The geodesic boundary value problem and its solution on a triaxial ellipsoid

Abstract

The geodesic problem on a triaxial ellipsoid is solved as a boundary value problem, using the calculus of variations. The boundary value problem consists of solving a non-linear second order ordinary differential equation, subject to the Dirichlet conditions. Subsequently, this problem is reduced to an initial value problem with Dirichlet and Neumann conditions. The Neumann condition is determined iteratively by solving a system of four first-order ordinary differential equations with numerical integration. The last iteration yields the solution of the boundary value problem. From the solution, the ellipsoidal coordinates and the angle between the line of constant longitude and the geodesic, at any point along the geodesic, are determined. Also, the constant in Liouville's equation is determined and the geodesic distance between the two points, as an integral, is computed by numerical integration. To demonstrate the validity of the method presented here, numerical examples are given. The geodesic boundary value problem and its solution on a biaxial ellipsoid are obtained as a degenerate case.