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

Abstract

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 of2 . Finally, we arrive by induction at an algorithm for generating uniform random points on a unit d-sphere.

The equal spacing of N points on a sphere with application to partition-of-unity wave diffraction problems

“…where (19) is used to normalise the position vectors to relocate the particles back onto S. Equations (14) to (19) are repeated in a time-stepping scheme to convergence. An appropriate measure of the performance of the method is the minimum angle, ρ, between any two vectors u i and u j , i.e.…”

“…Given a suitable normal distribution, this method has a lower ratio of rejected to accepted points compared to taking points from the uniform distibution. A family of methods, using the beta distribution, were developed for higher dimensional spheres [10,11,12,13,14]. The relationship between these efficient methods was presented by Harman and Vladimir [15].…”

On decompositional algorithms for uniform sampling from n-spheres and n-balls

“…i.e., for uniform sampling from the n-ball, Algorithm YPHL corresponds to the method proposed by Yang et al [31], which uses the so-called patchwork method for generating from the (atomic) distribution U B 2 .…”

“…The algorithms in the third group, which circumvent generating normal variates, are essentially based on the properties of the marginals of the uniform distribution on n-spheres and n-balls; see the papers by Hicks and Wheeling [16], Sibuya [27], Marsaglia [19], Tashiro [29], Guralnik et al [14], Fang et al [11], Yang et al [31]. It is the purpose of this paper to clarify mutual relations of these algorithms as well as to use the general point of view to propose alternative methods for generating uniformly from n-spheres and n-balls of dimensions n = 5, 6, 7.…”