基于点间距离求球体半径，支持误差与形变下的高精度估计。
Determining Sphere Radius through Pairwise Distances
- 仅用4个必要点和任意多点构建距离矩阵求解半径。
- 给出半径估计标准差公式，量化测量误差与形状偏差影响。
- 发现最优点布局，使半径估计方差最小，适合精密测量场景。
本文提出一种新方法，基于球面上点之间的距离测量来确定球体半径。考虑了最一般情形：距离测量存在误差，且球体存在随机偏离理想形状的情况。采用最少必要四点及任意数量的点进行计算，推导出通过点对距离矩阵求解球体半径的闭式表达式。同时，给出了由测量误差与形状偏差导致的半径估计标准差。研究还找到了能最小化半径估计标准差的最优点配置。本文完整描述了解法并提供所有数学推导，代码已开源至 https://github.com/boris-sukhovilov/Sphere_Radius。
原文摘要 · Abstract (English)
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.
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