Section 3 of this paper explains this topic pretty well in N-dimensions: http://jmlr.org/papers/volume17/blaser16a/blaser16a.pdf http://jmlr.org/papers/volume1

Section 3 of this paper explains this topic pretty well in N-dimensions: http://jmlr.org/papers/volume17/blaser16a/blaser16a.pdf http://jmlr.org/papers/volume17/blaser16a/blaser16a.pdf To get a random point within the sphere (rather than on the surface of the sphere, as described in the paper) you also need to pick a random distance from the origin to randomly scale the point on the surface.