标题： 通过成对距离确定球体半径 显示英文标题

标题： Determining Sphere Radius through Pairwise Distances

摘要： 我们提出了一种基于测量球面上点之间距离来确定球面半径的新方法。 我们考虑了最普遍的情况，即当测量存在误差且球体偏离理想形状时，如何确定其半径。 为了解决这个问题，我们使用了最少必要的四个点以及任意数量的N个点。 我们通过点对距离矩阵给出了球体半径的一种新的封闭形式解。 同时，我们还确定了由测量误差和球体偏离理想形状引起的半径估计的标准偏差。 我们找到了球体上点的最优配置，这些配置可以最小化半径估计的标准偏差。 本文描述了我们的解决方案，并提供了所有数学推导过程。 我们在 https://github.com/boris-sukhovilov/Sphere_Radius 共享了我们方法的开源代码。 显示英文摘要

摘要： We propose a novel method for determining the radius of a spherical surface based on the distances measured between points on this surface. We consider the most general case of determining the radius when the distances are measured with errors and the sphere has random deviations from its ideal shape. For the solution, we used the minimally necessary four points and an arbitrary N number of points. We provide a new closed form solution for the radius of the sphere through the matrix of pairwise distances. We also determine the standard deviation of the radius estimate caused by measurement errors and deviations of the sphere from its ideal shape. We found optimal configurations of points on the sphere that provide the minimum standard deviation of the radius estimate. This paper describes our solution and provides all the mathematical derivations. We share the implementation of our method as open source code at https://github.com/boris-sukhovilov/Sphere_Radius.