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 X-d with uniform distribution on a unit d-sphere into X-d = (Xd-2, root 1 - parallel to Xd-2 parallel to(2)(Xd-1, X-d)), such that the random vector (Xd-1, X-d) is uniformly distributed on a unit 2-sphere and Xd-2 has conditional uniform distribution on a (d - 2)-sphere of radius root 1 - V, given V = v with V having the p.d.f. d/2 (1 - v) d-2/2. Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere. (c) 2004 Elsevier Inc. All rights reserved.