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 = ( X d - 2 , 1 - ∥ X d - 2 ∥ 2 ( X d - 1 , X d ) ) , such that the random vector ( X d - 1 , X d ) is uniformly distributed on a unit 2-sphere and X d - 2 has conditional uniform distribution on a ( d - 2 ) -sphere of radius 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.