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.

An approach formulated by vector algebra is proposed to deal with great circle sailing problems. Using the technique of the fixed coordinates system and relative longitude concept, derivations of formulae for this approach are simpler than those of the conventional methods. Due to fixing the initial great circle course, the great circle track (GCT) is determined. Since the course is fixed (known as “COFI” in this paper), the proposed approach, which we have named the “COFI method”, can directly calculate the waypoints along the GCT. It is considered that the COFI method is a more understandable and straightforward method to solve waypoint problems than older approaches in the literature. Based on the COFI method, a program has been developed for the navigator. In addition, the spherical triangle method with respect to the equator crossing point (STM-E) is developed by supplemental theorem. Several examples are demonstrated to validate the proposed COFI method and STM-E.

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.

The present paper is concerned with the solution of a series of practical problems relevant to great circle navigation, including the determination of the true course at any point on the great circle route and the determination of the lateral deviation from a desired great circle route. Intersection between two great circles or between a great circle and a parallel is also analyzed. These problems are approached by means of vector analysis, which yields solutions in a very compact form that can be computed numerically in a very straightforward manner. This approach is thus particularly appealing for performing computer-aided great circle navigation.

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.

Traditionally, on a great circle, the latitude or longitude of a waypoint is found by inspection. In this paper, using an elementary knowledge of vector algebra including linear combination of vectors and vector basis, we provide an easy method for finding the equation of a great circle path as a parameterized curve. By use of this vector function of distance travelled, the latitude and longitude of waypoints can be found based on the distance from departure point along a great circle. The approach is intended to appeal to the navigator who is interested in the mathematics of navigation and who, nowadays, solves his navigation problems with a personal computer.

I am grateful to both Dr Ponsonby and Sqn Ldr Hoare for their responses to my original article and would like to thank them for replying. It is interesting that they have come up with such different approaches, one based on solid geometry (but not spherical trigonometry) and the other based on a map projection, which both give exact or near-exact answers.