The most uniform distribution of points on the sphere

TL;DR

This paper introduces a new physical measure of uniformity for distributing points on a sphere, compares existing algorithms using this measure, and optimizes point placement through gradient descent to achieve more uniform distributions.

Contribution

It proposes a novel measure of uniformity based on point distances and evaluates various algorithms and potentials for optimal point distribution on the sphere.

Findings

New measure effectively characterizes uniformity.

Gradient descent improves distribution uniformity.

Identifies the most effective algorithms and potentials.

Abstract

How to distribute a set of points uniformly on a spherical surface is a very old problem that still lacks a definite answer. In this work, we introduce a physical measure of uniformity based on the distribution of distances between points, as an alternative to commonly adopted measures based on interaction potentials. We then use this new measure of uniformity to characterize several algorithms available in the literature. We also study the effect of optimizing the position of the points through the minimization of different interaction potentials via a gradient descent procedure. In this way, we can classify different algorithms and interaction potentials to find the one that generates the most uniform distribution of points on the sphere.