Abstract

A variant of the Thomson problem, which is about placing a set of points uniformly on the surface of a sphere, is that of generating uniformly distributed points on the sphere that are endowed with antipodal symmetry, i.e., if x is an element of the point set then −x is also an element of that point set. Point sets with antipodal symmetry are of special importance to many scientific and engineering applications. Although this type of point sets may be generated through the minimization of a slightly modified electrostatic potential, the optimization procedure becomes unwieldy when the size of the point set increases beyond a few thousands. Therefore, it is desirable to have a deterministic scheme capable of generating this type of point set with near uniformity. In this work, we will present a simple deterministic scheme to generate nearly uniform point sets with antipodal symmetry.

中文翻译：

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在单位球面上生成近乎均匀分布的对映对称点的简单方案。

摘要

汤姆森问题的一个变体，即在球面上均匀地放置一组点，是在球面上生成均匀分布的点，这些点具有对映对称性，即，如果x是点的元素设置然后 - x也是那个点集的一个元素。具有对映对称性的点集对许多科学和工程应用具有特别重要的意义。尽管可以通过最小化稍微修改的静电势来生成这种类型的点集，但是当点集的大小增加到超过几千个时，优化过程变得笨拙。因此，希望有一种确定性方案能够以接近均匀的方式生成这种类型的点集。在这项工作中，我们将提出一个简单的确定性方案来生成具有对映对称性的几乎均匀的点集。