Towards Uniform Online Spherical Tessellations

6^th March 2019, 14:00
Dr Paul Bell
Liverpool John Moores University

Abstract

We study the problem of uniformly placing a set of N points onto the 2-sphere in the online setting, measured by studying the gap ratio from discrepancy theory. We start with some applications of this problem in various domains. We then propose a two-phase online algorithm, which uses a regular icosahedron in phase 1,
and a recursive triangular dissection of each independent face of the icosahedron in phase 2 to improve the previous upper-bound for the gap ratio from 3.69 to 2.84. We also prove the first lower bound for the problem of 1.618 (the golden ratio). Our proof ideas use techniques from spherical trigonometry, geometry and spherical calculus.
We conclude with some open problems.