The Great Elliptic Sailing (GES), which can reduce sailing distance, is important to navigators. Whether a Great Ellipse (GE) is worth using depends on whether the distance saved is significant. Otherwise, the Rhumb Line (RL) is easier to steer. We propose a simple criterion to evaluate the difference in distance between the GE and the RL. The criterion is that the GE is worth using when the vertex lies between the departure and destination. In order to take the advantage of shorter distance, the GE is usually approximated as a series of waypoints. Unlike currently practised methods, we propose the Longitude Bisection Method (LBM) which determines waypoints with varying intervals. This approach can establish the appropriate number of waypoints to approximate the GE effectively. The proposed criterion and the LBM are demonstrated in practical examples.

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

In this paper, formulae drawn from the discipline of map projections are applied to provide simple and accurate solutions for the Inverse and Direct problems on the Great Ellipse. Distance along the meridional arc of the spheroid as a function of geodetic latitude is defined in terms of an elliptic integral which will be replaced here with a compact harmonic series approximation possessing simplicity and high accuracy. Latitude as a function of distance along the meridional arc will also be obtained via another equally simple inversion series that also possesses high accuracy. When these two series are applied with their constants modified to suit the section ellipse, they will be shown to provide accurate solutions to the inverse and direct navigation problems pertaining to the Great Ellipse, and thereby provide a complete solution that is also simple to implement.

Vector based methods applied to navigational problems offer some advantages in the calculation of distance and azimuth. In this paper, the methods of vector analysis are used to develop a variety of expressions for azimuth that are suited to both the syntax of computer algorithms and commercial mathematical software. The solutions presented here do not require recourse to the rules of spherical trigonometry - a distinct advantage when dealing with the spheroid. In the process of preparing this paper, an error of principle that occurred in a previous work was revealed and is repaired here.