The easiest way to generate random points on a (hyper)sphere, at least with respect to most sampling APIs, is to sample from an isotropic normal distribution of the appropriate dimension (for a three-dimensional sphere, just call randn() three times), then normalize the result to be on the surface of the sphere.
This works because the normal distribution is rotationally symmetric, so each sample is essentially a random unit vector scaled by a random (normally distributed) radius. Once you remove the variation in radius you just get back the random unit vector, i.e. a point on the unit sphere.
This works because the normal distribution is rotationally symmetric, so each sample is essentially a random unit vector scaled by a random (normally distributed) radius. Once you remove the variation in radius you just get back the random unit vector, i.e. a point on the unit sphere.