Online Uniformly Inserting Points on the Sphere

21^st April 2017, 13:00 Ashton Lecture Theatre
Prof. Sheung Hung Poon
School of Computing and Informatics
University of Technology Brunei (UTB)

Abstract

In many scientific and engineering applications, there are occasions where points need to be inserted uniformly onto a sphere. Previous works on uniform point insertion mainly focus on the offline version, i.e., to compute N positions on the sphere for a given integer N with the objective to distribute these points as uniformly as possible. An example application is the Thomson problem where the task is to find the minimum electrostatic potential energy configuration of N electrons constrained on the surface of a sphere. In this paper, we study the online version of uniformly inserting points on the sphere, where the number of inserted points is not known in advance. This means that the points are inserted one at a time and the insertion algorithm does not know beforehand how many points are going to be inserted. The objective is again to achieve a distribution of the points on the sphere that is as uniform as possible at each step. The uniformity is measured by the gap ratio, which is the ratio between the maximal gap and the minimal gap of any pair of inserted points. We give a two-phase algorithm by making use of the structure of the regular dodecahedron, of which the gap ratio is upper bounded by 5.99. This is the first result for the problem of online uniform point insertion on the sphere.

PS. This is a joint work with The University of Hong Kong, and Chinese Academy of Sciences.