Online inserting points uniformly on the sphere

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 interger 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. The number of inserted points is not known in advance, which means the points are inserted one at a time and the insertion algorithm does not know when to stop. As before, the objective is achieve a distribution of the points that is as uniform as possible at each step. The uniformity is measured by the gap ratio, the ratio between the maximal gap and the minimal gap of any pair of inserted points. We give a two-phase algorithm by using the structure of the regular dodecahedron, of which the gap ratio is upper bounded by 5.99. This is the first result for online uniform point insertion on the sphere.