Towards Uniform Online Spherical Tessellations

• Main Authors: Bell, Paul C, Potapov, Igor
• Format: Journal Article
• Language: English
• Published: 03-10-2018
• Subjects: Computer Science - Computational Geometry
• DOI: 10.48550/arxiv.1810.01786

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.