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. 2011 Mar 1;2(1):88-91.
doi: 10.1016/j.jocs.2010.12.003.

Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere

Cheng Guan Koay  1
Affiliations

Analytically exact spiral scheme for generating uniformly distributed points on the unit sphere

Cheng Guan Koay. J Comput Sci. .

Abstract

The problem of constructing a set of uniformly-distributed points on the surface of a sphere, also known as the Thomson problem, has a long and interesting history, which dates back to J.J. Thomson in 1904. A particular variant of the Thomson problem that is of great importance to biomedical imaging is that of generating a nearly uniform distribution of points on the sphere via a deterministic scheme. Although the point set generated through the minimization of electrostatic potential is the gold standard, minimizing the electrostatic potential of one thousand points (or charges) or more remains a formidable task. Therefore, a deterministic scheme capable of generating efficiently and accurately a set of uniformly-distributed points on the sphere has an important role to play in many scientific and engineering applications, not the least of which is to serve as an initial solution (with random perturbation) for the electrostatic repulsion scheme. In the work, we will present an analytically exact spiral scheme for generating a highly uniform distribution of points on the unit sphere.

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Figures

Figure 1
Figure 1
A surface element and a line element on the unit sphere.
Figure 2
Figure 2
The new spiral point set of 88 points and its Voronoi tessellation.
Figure 3
Figure 3
(A) Voronoi area and (B) circumference of each of the 88 points shown in Figure 2.
Figure 4
Figure 4
Performance evaluation of the fixed point method and Newton’s method.
See this image and copyright information in PMC

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