Cовершенствование математических моделей проекции Гаусса – Крюгера для вычисления плоских прямоугольных координат по геодезическим координатам

Abstract

Algorithms with improved convergence for the calculation of rectangular coordinates in the Gauss – Kruger coordinate system according to the parameters of any ellipsoid were designed. The approach of definition the spherical components in the classic series defined variables x, y, represented by the difference between the degrees of longitude l, followed by the replacement of their sums by formulas of spherical trigonometry. For definition of the amounts of spherical components of the relevant decompositions patterns of transverse-cylindrical sphere plane projection in the condition of the initial data equality on the ellipsoid and sphere radius N were used. Analysis of othertransformation methods of classical expansions in series, used in derivation of both logarithmical and non-logarithmical working formulas is carried outfor comparison with developed algorithms. The technique of algorithms development with usage of hyperbolic tangent function, applied by L. Kruger, Yu. Karelin, A. Schödlbauer is considered and their analysis is carried out. Advantages of Krasovskii – Isotov formulas for six-degree strips are pointed out. The usage of the spherical function sin τ in the expansion made it possible not only to obtain a rapidly convergent series, but also to represent the spherical part of the solution of the problem with the help of trigonometric identities in different types. It is proved that derived for the calculation algorithms with the proposed estimates of their accuracy, are optimal in removing points from the central meridian to l ≤ 6°. For the difference of longitudes l > 6°, the expansions of the unknown quantities into Fourier series should be applied. An example of the calculation of coordinates in the system SK-2011 is given. Theoretical studies have been carried out and shortened formulas with a reliability estimate for the determination of coordinates in the area l ≤ 3° have been proposed.