Towards Uniform Online Spherical Tessellations

TL;DR

This paper introduces a new efficient tessellation method based on the icosahedron for uniform point placement on spheres, improving the online gap ratio bounds and establishing new lower bounds for small point sets.

Contribution

It presents a novel tessellation technique using the icosahedron that improves the upper bound for online uniform point placement on spheres and establishes new lower bounds for small point counts.

Findings

Upper bound for gap ratio improved to approximately 2.84

Lower bound for three points is the golden ratio (~1.618)

Lower bound for four points is at least 1.726

Abstract

The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as "incremental generation") plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is the golden ratio, approx. 1.618, and for at least four points is no less than 1.726.