Jacobi's solution for geodesics on a triaxial ellipsoid

TL;DR

This paper discusses a numerical implementation of Jacobi's classical solution for geodesics on a triaxial ellipsoid, enabling accurate computation of geodesic paths and distances, including the inverse problem of shortest path determination.

Contribution

It provides a detailed numerical method for evaluating Jacobi's integrals and solving the coupled equations for geodesics on a triaxial ellipsoid, extending previous solutions.

Findings

Accurate evaluation of integrals for geodesic calculations.

Method for solving the inverse problem of shortest paths.

Implementation of coupled system of equations for geodesics.

Abstract

On Boxing Day, 1838, Jacobi found a solution to the problem of geodesics on a triaxial ellipsoid, with the course of the geodesic and the distance along it given in terms of one-dimensional integrals. Here, a numerical implementation of this solution is described. This entails accurately evaluating the integrals and solving the resulting coupled system of equations. The inverse problem, finding the shortest path between two points on the ellipsoid, can then be solved using a similar method as for biaxial ellipsoids.