[Paper Review] A Comparison of Popular Point Configurations on $\mathbb{S}^2$

Douglas P. Hardin, T. J. Michaels+1 more|arXiv (Cornell University)|Jul 13, 2016

Advanced Differential Geometry Research49 references8 citations

TL;DR

This paper compares popular spherical point configurations on $\mathbb{S}^2$ for numerical analysis applications, evaluating equidistribution, separation, covering, and Riesz potential energy. It introduces a new 'equal area icosahedral points' configuration with low mesh ratio and provides asymptotic energy analyses for $N < 50,000$, offering new conjectures on energy scaling and lattice structure dominance.

ABSTRACT

There are many ways to generate a set of nodes on the sphere for use in a variety of problems in numerical analysis. We present a survey of quickly generated point sets on $\mathbb{S}^2$, examine their equidistribution properties, separation, covering, and mesh ratio constants and present a new point set, equal area icosahedral points, with low mesh ratio. We analyze numerically the leading order asymptotics for the Riesz and logarithmic potential energy for these configurations with total points $N&lt;50,000$ and present some new conjectures.

Motivation & Objective

• To evaluate and compare widely used spherical node configurations for numerical analysis applications.
• To analyze key geometric and potential energy properties: equidistribution, separation, covering, and mesh ratio.
• To introduce and evaluate a new point set, 'equal area icosahedral points,' with improved mesh ratio.
• To numerically investigate leading-order asymptotics of Riesz and logarithmic potential energies for $N < 50,000$.
• To propose new conjectures on energy scaling and lattice structure dominance in minimal energy configurations.

Proposed method

• Survey and generate 10 popular spherical node sets: Fibonacci/spiral, projections from low-discrepancy grids, zonal equal area, HEALPix, polygonal (icosahedral, cubed sphere, octahedral), minimal energy, maximal determinant, random, and mesh icosahedral equal area nodes.
• Define and compute key metrics: separation $\delta(\omega_N)$, covering radius $\eta(\omega_N)$, mesh ratio $\gamma(\omega_N) = \eta(\omega_N)/\delta(\omega_N)$, and spherical cap discrepancy $DC(\omega_N)$.
• Compute Riesz energy $E_s(\omega_N) = \sum_{x \neq y} |x - y|^{-s}$ and logarithmic energy $E_{\log}(\omega_N)$ for $s > 0$ and $s < 0$.
• Use geometric and asymptotic analysis to bound separation and covering radii, particularly for the new equal area icosahedral configuration.
• Apply Voronoi cell decomposition to study local structure and relate to hexagonal lattice patterns.
• Prove that the mesh ratio of the equal area icosahedral points is bounded, establishing quasi-uniformity.

Experimental results

Research questions

• RQ1 How do different spherical node configurations compare in terms of equidistribution, separation, and covering radius?
• RQ2 What is the mesh ratio behavior of the newly proposed equal area icosahedral point set, and does it achieve quasi-uniformity?
• RQ3 What are the leading-order asymptotics of Riesz and logarithmic potential energies for $N < 50,000$?
• RQ4 Can new conjectures be formulated about the dominant term in the energy expansion for $s > 2$?
• RQ5 How do the geometric properties of the new equal area icosahedral nodes compare to existing configurations like HEALPix or Fibonacci spirals?

Key findings

• The new 'equal area icosahedral points' configuration achieves a low mesh ratio, with $\liminf_{N\to\infty} \delta(\omega_N)\sqrt{N} \geq \sqrt{8}$, indicating strong quasi-uniformity.
• The mesh ratio $\gamma(\omega_N)$ for the equal area icosahedral points is bounded, confirming quasi-uniformity, and the covering radius satisfies $\eta(\omega_N) \leq \sqrt{4 + \pi^2/(8k^2)}$ for $N = 4k^2 + 2$.
• For $N < 50,000$, the Riesz energy $E_s(\omega_N)$ for the new point set shows favorable scaling, supporting its use in energy-minimization problems.
• The authors conjecture that for $s > 2$, the dominant term in the asymptotic expansion of $E_s(N)$ is related to the Epstein-Zeta function of the hexagonal lattice.
• The separation and covering radii of the equal area icosahedral points are shown to be $O(1/\sqrt{N})$, matching optimal asymptotic rates.
• The Voronoi decomposition of nearly optimal energy configurations exhibits a mix of spherical hexagons, heptagons, and pentagons, consistent with icosahedral and hexagonal lattice structures.