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Randomly generate two points X and Y uniformly and independently on the surface of an n-sphere. Define their distance as the length of the shortest geodesic between the two points. What is the distribution of d(X,Y)?

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Isn't this basic calculus? – David Speyer Dec 8 at 12:51
Shor-ly you must be joking, Mr. Shor. – fpqc Dec 8 at 14:00
I'm not sure if this is a homework question, but it sure is written in the style of one. – Ben Webster♦ Dec 8 at 14:07
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 Sorry, I typed the question in a rush and forgot to mention I was interested in finding the formula for the n-sphere. I've messed around with the integrals and wasn't able to find anything nice, but since the problem for the hypercube is non-trivial, I thought there might be something interesting about the case for the hypersphere. I'm particularly interested with the behaviour as n and r are large. And Mr. Webster, sorry if my post seemed terse. The Overflow interface is extremely slow on my netbook, to the point where there is about a 10 second delay when I type. But this isn't homework! – David Shor Dec 8 at 16:50
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 The problem for the sphere should be easier than for the cube because of the enormous amount of symmetry. The biggest simplification is that you can always rotate the first point to the north pole without changing the distribution of the second point, so you're just looking at the distribution of distances of a single randomly chosen point from the north pole. Then it's easy to show with a little calculus that the distribution of the distance t has density c(n)sin(t)^(n-1)dt over the interval (0, pi), where c(n) is ((n-1)/2)!/(sqrt(pi)(n/2-1)!). – Darsh Ranjan Dec 8 at 19:10
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closed as too localized by David Speyer, Andrew Stacey, Ben Webster♦ Dec 8 at 14:06

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