Calculation of latitude by secant method on the basis of spatial orthogonal coordinates

Abstract

This article discusses the importance of conversion of orthogonal coordinates X, Y, Z into curvilinear geodesic coordinates: latitude B, longitude L, and altitude H. Geodesic latitude in tangent function is calculated by transcend equation with variable coefficients which is meaningless when the point is located on rotation axis of Earth ellipsoid. For this case the algorithm for calculation of latitude, altitude, and longitude is provided. The algorithms are developed and analyzed by solution of less complicated transcend equation with the reduced latitude u using root isolating. Reasonability of development of non-iterative algorithms is substantiated. Root isolating for tgu made it possible to apply the secant method in the segment [T1, T1 ] according to which the latitude is calculated with the error of ∆u ≤ 0, 0011″ (T1 = tgu1; T1 = tgu1). In order to improve accuracy according to the requirements of interstate standard: ∆B ≤ 1, 1111″, the segment [T1, T1 ] was reduced by two methods from left side. In order to develop non-iterative algorithm, the segment [T1, T1 ] was selected where the latitude was determined not only at H > 1, but also at H < 1. Using auxiliary variables, the general secant equation was converted into convenient form for calculations of latitude. Its error has been estimated which was analyzed for extremum. According to the proposed algorithm the latitude is determined with the error of ∆B ≤ 1″•10−10. This method can be applied for development of non-iterative algorithms in other segments of root isolating with subsequent comparative analysis.