Distance distributions

New 2010-06-21: Distance distribution of two points in the unit ball by Khanh-Dang Nguyen Thu Lam.

If two points are uniformly and independently distributed in a 1d, 2d, or 3d region, what is the distribution of the distance d between them? Here I give the density (pdf) of the distance for certain simple regions, and the mean. [] is the indicator function; i.e. [x] is 1 if x is true, else 0. Please email me if you know any other exactly soluble cases.

one dimension

region	pdf	mean
unit interval	2(1-d)[0≤d≤1]	1/3
unit 1-torus	2[0≤d≤1/2]	1/4

two dimensions

region	pdf	mean
unit 2-torus	2πd[0≤d≤1/2]+2d(π-4arcsec(2d))[1/2≤d≤1/√2]	(√(2)+log(1+√(2)))/6
unit-radius disk	d/π(4arctan(√(4-d2)/d)-d√(4-d2))[0≤d≤2]	128/(45π)
unit square	4d(π/2-2d+d2/2)[0≤d≤1]+(arcsin(1/d)-arccos(1/d)+√(d2-1)-(d2+2)/2)[1≤d≤√2])	(2+√(2))/15-log(7+5√(2))/2

three dimensions

region	pdf	mean
interior of unit sphere	?	36/35
interior of unit cube	see [1]	(4+17√2-16√3-7π+2log((1+√2)(7+4√3)))/105