Towards Uniform Online Spherical Tessellations

• Published in: Discrete & computational geometry Vol. 67; no. 4; pp. 1124 - 1146
• Main Authors: Bell, Paul C., Potapov, Igor
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-06-2022; Springer Nature B.V
• Subjects: Aeronautical engineering; Aeronautics; Algorithms; Combinatorics; Computational Mathematics and Numerical Analysis; Computer graphics; Computer science; Crystallography; Geometry; Icosahedrons; Lower bounds; Mathematics; Mathematics and Statistics; Quaternions; Robotics; Tessellation; Upper bounds; Online algorithms; Computational geometry; Discrepancy theory; Spherical trigonometry; 52C45; 11K38; 68W27; 52C35; 68U05; Uniform point placement
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-022-00384-x

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
(
1
+
5
)
/
2
≈
1.618
and for at least four points is no less than 1.726.