Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

TL;DR

This paper presents a numerical method for solving the direct geodesic problem on an oblate spheroid in both geodetic and Cartesian coordinates, emphasizing stability, accuracy, and computational efficiency.

Contribution

It introduces a stable and precise numerical solution for geodesic equations in Cartesian coordinates, validated against existing methods.

Findings

The method achieves high accuracy and stability in Cartesian coordinates.

Validation shows comparable or improved performance over Karney's method.

The approach allows computation of geodesic coordinates and azimuths at any point.

Abstract

The direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically in geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not assumed known but it is computed, allowing to check the precision of the method. An extended data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney's method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.