Geodesics on a spheroid

The shortest path between two points on a (smooth) surface is called a geodesic curve on the surface. On a flat surface the geodesics are straight lines, on a sphere they are great circles. Remarkably, the path taken by a particle sliding without friction on a surface will always be a geodesic. This is because a defining characteristic of a geodesic is that at each point on its path, the local center of curvature always lies in the direction of the surface normal, that is, in the direction of any constraint force required to keep the particle on the surface. There are thus no forces in the local tangent plane of the surface to deflect the particle from its geodesic path.

There is a general procedure, using the calculus of variations[2], 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 $m$ sliding on the surface of a spheroid. The constraint forces, normal to the surface, do no work, so the particle's kinetic energy $mv^2/2$, and thus its speed $v$, 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 $mv\sin\alpha\times OD$, where $vsin\alpha$ is the azimuthal component of the particle's velocity and $OD$ is the distance of the particle from the polar axis $OB$. Thus, using Eqn. (4) we obtain:

$$\sin\alpha\cos\theta=\sin\alpha\cos\phi/(1-e^2\sin^2\phi^2)^{1/2}=\mathrm{constant}$$ (16)

If we refer to the azimuth of the geodesic as it crosses the equator ($\theta=\phi=0$) as $\alpha_0$, we can evaluate the above constant, obtaining:

$$\sin\alpha\cos\theta=\sin\alpha\cos\phi/(1-e^2\sin^2\phi^2)^{1/2}=\sin\alpha_0$$ (17)

This equation takes its simplest form when reduced latitudes, $\theta$ are used, so geodesic calculations are generally done in $(\theta, L)$ coordinates, with necessary conversions back and forth to geodetic coordinates being performed using Eqn. (2). In fact, the relationship between the azimuth $\alpha$ and the reduced latitude $\theta$ 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[3] with a common value of $\alpha_0$. At each (reduced) latitude, $\theta$ the geodesics have the same azimuth, $\alpha$. However, distances and longitude differences differ by $O(e^2)$ 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.