Abstract
The article gives a brief analysis of methods and algorithms for the transformation of spatial rectangular coordinates to curvilinear coordinates - geodetic latitude, geodetic longitude, geodetic height. Two algorithms for solving the equation for determining longitude are considered. Three formulas used to calculate the height are analyzed, with an estimate of their errors due to the approximate latitude. The shortcomings of mathematical solutions to these problems are revealed. A study of different approaches and methods for solving the transcendental equation for determining the latitude, based on the theory of separation of the root of the equation, is performed. Using this technique, iterative processes were performed to calculate the reduced latitude , using trigonometric identities, by introducing an auxiliary angle and transforming it to an algebraic quartic equation, which Borkowski solves by the Ferrari's method. The determination of the root isolation interval allowed using the chord method (proportional parts) to determine the latitude. In all cases, estimates of the convergence of the iterative processes that facilitate the comparative analysis of the proposed solutions are obtained. By further decreasing the separation interval of the root, the accuracy of the non-iterative determination of the latitude is improved by the Newton method.
Highlights
When solving geodetic problems in space, it is necessary to move from the rectangular coordinates X,Y, Z of the point Q to its curvilinear coordinates: geodetic latitude B, geodetic longitude L, geodetic altitude H, measured along the normal from the surface of the ellipsoid
A large number of publications are devoted to the solution of this problem [1,2,3,4,5]. Algorithms for such a transition are obtained by transforming a system of equations
U B at each iteration, determining its marginal errors u depending on the latitude and altitude H, performing a comparative analysis by the analytical methods of the proposed solutions of equations (10) and (11)
Summary
When solving geodetic problems in space, it is necessary to move from the rectangular coordinates X ,Y , Z of the point Q to its curvilinear coordinates: geodetic latitude B , geodetic longitude L , geodetic altitude H , measured along the normal from the surface of the ellipsoid. A large number of publications are devoted to the solution of this problem [1,2,3,4,5]. Algorithms for such a transition are obtained by transforming a system of equations. The problem of the reverse transition is more difficult to solve. Let us analyze briefly the applied methods of transformations of the system (1) and the results of the research
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