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
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.