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| {{Geodesy}}
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| In [[geodesy]], a '''reference ellipsoid''' is a mathematically-defined surface that approximates the [[geoid]], the truer [[figure of the Earth]], or other planetary body.
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| 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.
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| == Ellipsoid parameters ==
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| In 1687 [[Isaac Newton]] published the [[Philosophiæ Naturalis Principia Mathematica|Principia]] in which he included a proof<ref name=newton>Isaac Newton:''Principia'' Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [http://books.google.com/books?id=6EqxPav3vIsC&pg=PA239]</ref>{{Failed verification|date=February 2012}} that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate [[ellipsoid]] of revolution which he termed an [[Oblate spheroid|oblate]] [[spheroid]]. Current practice (2012)<ref name=torge>Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8</ref><ref name=flattening>{{cite book | author=Snyder, John P. |title=Flattening the Earth: Two Thousand Years of Map Projections | publisher =University of Chicago Press|year=1993|isbn=0-226-76747-7 | page=82}}</ref> uses the word 'ellipsoid' alone in preference to the full term 'oblate ellipsoid of revolution' or the older term 'oblate spheroid'. In the rare instances (some [[asteroid]]s and [[planet]]s) where a more general ellipsoid shape is required as a model the term used is [[Triaxial ellipsoid|triaxial]] (or scalene) ellipsoid. A great many ellipsoids have been used with various sizes and centres but modern (post [[GPS]]) ellipsoids are centred at the actual [[center of mass]] of the Earth or body being modeled.
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| The shape of an (oblate) ellipsoid (of revolution) is determined by the shape parameters of that [[ellipse]] which generates the ellipsoid when it is rotated about its minor axis. The [[semi-major axis]] of the ellipse, ''a'', is identified as the equatorial radius of the ellipsoid: the [[semi-minor axis]] of the ellipse, ''b'', is identified with the [[Geographic pole|polar]] distances (from the centre). These two lengths completely specify the shape of the ellipsoid but in practice geodesy publications classify reference ellipsoids by giving the semi-major axis and the ''inverse '' [[flattening]], ''1/f'', The flattening, ''f'', is simply a measure of how much the symmetry axis is compressed relative to the equatorial radius:
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| :<math>
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| \begin{align}
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| f&=\frac{a-b}{a}.
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| \end{align}
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| </math>
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| For the [[Earth]], <math>f\,\!</math> is around 1/300 corresponding to a difference of the major and minor semi-axes of approximately 21 km. Some precise values are given in the table below and also in [[Figure of the Earth]]. 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|Saturn's]] triaxial moons, [[Telesto (moon)|Telesto]], is nearly 1/3 to 1/2.
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| A great many other parameters are used in [[geodesy]] but they can all be related to one or two of the set ''a'', ''b'' and ''f''. They are listed in [[ellipse]].
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| == Coordinates ==
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| {{Unreferenced section|date=October 2011}}
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| {{main|Latitude|Longitude}}
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| 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).
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| For this purpose it is necessary to identify a ''zero [[meridian (geography)|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.
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| 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.
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| 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 [[Surface normal|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.
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| 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. See [[Geodetic system]] for more detail. <!--If these coordinates, i.e., latitude <math>\phi\,\!</math>, longitude <math>\lambda\,\!</math> and height ''h'', are given, one can compute the ''geocentric rectangular coordinates'' of the point as follows:
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| : <math> \begin{align}
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| X & = \left( N(\phi) + h\right)\cos{\phi}\cos{\lambda} \\
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| Y & = \left( N(\phi) + h\right)\cos{\phi}\sin{\lambda} \\
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| Z & = \left( N(\phi) (1-e^2) + h\right)\sin{\phi}
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| \end{align}
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| </math>
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| where
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| : <math>
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| N(\phi) = \frac{a}{\sqrt{1-e^2\sin^2 \phi }},
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| </math><br />
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| and <math>a</math> and <math>e^2</math> are the [[semi-major axis]] and the square of the first numerical [[eccentricity (mathematics)|eccentricity]] of the ellipsoid respectively. N is the '''''[[radius of curvature (applications)|radius of curvature]]''' in the [[prime vertical]]''.
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| In contrast, extracting <math>\phi\,\!</math>, <math>\lambda\,\!</math> and ''h'' from the rectangular coordinates usually requires [[Iterative method|iteration]]. A straightforward method is given in an [[OSGB]] publication<ref name=osgb>A guide to coordinate systems in Great Britain. This is available as a pdf document at
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| [http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents]] Appendices B1, B2</ref> and also in web notes.<ref name=osborne>Osborne, P (2008). [http://mercator.myzen.co.uk/mercator.pdf The Mercator Projections] Section 5.4</ref> More sophisticated methods are outlined in [[Geodetic system#From ECEF to geodetic|Geodetic system]].
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| -->
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| == Historical Earth ellipsoids ==
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| {{Main|Earth ellipsoid#Historical Earth ellipsoids}}
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| Currently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is the one defined by [[WGS 84]].
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| Traditional reference ellipsoids or ''[[geodetic datum]]s'' are defined regionally and therefore non-geocentric, e.g., [[ED50]]. Modern geodetic datums are established with
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| the aid of [[GPS]] and will therefore be geocentric, e.g., WGS 84.
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| == Ellipsoids for other planetary bodies ==
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| 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.
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| 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 [[Oval (geometry)|egg shaped]], where its north and south polar radii differ by approximately 6 [[kilometer|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.
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| For gaseous planets like [[Jupiter]], an effective surface for an ellipsoid is chosen as the equal-pressure boundary of one [[Bar (unit)|bar]]. Since they have no permanent observable features the choices of prime meridians are made according to mathematical rules.
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| Small moons, asteroids, and comet nuclei frequently have irregular shapes. For some of these, such as Jupiter's [[Io (moon)|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 [[convex set|non-convex]] bodies, such as [[433 Eros|Eros]], in that latitude and longitude don't always uniquely identify a single surface location.
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| == See also ==
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| * [[Earth ellipsoid]]
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| * [[Earth radius]]
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| * [[Meridian arc]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * 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.
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| **Web address: http://astrogeology.usgs.gov/Projects/WGCCRE
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| * ''OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture'', Annex B.4. 2005-11-30
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| **Web address: http://www.opengeospatial.org
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| == External links ==
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| *[http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs.html Coordinate System Index]
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| *[http://publib.boulder.ibm.com/infocenter/db2luw/v8/topic/com.ibm.db2.udb.doc/opt/csb3022a.htm Geographic coordinate system]
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| *[http://www.spenvis.oma.be/help/background/coortran/coortran.html Coordinate systems and transformations] ([[SPENVIS]] help page)
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| *[http://www.agnld.uni-potsdam.de/~shw/3_References/0_GPS/GPSHelmert1.html Coordinate Systems, Frames and Datums]
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| {{DEFAULTSORT:Reference Ellipsoid}}
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| [[Category:Geodesy]]
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| [[Category:Global Positioning System]]
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| [[Category:Navigation]]
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| [[Category:Geophysics]]
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| [[Category:Surveying]]
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