论文标题
单位$ n $ -sphere的点之间的平方距离总和
Sums of Squared Distances between Points on a Unit $n$-Sphere
论文作者
论文摘要
在本文中，我们证明了两个定理有关单位$ n $ sphere上点之间平方距离的总和，这些总和概括了以前知道的两个事实，即这些点是常规多边形的顶点。第一个定理是，鉴于单位$ n $ -sphere的多v $点，这些点之间的平方距离的总和为$ v^2（1-d^2）$，其中$ d $是点的质心和单元$ n $ sphere的中心之间的距离（对于任何$ n $ n \ geq 2 $）。第二个是，鉴于以原点为中心的单元$ n $ sphere上的一组有限的积分，因此该点集的点是对称的，而点集的对称组则在集合上实现，因此这些点之间的平方差距为$ 2K + 2 $ k $之间的差距为$ 2 $ k $，是$ 2 $ k $的差异（对于任何$ n $ n $ n $ n $ n \ geq 2 $ n \ geq）。使用第一定理，我们找到了一种计算有限归一化帧的势能函数的新方法。
In this paper, we prove two theorems concerning the sums of squared distances between points on a unit $n$-sphere that generalize two facts previously known about the case where the points are the vertices of a regular polygon. The first theorem is that, given a multiset of $V$ points on a unit $n$-sphere, the sum of the squared distances between these points is $V^2 ( 1 - d^2 )$ where $d$ is the distance between the centroid of the points and the center of the unit $n$-sphere (for any $n \geq 2$). The second is that, given a finite set of points on the unit $n$-sphere centered at the origin such that the point set is symmetric about the origin and the symmetry group of the point set acts transitively on the set, the sum of the squared distinct distances between these points is $2k + 2$ where $k$ is the number of distinct distances between the points (for any $n \geq 2$). Using the first theorem, we find a new way to calculate the potential energy function of a finite normalized frame.
