Spheroid Geometry

For many purposes, it is entirely adequate to model the earth as a sphere. However, in reality, the earth's mean sea-level surface is better approximated by a different geometric shape, an oblate spheroid- the surface created by rotating an ellipse about its polar axis. Compared to a sphere, an oblate spheroid is flattened at the poles. The earth's flattening is quite small, about 1 part in 300, and navigation errors induced by assuming the earth is spherical, for the most part, do not exceed this, and so for many purposes a spherical approximation may be entirely adequate. On a sphere, the commonly used coordinates are latitude and longitude, likewise on a spheroid, however on a spheroid one has to be more careful about what exactly one means by latitude.

Figure 1: The meridional ellipse

In Figure 1 we depict a cross-section of the spheroid through the poles. The point O is the center of the earth. B is the North Pole. The major (equatorial) axis, OA, of the meridional ellipse has length $a$, the minor (polar) axis, OB has length $b$. A point $P$ on the ellipse has coordinates $(a\cos\theta,b\sin\theta)$ where the angle $\angle AOP^\prime$ is called the reduced or parametric latitude. The point $P^\prime$ is the point on the circumscribing circle (of radius $a$) the same distance from the polar axis as P. The angle $\angle AOP$ is called the geocentric latitude.

However, the latitude used in navigation and geodesy is the geodetic or astronomical latitude, which is defined to be the angle between the northerly horizon at P and the polar axis. It is equal to the angle $\phi$ in Figure 1.

Longitude, $L$, is defined in exactly the same way on the spheroid as on the sphere, namely as the angle between the meridian and the prime meridian. We use here the standard convention of North longitudes and East latitudes as positive.

In three dimensional Cartesian coordinates, points on the spheroid have coordinates $(a\cos\theta\cos{}L,a\cos\theta\sin{}L,b\sin\theta)$.

There are alternative ways of specifying the dimensions of the spheroid other than by its major and minor radii $a$ and $b$. The flattening, $f$, is defined by $f=1-b/a$, and the eccentricity $e$ by $e^2=1-b^2/a^2$. For the WGS84 spheroid, $a=6378.137$ km and $f=1/298.257223563$. The eccentricity and flattening are thus related by:

$$e^2=f(2-f)$$ (1)