This paper offers a precise and almost final solution to one of the most debated problems connected with the astrogeodetic ‘fix’ by azimuth differences. The rectangular and linear solution on the gnomonic plane, tangent to the celestial globe and the dead reckoning position, enables the use of azimuth differences and is connected to Pothenot's solution. Thus this problem may be solved with simple linear algorithms that require neither differential modes, nor the introduction of the unknown origin error.

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.