Earth section paths. Solution to the inverse and direct problems, and waypoints without iterations

Abstract

The use of central elliptical sections in the calculation of air and sea routes and normal sections in Geodesy is common. Elliptic sections do not represent the shortest path between two points, although are often used in navigation to replace the geodesic lines. All developments include some kind of iteration to solve one of the problems, direct or inverse. When using vector algebra methods and perturbed series, the problems can be solved using an equidistant circle that represents the path of the elliptical section. This is possible because the flattening of the elliptical section is less than or equal to the Earth’s flattening, which implies that the series calculated for the terrestrial ellipsoid, used in the section, always converge. Three direct methods are described in order to calculate: the distance, the azimuth, the coordinates of a point and intermediate positions of an elliptical section. Those algorithms provide solutions to the inverse and direct algorithms with a consistency of the order of truncation error of double-type numbers.

Keyword : elliptical sections, routes, waypoints, great ellipse, direct formula, rectified angle, elliptic arc