ABSTRACT
This chapter focuses on the geometry of the surface of a revolving body, more specifically the ellipsoid of revolution as an approximation to the figure of the Earth. Discussed are the calculation the geodesic on the surface of the ellipsoid by solving a system of ordinary differential equations in arc length, and how to use this to solve the forward and reverse geodetic problem on the ellipsoid. The theoretically interesting Clairaut's equation and invariant is presented. Then the meridian ellipse is studied. Three different kinds of latitude are identified, with conversions between them. The eccentricity of the ellipse is defined and connected with the flattening. Equations expressing geocentric rectangular co-ordinates into geodetic latitude, longitude and height on the ellipsoid of revolution are derived, and a simple iterative method for the inverse calculation is presented. Then, the calculation of the length of a meridian arc is discussed. Finally we talk about how choosing the reference ellipsoid also implies choosing an approximation or model for the gravity field called the normal gravity field. This leads in a natural way to Claraut's beautiful identity connecting geometric and gravimetric Earth parameters, and the gravity formula of Somigliana--Pizzetti valid on the surface of the ellipsoid.
