Distributing Points on the Sphere, I
نویسندگان
چکیده
The problem of “evenly” distributing points on a sphere has a long history. Albeit its intuitive meaning, it is necessary to define “even distribution” mathematically. Various metrics, θs, 0 ≤ s ≤ ∞ (see below for definitions), whose extrema may correspond to even distributions have been proposed. The starting point of this study is Problem 7 in [Smale 00] where the implications of the distribution of points on the sphere, corresponding to the global minimum of θ◦, for numerical analysis are discussed. A different version of the problem based on minimizing the Coulomb potential of electrons distributed on a sphere is generally known as the Thomson problem in spite of the fact that the problems considered in [Thomson 04] are quite different. [Altschuler et al. 97] reports on a numerical study of Thomson’s problem where points are distributed on the sphere according to a number theoretic algorithm. In [Sarnak 90], the issue of constructing -good sets {R1, · · · , Rn} ⊂ SO(3) and its relation to the Ruziewicz problem is investigated. The explicit construction in [Sarnak 90] gives a method for distributing points on S by applying Rjs (or words in Rjs) to a random point in S . In [Rakhmanov et al. 94] and [Kuijlaars and Saff 98], various notions of energy (or metric) for points on a sphere and some theoretical results are discussed. In this work, we numerically analyze four different methods for distributing points on S. Besides the methods given in [Altschuler et al. 97] and [Sarnak 90], we discuss two simple geometric algorithms, the “subdivision” and “polar coordinates” methods, the details of which appear in the next section. The analysis of the merits of the algorithms is based on the calculation of the metrics θ◦, and θ1, and we will make some remarks about
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ورودعنوان ژورنال:
- Experimental Mathematics
دوره 12 شماره
صفحات -
تاریخ انتشار 2003