Cube Line Picking--Face and Interior


Consider the distribution of distances l between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form

 P(l)={4/3l^4-2pil^2(l-1)   for 0<=l<=1; 2l{1/3[(3-4sqrt(l^2-1))l^2-8sqrt(l^2-1)-6pil+6pi-1]+4l^2sec^(-1)l}   for 1<l<=sqrt(2); 2l{1/3[(2sqrt(l^2-2)+3pi-3)l^2+8sqrt(l^2-2)+6pi-5] ; -4(l^2+2)tan^(-1)(sqrt(l^2-2))+4ltan^(-1)(lsqrt(l^2-2))-4lcsc^(-1)(sqrt(2-2l^(-2)))}   for sqrt(2)<l<=sqrt(3).

The first even raw moments mu_n^' for n=0, 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395, ....

See also

Cube Line Picking, Cube Line Picking--Face and Face

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Mathai, A. M.; Moschopoulos, P.; and Pederzoli, G. "Distance between Random Points in a Cube." J. Statistica 59, 61-81, 1999.

Cite this as:

Weisstein, Eric W. "Cube Line Picking--Face and Interior." From MathWorld--A Wolfram Web Resource.

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