Validation of Vincenty's Formulas for the Geodesic Using a New Fourth-Order Extension of Kivioja's Formula

Abstract

Vincenty’s (1975) formulas for the direct and inverse geodetic problems (i.e., in relation to the geodesic) have been verified by comparing them with a new formula developed by adapting a fourth-order Runge-Kutta scheme for the numerical solution of ordinary differential equations, advancing the work presented by Kivioja in 1971. A total of 3,801 lines of varying distances (10 to 18,000km) and azimuths (0 to 90°, because of symmetry) were used to compare these two very different techniques for computing geodesics. In every case, the geodesic distances agreed to within 0.115mm, and the forward and reverse azimuths agreed to within 5 × 10 −6 seconds of arc, thus verifying Vincenty’s formula. If one wishes to plot the trajectory of the geodesic, however, the fourth-order Runge-Kutta extension of Kivioja’s formula is recommended as a numerically efficient and convenient approach.