On a combination method of VDR and patchwork for generating uniform random points on a unit sphere

In this paper, we use a combination of VDR theory and patchwork method to derive an efficient algorithm for generating uniform random points on a unit d-sphere. We first propose an algorithm to generate random vector with uniform distribution on a unit 2-sphere on the plane. Then we use VDR theory to reduce random vector Xd with uniform distribution on a unit d-sphere into Xd=(Xd-2,1-∥Xd-2∥2(Xd-1,Xd)), such that the random vector (Xd-1,Xd) is uniformly distributed on a unit 2-sphere and Xd-2 has conditional uniform distribution on a (d-2)-sphere of radius 1-V, given V=v with V having the p.d.f. d2(1-v)d-22. Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.