Distance and course to steer along part of a great ellipse can be readily determined with the use of vector analysis and differential geometry. Although the resulting integral for distance is not amenable to direct solution, it can nevertheless be handled efficiently and quickly by available commercial mathematical software. In this paper, a method of computing distance along with course to steer is presented and is one that has been prepared with the syntax of a particular commercial mathematics package in mind. This approach was intended to appeal to the navigator who is interested in the mathematics of navigation, and who nowadays solves his problems with a portable computer.

The normal Great Circle method of computing the shortest distance between two positions on the Earth – e.g. from an aircraft's present position (PP) to a waypoint (WP) – is not accurate enough to meet present-day requirements for aircraft Nav–Attack systems.

On the surface of an Ellipsoid (or Spheroid), the true ‘shortest distance’ is along a geodesic curve between the two points, but the computation of this curve is complex, and as shown by R. Williams at Reference, the difference between the geodesic and Great Ellipse distances between two points is negligible (<0·01 nm).

The Great Ellipse through two points on a spheroid is defined as the ellipse that passes through the two points and the centre of the spheroid; it therefore has a major axis equal to the Earth's, and a minor axis that is between the Earth's major axis (for two points on the Equator) and minor axis (for two points on the same, or diametrically opposite, longitudes). Thus the problem of deciding on which Great Ellipse the two points lie is equivalent to determining the magnitude of the minor axis β of the ellipse on which they both lie.

This article addresses the mathematical foundations of rhumblines or loxodrome curves. These curves are critical to navigation and small-scale charting by virtue of the fact that they provide an efficient routeing from one point on a surface to another by means of a constant ‘course angle‘. This article will develop the necessary mathematical relations for the construction of such a curve, then apply the relations to both spherical and oblate-spheroidal surfaces. The purpose of this article is to produce a superior oblate-spheroidal loxodrome curve, which better models curves or routes of constant course on the actual, approximately oblatespheroidal, Earth.

Lambert, in his paper entitled The distance between two widely separated points on the surface of the Earth, published in 1942, having given a solution which was applicable in most cases, stated that there was a particular problem in determining the shortest distance between two nearly antipodean points on the surface of the spheroidal Earth but this was not fully analysed in that paper.