
the origin, in the hyperbolic plane. Let P be
a uniformly distributed random point in D. It
is known that the hyperbolic distance between
P and the origin has probability density [1]

(1/(cosh(R)-1))*sinh(x), where 0 <= x <= R.

Now let P, Q be independent uniformly distributed
random points in D. Consider the hyperbolic
distance between P and Q. What is its probability
density?

Pointers to the literature would be appreciated!

Thank you,

Steve Finch
http://algo.inria.fr/bsolve/

Reference

1. Y. Isokawa, Geometric probabilities concerning
large random triangles in the hyperbolic plane,
Kodai Math. J. 23 (2000) 171-186; available online
at

http://projecteuclid.org/euclid.kmj/1138044209