pick a random point on the surface of a unit sphere;mathematica

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates and from uniform distributions and , since the area element is a function of , and hence points picked in this way will be "bunched" near the poles (left figure above).

To obtain points such that any small area on the sphere is expected to contain the same number of points (right figure above), choose and to be random variates on . Then

			(1)
			(2)

gives the spherical coordinates for a set of points which are uniformly distributed over . This works since the differential element of solid angle is given by

Weisstein, Eric W. "Sphere Point Picking." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SpherePointPicking.html