Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere

Abstract

The problem of constructing a set of uniformly distributed points on the surface of a sphere, also known as the Thomson problem, has a long and interesting history, which dates back to J.J. Thomson in 1904. A particular variant of the Thomson problem that is of great importance to biomedical imaging is that of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential is the gold standard, minimizing the electrostatic potential of one thousand points (or charges) or more remains a formidable task. Therefore, a deterministic scheme capable of generating efficiently and accurately a set of uniformly distributed points on the sphere has an important role to play in many scientific and engineering applications, not the least of which is to serve as an initial solution (with random perturbation) for the electrostatic repulsion scheme. In this work, we will present an analytically exact spiral scheme for generating a highly uniform distribution of points on the unit sphere.