Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 231

Fig. 5.39 Geodesic polar
coordinate system
Fig. 5.40 Solution of
geodetic problems
mathematical properties of the ellipsoid than the plane. Because of this complexity,
there are various formulae, dozens at present, for solving geodetic problems. Based
on distance, the formulae can generally be categorized into short-distance (within
400 km), mid-distance (400-1,600 km), and long-distance (1,000-20,000 km).
Based on accuracy, they can be sorted into precise formulae and approximate
formulae.
Theoretically, the formulae for solutions of geodetic problems are mostly based
on the three differential equations of geodesics, although their forms and methods
of derivations vary. For solution of long-distance geodetic problems, Clairaut's
equation for geodesics should also be applied. The formulae for solutions can
basically be classified into the following three categories in light of the methods
for solution.
1. Based on the geodesic on the ellipsoid and its three differential equations, we
expand the differences in geodetic longitude l, geodetic latitude b, and geodetic
azimuth a of the two endpoints of the geodesic into the ascending power series of