Bell, PC ORCID: https://orcid.org/0000-0003-2620-635X and Potapov, I (2022) Towards Uniform Online Spherical Tessellations. Discrete & Computational Geometry, 67 (4). 1124 - 1146.

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Abstract

The problem of uniformly placing 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 <jats:italic>gap ratio</jats:italic> 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 and for at least four points is no less than 1.726.</jats:p>

Item Type: Article
Additional Information: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Uncontrolled Keywords: online algorithms; discrepancy theory; spherical trigonometry; uniform point placement; computational geometry
Subjects: Q Science > Q Science (General)
Q Science > QA Mathematics
Divisions: Faculty of Natural Sciences > School of Computing and Mathematics
Depositing User: Symplectic
Date Deposited: 30 Jun 2022 14:55
Last Modified: 30 Jun 2022 14:55
URI: https://eprints.keele.ac.uk/id/eprint/11046

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