Reference ellipsoid
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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 preferred surface on which geodetic network computations are performed and point co-ordinates such as latitude, longitude, and elevation are defined.
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Ellipsoid properties
Mathematically, a reference ellipsoid is usually an oblate (flattened) spheroid with two different axes: an equatorial radius (the semi-major axis <math>a</math>), and a polar radius (the semi-minor axis <math>b</math>). More rarely, a scalene ellipsoid with three axes (triaxial) is used, usually for modeling some non-Earth bodies. 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.
Due to rotational forces, the equatorial radius is usually larger than the polar radius. This ellipticity or flattening, <math>f</math>, determines how close to a true sphere an oblate spheroid is, and is defined as
- <math>
f = \frac{a-b}{a} </math>
which is related to the eccentricty, <math>e</math>, of the cross-sectional ellipse by
- <math>
e^2=\frac{a^2-b^2}{a^2} = f(2-f). </math>
For the Earth, <math>f</math> 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.
It is traditional when defining a reference ellipsoid to specify the semi-major equatorial radius <math>a</math> (usually in meters) and the inverse of the flattening ratio <math>1/f</math>. The semi-minor polar radius is then easily derived.
Co-ordinates
A primary use of reference ellipsoids is to serve as a basis for a co-ordinate 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 co-ordinate 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 geographic 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 than the geocentric 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 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 <math>\varphi</math>, longitude <math>\lambda</math> and height h, are given, one can compute the geocentric rectangular co-ordinates of the point as follows:
- <math>
\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} </math>
where
- <math>
N = N(\varphi) = \frac{a}{\sqrt{1-e^2\sin^2\varphi}} </math>
is the so called radius of curvature in the prime vertical.
The meridional radius of curvature of the ellipsoid is given by the formula
- <math>
M(\varphi) = \frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^{3/2}}. </math>
These formulae have a closed-form inverse, though the algebra is rather involved. It can be shown that
- <math>
\begin{matrix} \lambda &=& \arctan y/x \\ \varphi &=& \arctan \frac{z+e'^2 b \sin^3 \theta}{p-e^2 a \cos^3 \theta} \\ h &=& p/\cos \varphi - N \end{matrix} </math>
where <math>p</math>, <math>e'</math> and <math>\theta</math> are defined by
- <math>
\begin{matrix} p &=& \sqrt{x^2+y^2} \\ e' &=& \frac{a^2-b^2}{b^2} \\ \theta &=& \arctan \frac{z a}{p b} \end{matrix} </math>
Due to the complexity of these expressions, inversion is often achieved through an iterative process known as Bowring's method. [1] [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 (<math>1/f</math>) |
|---|---|---|
| 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 its 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
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 [3]de:Referenzellipsoid