Analysis of Geodetic Latitude Calculation Algorithms Using Chord Method

Abstract

This article is an extension of investigation performed by the authors earlier, on the transformation of spatial rectangular coordinates X, Y, Z into geodetic curvilinear ones: latitude B, longitude L, height H. To calculate geodetic latitude B and reduced latitude U by solving a transcendental equation with variable coefficients f(t)=0, where t=tgU, a chord method was applied for the segment [T 1 , T 2 ]; T 1 =tgu 1 ; T-=tgu2. It is mathematically proved that the algorithm constructed on this segment using chord method will be valid only for points on the earth's surface with heights above sea level and with maximum error ΔB=0. 0017". To expand the limits of algorithm and to increase its accuracy, the segment of root location [T 1 , T 2 ] is reduced on the right to the value of [T 1 , T 6 ] where the latitude is determined not only for H>0 but also for H<0. The general formula of chords is transformed to a convenient for calculations form by introducing auxiliary quantities. Its accuracy was estimated and a comparative analysis with the previously obtained algorithm on segment [T 3 , T 2 ] was performed. As a result of a two-sided reduction of interval [T 1 , T 2 ], segment [T3, T6] is obtained where the root of equation will be located only for H>0. It was proved that in order to calculate latitude for points of the earth's surface for |H| ≤ 10 km in non-iterative way using chord method, it was reasonable to use an algorithm constructed on segment [T3, T2], and for near-Earth space points at H>a, calculations should be carried out according to the algorithm constructed on segment [T1, T6].