# Reference ellipsoid

In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body. Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude, longitude, and elevation are defined.

## Ellipsoid properties

Mathematically, a reference ellipsoid is usually an oblate (flattened) spheroid with two different axes: An equatorial radius (the semi-major axis $a\,\!$), and a polar radius (the semi-minor axis $b\,\!$). More rarely, a scalene ellipsoid with three axes (triaxial——$a_x,\,a_y,\,b\,\!$) is used, usually for modeling the smaller, irregularly shaped moons and asteroids. The polar axis here is the same as the rotational axis, and is not the magnetic or orbital pole. The geometric center of the ellipsoid is placed at the center of mass of the body being modeled, and not the barycenter in a multi-body system.

In working with elliptic geometry, several parameters are commonly utilized, all of which are trigonometric functions of an ellipse's eccentricity, $o\!\varepsilon\,\!$:

$o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!$

Due to rotational forces, the equatorial radius is usually larger than the polar radius. This ellipticity or flattening, $f\,\!$, determines how close to a true sphere an oblate spheroid is, and is defined as

$f=\operatorname{ver}(o\!\varepsilon)=2\sin^2\left(\frac{o\!\varepsilon}{2}\right)=1-\cos(o\!\varepsilon)=\frac{a-b}{a}.\,\!$

For Earth, $f\,\!$ is around 1/300, and is very gradually decreasing over geologic time scales. For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825, while Jupiter is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto, is nearly 1/3 to 1/2!

Such flattening is related to the eccentricity, $e\,\!$, of the cross-sectional ellipse by

$e^2=f(2-f)=\sin^2(o\!\varepsilon)=\frac{a^2-b^2}{a^2}.\,\!$

It is traditional when defining a reference ellipsoid to specify the semi-major equatorial radius $a\,\!$ (usually in meters) and the inverse of the flattening ratio $1/f\,\!$. The semi-minor polar radius is then easily derived.

## Coordinates

A primary use of reference ellipsoids is to serve as a basis for a coordinate system of latitude (north/south), longitude (east/west), and elevation (height). For this purpose it is necessary to identify a zero meridian, which for Earth is usually the Prime Meridian. For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater Airy-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid.

The longitude measures the rotational angle between the zero meridian and the measured point. By convention for the Earth, Moon, and Sun it is expressed as degrees ranging from −180° to +180° For other bodies a range of 0° to 360° is used.

The latitude measures how close to the poles or equator a point is along a meridian, and is represented as angle from −90° to +90°, where 0° is the equator. The common or geodetic latitude is the angle between the equatorial plane and a line that is normal to the reference ellipsoid. Depending on the flattening, it may be slightly different from the geocentric (geographic) latitude, which is the angle between the equatorial plane and a line from the center of the ellipsoid. For non-Earth bodies the terms planetographic and planetocentric are used instead.

The coordinates of a geodetic point are customarily stated as geodetic latitude and longitude, i.e., the direction in space of the geodetic normal containing the point, and the height h of the point over the reference ellipsoid. If these coordinates, i.e., latitude $\phi\,\!$, longitude $\lambda\,\!$ and height h, are given, one can compute the geocentric rectangular coordinates of the point as follows:

$X_t=[N+h]\cos(\phi)\cos(\lambda);\,\!$
$Y_t=[N+h]\cos(\phi)\sin(\lambda);\,\!$
$Z_t=[\cos(o\!\varepsilon)^2N+h]\sin(\phi);\,\!$

where

$N=N(\phi)=\frac{a}{\sqrt{1-(\sin(\phi)\sin(o\!\varepsilon))^2}}\,\!$

is the radius of curvature in the prime vertical.

In contrast, extracting $\phi\,\!$, $\lambda\,\!$ and h from the rectangular coordinates usually requires iteration:

Letting $\phi_c=\arctan(\sec(o\!\varepsilon)^2\tan(\psi_t))\;\,\!$, $\phi_p=\phi_c:\;\phi_c=\arctan\!\left(\frac{\qquad\;\;a^2Z_t\quad\,+\frac{1}{4}[N(\phi_p)\sin(\phi_p)]^3\sin(2o\!\varepsilon)^2}{\!\!\!\!\!a^2\sqrt{X_t^2+Y_t^2}\,-[N(\phi_p)\cos(\phi_p)]^3\sin(o\!\varepsilon)^2}\right);\,\!$

Repeat until $\phi_c=\phi_p\,\!$: $\phi=\phi_c.\,\!$

Or, introducing the geocentric, $\psi\,\!$, and parametric, or reduced, $\beta\,\!$, latitudes:

$\psi_t=\arctan\left(\frac{Z_t}{\sqrt{X_t^2+Y_t^2}}\right)\;$ and $\beta_c=\arctan(\sec(o\!\varepsilon)\tan(\psi_t))\;\,\!$,

$\phi_p=\phi_c:\;\phi_c=\arctan\!\left(\frac{\qquad\,Z_t\qquad+b\sin(\beta_c)^3\tan(o\!\varepsilon)^2}{\sqrt{X_t^2+Y_t^2}\;-a\cos(\beta_c)^3\sin(o\!\varepsilon)^2}\right);\,\!$

$\beta_p=\beta_c:\;\beta_c=\arctan\left(\cos(o\!\varepsilon)\tan(\phi_c)\right);\;\,\!$

Repeat until $\phi_c=\phi_p\,\!$ and $\beta_c=\beta_p\,\!$:

$\phi=\phi_c;\quad\beta=\beta_c;\quad\psi=\arctan(\cos(o\!\varepsilon)\tan(\beta)).\,\!$

Once $\phi\,\!$ is determined, then h can be isolated:

$h=\sec(\phi){\color{white}\dot{{\color{black}\sqrt{X_t^2+Y_t^2}}}}-N\;=\;\csc(\phi)Z_t-\cos^2(o\!\varepsilon)N,\,\!$
${}_{\color{white}8.}=\cos(\phi){\color{white}\dot{{\color{black}\sqrt{X_t^2+Y_t^2}}}}\,+\,\sin(\phi)\left[Z_t+\sin^2(o\!\varepsilon)N\sin(\phi)\right]-N.\,\!$

## Common reference ellipsoids for the Earth

Currently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is WGS 84.

Traditional reference ellipsoids or ms are defined regionally and therefore non-geocentric, e.g., ED50. Modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e.g., WGS 84.

The following table lists some of the most common ellipsoids:

Name Equatorial axis (m) Polar axis (m) Inverse flattening,
$1/f\,\!$
Clarke 1866 6 378 206.4 6 356 583.8 294.978 698 2
Bessel 1841 6 377 397.155 6 356 078.965 299.152 843 4
International 1924 6 378 388 6 356 911.9 296.999 362 1
Krasovsky 1940 6 378 245 6 356 863 298.299 738 1
GRS 1980 6 378 137 6 356 752.3141 298.257 222 101
WGS 1984 6 378 137 6 356 752.3142 298.257 223 563
Sphere (6371 km) 6 371 000 6 371 000 $\infty$

See Figure of the Earth for a more complete historical list.

## Ellipsoids for non-Earth bodies

Reference ellipsoids are also useful for geodetic mapping of other planetary bodies including planets, their satellites, asteroids and comet nuclei. Some well observed bodies such as the Moon and Mars now have quite precise reference ellipsoids.

For rigid-surface nearly-spherical bodies, which includes all the rocky planets and many moons, ellipsoids are defined in terms of the axis of rotation and the mean surface height excluding any atmosphere. Mars is actually egg shaped, where its north and south polar radii differ by approximately 6 km, however this difference is small enough that the average polar radius is used to define its ellipsoid. The Earth's Moon is effectively spherical, having no bulge at its equator. Where possible a fixed observable surface feature is used when defining a reference meridian.

For gaseous planets like Jupiter, an effective surface for an ellipsoid is chosen as the equal-pressure boundary of one bar. Since they have no permanent observable features the choices of prime meridians are made according to mathematical rules.

Small moons, asteroids, and comet nuclei frequently have irregular shapes. For some of these, such as Jupiter's Io, a scalene (triaxial) ellipsoid is a better fit than the oblate spheroid. For highly irregular bodies the concept of a reference ellipsoid may have no useful value, so sometimes a spherical reference is used instead and points identified by planetocentric latitude and longitude. Even that can be problematic for non-convex bodies, such as Eros, in that latitude and longitude don't always uniquely identify a single surface location.
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