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

Contents

1 Ellipsoid properties

2 Co-ordinates

3 Common reference ellipsoids for the Earth

4 Ellipsoids for non-Earth bodies

5 See also

6 References

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) is used, usually for modeling some non-Earth bodies.

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 the ellispoid is, and is defined as

$f = \frac{a-b}{a}$

and the square of the ellipticity (or first eccentricity) $e$ by

$e^2=\frac{a^2-b^2}{a^2} = f(2-f).$

For the Earth, $f$ is around 1/300, and is very gradually decreasing over geologic time scales. For comparison, the Moon is nearly spherical with a flattening of 0, while Jupiter is visibly oblate at about 1/15.

Co-ordinates

The co-ordinates 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 co-ordinates, i.e., latitude $\varphi$ , longitude and height h, are given, one can compute the geocentric rectangular co-ordinates of the point as follows:

$\begin{matrix} x &=& (N + h)\cos \varphi \cos \lambda \\ y &=& (N + h) \cos \varphi \sin \lambda \\ z &=& (N (1-e^2) + h) \sin \varphi \end{matrix}$

where

$N = N(\varphi) = \frac{a}{\sqrt{1-e^2\sin^2\varphi}}$

is the so called radius of curvature in the prime vertical.

The meridional radius of curvature of the ellipsoid is given by the formula

$M(\varphi) = \frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^{3/2}}.$

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 geodetic datums 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) Inverse flattening ( $1 / f$ )

Clarke 1886 6 378 206.4 294.978 698 2

GRS 1980 6 378 137 298.257 222 101

International 1924 6 378 388 297.0

WGS 1984 6 378 137 298.257 223 563

Sphere (6371 km) 6 371 000 0

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 it's 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 it's ellipsoid. The Earth's Moon is effectively spherical, having no bulge at it's 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 permament 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 ellipsoid. 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.

See also

Earth radius

Figure of the Earth

Geoid

References

P. K. Seidelmann (Chair), et. al. (2005), Report Of The IAU/IAG Working Group On Cartographic Coordinates And Rotational Elements: 2003, Celestial Mechanics and Dynamical Astronomy, 91, pp. 203-215. Web address http://astrogeology.usgs.gov/Projects/WGCCRE/

OpenGIS® Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture, Annex B.4. 2005-11-30 [1]

Category: Geodesy

This article based on this article: Reference ellipsoid from the free encyclopedia Wikipedia and work with the GNU Free Documentation License. In Wikipedia is this list of the authors.