GPS Satellite Surveying, 4th Edition

Appendix BThe Ellipsoid

The ellipsoid of rotation is a geometric structure for mathematical formulations and computations. For example, the observables of the 3D geodetic model refer to the ellipsoidal normal and the geodetic horizon, whereas the observables of the ellipsoidal model are the angle between geodesics and the length of the geodesic on the ellipsoidal surface. In the case of the conformal mapping model, the ellipsoidal surface is mapped conformally. Details of these mathematical models are given in Sections 4.5 and 4.6. Because the ellipsoid is important as a computational reference and as a means to express position coordinates, the ellipsoid and the related geometry are summarized here. Since only ellipsoids of rotation have been adopted in practical geodesy and triaxial ellipsoids have been limited to theoretical studies, we will use the term ellipsoid for reasons of brevity to mean ellipsoid of rotation. Such an ellipsoid is generated when rotating an ellipse around the semiminor axis.

The expressions for computing on the ellipsoidal surface and on the conformal mapping plane are deeply rooted in differential geometry. Working expressions typically utilize series expansions that are simplified by truncating insignificant terms (having specific applications in terms of position accuracy and area in mind). The algebraic work necessary to arrive at working expressions is considerable and not at all obvious to the novice. Prior to the introduction of electronic computers, ...