Solving the Direct and Inverse Geodetic Problems on the Ellipsoid by Numerical Integration

Taking advantage of numerical integration, we solve the direct and inverse geodetic problems on the ellipsoid. In general, the solutions are composed of a strict solution for the sphere plus a correction to the ellipsoid determined by numerical integration. Primarily the solutions are integrals along the geodesic with respect to the reduced latitude or azimuth, but these techniques either have problems when the integral passes a vertex (i.e., point with maximum/minimum latitude of the arc) or a singularity at the equator. These problems are eliminated when using Bessel’s idea of integration along the geocentric angle of the great circle of an auxiliary sphere. Hence, this is the preferred method. The solutions are validated by some numerical comparisons to Vincenty’s iterative formulas, showing agreements to within $2\times {10}^{-10}$ of geodesic length (or 3.1 mm) and $4\times {10}^{-5}$ as seconds of azimuth and position for baselines in the range of 19,000 km.