Consider the distribution of distances 
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).](/images/equations/CubeLinePickingFaceandInterior/NumberedEquation1.svg) |
(1)
|
The first even raw moments 
for 
, 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395,
....
