Method for determining the radius spherical surface

The theoretical positions of the pairwise distance method have been developed as applied to the determination of the radius of a spherical surface. To implement the method, the reference points are placed on the investigated surfaces and linear distances are measured between them. The implementation of the method for determining the radius of a spherical surface from distances measured between 4 points located on the surface is presented. The specified number of points is the minimum necessary to solve the problem. An analytical expression is obtained for the radius of a spherical surface through the distances between pairs of points. The root-mean-square deviation of the error in the estimate of the radius caused by the errors in the measurement of distances is calculated. Optimal configurations of 4 points on the sphere are determined, which ensure the minimum dispersion of the estimate of the radius. For these optimal configurations, the opposite edges of tetrahedra formed by the corresponding points are equal to each other. The method was applied in a complex of works on the evaluation of the parameters of the alignment of a television-based measurement system for the angular position of a dynamic stand with a gas spherical bearing, equipped with reference point sources placed along the envelope of the spherical surface. Determination of the radius of such a spherical surface is complicated by the fact that it is discontinuous, covers a part that is less than half the sphere, there is no access to the center of the sphere, which is the center of rotation spherical bearing of the dynamic stand. The known indirect methods for determining the radius of a spherical surface, based on the results of linear or angular measurements with the help of special surface measuring devices based on measurements directly on the surface, the radius of curvature of which is determined, also prove to be unacceptable due to the absence in this case installation measuring base for the measuring device. The method makes it possible to exclude the use of expensive coordinate measuring machines to obtain a radius estimate, since a simple measuring instrument can be used to measure the pairwise distances.