The most uniform distribution of points on the sphere

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.

arXiv comment17 pages, 15 figures

References in corpus (16)

• HEALPix -- a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere
• Crystalline Order on a Sphere and the Generalized Thomson Problem
• An icosahedron-based method for pixelizing the celestial sphere
• Uncovering conformal symmetry in the $3D$ Ising transition: State-operator correspondence from a fuzzy sphere regularization
• Influence of Dislocations in Thomson's Problem
• Vortices in a Thin Film Superconductor with a Spherical Geometry
• Symmetric patterns of dislocations in Thomson's problem
• Density waves theory of the capsid structure of small icosahedral viruses
• Why charges go to the surface: a generalized Thomson problem
• Defect free global minima in Thomson's problem of charges on a sphere
• A new equal-area isolatitudinal grid on a spherical surface
• Universal amplitude-exponent relation for the Ising model on sphere-like lattices
• Absence of a Finite-Temperature Melting Transition in the Classical Two-Dimensional One-Component Plasma
• Spherical structure factor and classification of hyperuniformity on the sphere
• Hyperuniformity on spherical surfaces
• Analytical solution to Heisenberg spin glass models on sparse random graphs and their de Almeida-Thouless line

Cited by in corpus (3)

• Soft modes in vector spin glass models on sparse random graphs
• Linguistic Pattern Based Optimization of Economic and Spatial Uniformity Criteria in Facility Layout Problems
• Canonical equilibrium of mean-field $O(n)$~models in the presence of random fields