- Reference ellipsoid
In

geodesy , a**reference ellipsoid**is a mathematically-defined surface that approximates thegeoid , the truerfigure of the Earth , or other planetary body.Because of their relative simplicity, reference ellipsoids are used as a preferred surface on whichgeodetic network computations are performed and point coordinates such aslatitude ,longitude , andelevation are defined.**Ellipsoid properties**Mathematically, a reference ellipsoid is usually an

oblate (flattened)spheroid with two different axes: Anequator ial radius (thesemi-major axis $a,!$), and a polar radius (thesemi-minor axis $b,!$). More rarely, a scaleneellipsoid with three axes (triaxial——$a\_x,,a\_y,,b,!$) is used, usually for modeling the smaller, irregularly shaped moons andasteroid s. 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 thecenter 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 function s of an ellipse's, $o!varepsilon,!$:angular eccentricity ::$o!varepsilon=arccosleft(frac\{b\}\{a\}\; ight)=2arctanleft(sqrt\{frac\{a-b\}\{a+b\; ight);,!$

Due to rotational forces, the equatorial radius is usually larger than the polar radius. This ellipticity or "

flattening ", $f,!$, determines how close to a truesphere an oblate spheroid is, and is defined as:$f=operatorname\{ver\}(o!varepsilon)=2sin^2left(frac\{o!varepsilon\}\{2\}\; ight)=1-cos(o!varepsilon)=frac\{a-b\}\{a\}.,!$

For

Earth , $f,!$ is around 1/300, and is very gradually decreasing over geologic time scales. For comparison, Earth'sMoon is even less elliptical, with a flattening of less than 1/825, whileJupiter 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

meter s) 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), andelevation (height).For this purpose it is necessary to identify a "zero meridian", which for Earth is usually thePrime Meridian . For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the craterAiry-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 theprime vertical ".In contrast, extracting $phi,!$, $lambda,!$ and "h" from the rectangular coordinates usually requires iteration:

Letting $phi\_c=arctan(sec(o!varepsilon)^2\; an(psi\_t));,!$, $phi\_p=phi\_c:;phi\_c=arctan!left(frac\{qquad;;a^2Z\_tquad,+frac\{1\}\{4\}\; [N(phi\_p)sin(phi\_p)]\; ^3sin(2o!varepsilon)^2\}\{!!!!!a^2sqrt\{X\_t^2+Y\_t^2\},-\; [N(phi\_p)cos(phi\_p)]\; ^3sin(o!varepsilon)^2\}\; ight);,!$ Repeat until $phi\_c=phi\_p,!$: $phi=phi\_c.,!$

Or, introducing the

**geocentric**, $psi,!$, and**parametric**, or**reduced**, $eta,!$, latitudes::$psi\_t=arctanleft(frac\{Z\_t\}\{sqrt\{X\_t^2+Y\_t^2\; ight);$ and $eta\_c=arctan(sec(o!varepsilon)\; an(psi\_t));,!$, $phi\_p=phi\_c:;phi\_c=arctan!left(frac\{qquad,Z\_tqquad+bsin(eta\_c)^3\; an(o!varepsilon)^2\}\{sqrt\{X\_t^2+Y\_t^2\};-acos(eta\_c)^3sin(o!varepsilon)^2\}\; ight);,!$ $eta\_p=eta\_c:;eta\_c=arctanleft(cos(o!varepsilon)\; an(phi\_c)\; ight);;,!$ Repeat until $phi\_c=phi\_p,!$ and $eta\_c=eta\_p,!$: $phi=phi\_c;quadeta=eta\_c;quadpsi=arctan(cos(o!varepsilon)\; an(eta)).,!$

: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(o!varepsilon)^2N,,!::$\{\}\_\{color\{white\}8.\}=cos(phi)\{color\{white\}dot$color{black}sqrt{X_t^2+Y_t^2,+,sin(phi)left [Z_t+sin(o!varepsilon)^2Nsin(phi) ight] -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 "

geodetic datum s" are defined regionally and therefore non-geocentric, e.g.,ED50 . Modern geodetic datums are established withthe aid ofGPS and will therefore be geocentric, e.g., WGS 84.The following table lists some of the most common ellipsoids:

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 andMars 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.

**See also***

Earth radius

*Figure of the Earth

*Geoid

*Meridian arc **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

**Web address: http://www.opengeospatial.org**External links*** [

*http://www.apsalin.com/earth-ellipsoid-reference.aspx Earth Ellipsoids Reference*]

* [*http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs.html Coordinate System Index*]

* [*http://publib.boulder.ibm.com/infocenter/db2luw/v8/topic/com.ibm.db2.udb.doc/opt/csb3022a.htm Geographic coordinate system*]

* [*http://www.spenvis.oma.be/spenvis/help/background/coortran/coortran.html Coordinate systems and transformations*]

* [*http://www.agnld.uni-potsdam.de/~shw/3_References/0_GPS/GPSHelmert1.html Coordinate Systems, Frames and Datums*]

* [*http://www.mathworks.com/access/helpdesk/help/toolbox/aeroblks/ecefpositiontolla.html Aerospace Blockset: ECEF Position to LLA*]

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