Distributing many points on a sphere
Distributing many points on a sphere
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DOI：
10.1007/bf03024331
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发表时间：
1997-12-01
影响因子：
0.4
通讯作者：
Kuijlaars, A. B. J.
中科院分区：
文献类型：
--
作者：
Saff, E. B.;Kuijlaars, A. B. J.
-‘。在球面上均匀分布大量点的问题不仅启发了数学研究人员，而且引起了人们的关注~f：/t/~176：：t：u；hifi；L：/i~：L：i~t；一个简单的问题就是点在圆周上均匀分布的问题，等距点给出了一个明显的答案。因此，我们面临这样一个问题：球面上的哪几组点模仿单位圆上的统一根的作用？一种可以产生这样的点的方式是通过相对于适当的标准(例如“广义能量”)进行优化。虽然有大量且不断增长的文献研究了当N为“小”时N点的最佳球面构型，但这里我们将从渐近的角度(N--~~)集中讨论这个问题。稳定的碳-60分子(Kroto等人，1985年)*的发现使原子排列成球形(足球)，对当前的科学研究产生了相当大的影响。由Frk Chung、B.Kostant和S.Sternberg[5]揭示，对这种C60巴克敏斯特纤维烯的研究也有一个优雅的数学成分。现在，寻找更大的稳定碳分子的工作正在进行！虽然这种分子不会有严格的球形结构(由于成键的限制)，但这里感兴趣的是构建大的稳定的球形点构型，作为假设更复杂的分子网络结构的第一步。在静电学中，在球上定位相同的点电荷，使它们相对于库仑势能定律处于平衡状态是一个具有挑战性的问题，有时被称为稳定分子的对偶问题。当然，球面上均匀分布的多个点在计算领域有着重要的应用。事实上，求积公式依赖于适当选择的采样数据点，以便通过取这些点的平均值来近似面积积分。另一个例子出现在计算复杂性的研究中，M.Shub和S.Smer[22]遇到了确定球点以最大化其相互距离的乘积的问题。
-'. B. SAFF AND ABJ KUIJLAAR, c he problem of distributing a large number of points uniformly over the surface of a tsphere has not only inspired mathematical researchers, it has the attracted attention~ f:/t/~ 176:: t: u; hifi; l::::/i~: l: i~ t;~ ogous problem is simply that of uniformly distributing points on the circumference of a disk, and equally spaced points provide an obvious answer. So we are faced with this question: What sets of points on the sphere imitate the role of the roots of unity on the unit circle? One way such points can be generated is via optimization with respect to a suitable criterion such as" generalized energy." Although there is a large and growing literature concerning such optimal spherical configurations of N points when N is" small," here we shall focus on this question from an asymptotic perspective (N--~~). The discovery of stable carbon-60 molecules (Kroto, et al., 1985)* with atoms arranged in a spherical (soccer ball) pattern has had a considerable influence on current scientific pursuits. The study of this C60 buckminsterfifllerene also has an elegant mathematical component, revealed by FRK Chung, B. Kostant, and S. Sternberg [5]. Now the search is on for much larger stable carbon molecules! Although such molecules are not expected to have a strictly spherical structure (due to bonding constraints), the construction of large stable configurations of spherical points is of interest here, as an initial step in hypothesizing more complicated molecular net structures. In electrostatics, locating identical point charges on the sphere so that they are in equilibrium with respect to aCoulomb potential law is a challenging problem, sometimes referred to as the dual problem for stable molecules. Certainly, uniformly distributing many points on the sphere has important applications to the field of computation. Indeed, quadrature formulas rely on appropriately chosen sampled data-points in order to approximate area integrals by taking averages in these points. Another example arises in the study of computational complexity, where M. Shub and S. Smale [22] encountered the problem of determining spherical points that maximize the product of their mutual distances.