Determining Sphere Radius through Pairwise Distances

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.