On the coordinate systems transformation

Abstract

Determination of coordinate systems’ parameters transformation has always been and remains the core of surveyors’ practical and theoretical works. However, in most cases, the issue is interpreted under the assumption that the angular rotation of the coordinate axes and the similarity parameter are very small, which enables a significant simplifying of the algorithm. In this paper the strict theory of determining the 7 and 9 parameters of coordinate systems transformation within the framework of nonlinear method of least squares with limitations is considered. Strict algorithm is described and numerical experiments are performed. The connection of the coordinate systems transformation theory with the general one of mathematical statistics, known under the name of Procrust analysis, is specified. However, in general, artificial neural network theory is recommended for conversion of one coordinate system to another. This generally eliminates the need for a prior determination of the transformation’s appropriate parameters, inevitably associated with hypotheses for the transformation model, but quickly and accurately solves the main problem.