Pairwise distances method for determining radius of spherical surface with shape deviation

The theoretical positions of the pairwise distance method have been developed as applied to the determination of the radius of a spherical surface with shape deviation. To implement the method, the reference points are placed on the surface under investigation and linear distances are measured between them. The most general case of radius determination is considered, when pairwise distances are measured with errors, and the sphere has random deviations of shape. The considered approach is the development of the method for determining the radius of a spherical surface over distances measured between four points located on the surface. The specified number of points is the minimum necessary to solve the problem. The implementation of the method for determining the radius of a spherical surface over distances measured between an arbitrary, large, 4-number of points located on the surface is presented. In the framework of the developed method, analytical expressions for the radius of a spherical surface are obtained through distances between pairs of points and the root-mean-square deviations of errors in the estimates of the radius caused by errors in measuring distances and deviation of the shape are calculated. Conditions for the optimality of the configurations of a given number of points on the sphere that ensure the minimal variance of the estimate of the radius are determined. The method makes it possible to exclude the use of expensive coordinate machines to obtain a radius estimate, since a simple serial measuring tool can be used to measure pair distances. The evaluation of parameters by the pairwise distances method is suitable in the case of large discontinuous surfaces, when the measured surface is only a part of the total surface, direct measurements are impossible due to the inaccessibility of the measuring bases. The method was applied in a complex of works on the evaluation of the parameters of the alignment of a television system for measuring the angular position of a dynamic stand with a gas support.