Performance evaluation of the python Mpmath and math libraries in computing ellipsoidal distances using the inverse problem of geodesy

Since the works of Clairaut in the 18^th century and Bessel in the 19th century, the Direct and Inverse Problems of Geodesy have been central topics in this science, due to their relevance in both scientific and engineering applications that require high positional precision. Despite the vast literature on geodesic lines, studies comparing the performance and limitations of the different available numerical methods are still relatively scarce. Furthermore, the advancement of dynamic geodesy and the establishment of high-precision reference frames have imposed requirements on the solutions to these problems, which justifies the evaluation of modern computational tools. In this context, Python has stood out as a widely used language in scientific computing, including in Geodetic Sciences, due to its versatility and ability to integrate with numerical libraries. Thus, the objective of this work was to analyze Python’s performance in computing geodetic distances by solving the Inverse Problem. For this, Vincenty’s equations were implemented in two programs: one based on the math library, with fixed precision, and another on the mpmath library, with arbitrary precision. The results were validated against a set of one hundred thousand high-precision geodesic lines, with the largest discrepancies on the order of 10⁻⁵ m. When compared in a practical application, calculating distances between the Municipality of Paulista, in Pernambuco, Brazil, and ten globally distributed locations, the differences between the libraries reached at most 10⁻⁶ m, demonstrating numerical performance consistent with high-precision geodetic applications.