There is a general procedure, using the calculus of variations, to find the equation for geodesics given the metric of the surface (ie given Eqn. or (5)) (6). However, in this case, a simpler argument suffices.
Consider a particle of mass sliding on the surface of a spheroid. The
constraint forces, normal to the surface, do no work, so the
particle's kinetic energy , and thus its speed , remain
constant. In addition, because of the spheroid's axisymmetry, all its
surface normals pass through the polar axis. Thus the constraint
forces have zero moment about the polar axis. The angular momentum of
the particle around the polar axis is therefore conserved. Referring
to Figs. 1 and 3 we can
write this as
, where is the
azimuthal component of the particle's velocity and is
the distance of the particle from the polar axis . Thus, using
Eqn. (4) we obtain:
This equation takes its simplest form when reduced latitudes, are used, so geodesic calculations are generally done in coordinates, with necessary conversions back and forth to geodetic coordinates being performed using Eqn. (2). In fact, the relationship between the azimuth and the reduced latitude on a spheroidal geodesic is the same as on a spherical great circle. This sets up a correspondence between geodesics and great circles on an auxiliary sphere with a common value of . At each (reduced) latitude, the geodesics have the same azimuth, . However, distances and longitude differences differ by corrections. The great circle distances and longitude differences can be used as a first approximation to an iterative or perturbative evaluation of the corresponding quantities on the spheroidal geodesic.