{{Short description|Geographic coordinate system}} {{broader|Geographic coordinate system}} [[File:Geodetic coordinates.svg|thumb|right|upright=0.9|class=skin-invert-image|Geodetic coordinates {{math|P(''ɸ'',''λ'',''h'')}}]]
'''Geodetic coordinates''' are a type of [[curvilinear coordinate system|curvilinear]] [[orthogonal coordinate system]] used in [[geodesy]] based on a ''[[reference ellipsoid]]''. They include '''geodetic latitude''' (north/south) {{mvar|ϕ}}, ''[[longitude]]'' (east/west) {{mvar|λ}}, and '''ellipsoidal height''' {{mvar|h}} (also known as '''geodetic height'''<ref name="National Geodetic Survey (U.S.). National Geodetic Survey (U.S.) 1986 p. 107">{{cite book | author=National Geodetic Survey (U.S.). | author2=National Geodetic Survey (U.S.) | title=Geodetic Glossary | publisher=U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services | series=NOAA technical publications | year=1986 | url=https://books.google.com/books?id=sBlyBIfdHL8C&pg=PA107 | access-date=2021-10-24 | page=107}}</ref>). The triad is also known as '''Earth ellipsoidal coordinates'''<ref name="Awange Grafarend Paláncz Zaletnyik 2010 p. 156">{{cite book | last1=Awange | first1=J.L. | last2=Grafarend | first2=E.W. | last3=Paláncz | first3=B. | last4=Zaletnyik | first4=P. | title=Algebraic Geodesy and Geoinformatics | publisher=Springer Berlin Heidelberg | year=2010 | isbn=978-3-642-12124-1 | url=https://books.google.com/books?id=XrCBEVCwAewC&pg=PA156 | access-date=2021-10-24 | page=156}}</ref> (not to be confused with ''[[ellipsoidal-harmonic coordinates]]'').
==Definitions== {{further|Longitude|Latitude|Vertical position}}
Longitude measures the rotational [[angle]] between the zero meridian and the measured point. By convention for the Earth, Moon and Sun, it is expressed in degrees ranging from −180° to +180°. For other bodies a range of 0° to 360° is used. 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.
Geodetic latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, where 0° is the equator. The ''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 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 latitude]]'' and ''[[planetocentric latitude]]'' are used instead.
Ellipsoidal height (or ellipsoidal [[altitude]]), also known as geodetic height (or geodetic altitude), is the distance between the point of interest and the ellipsoid surface, evaluated along the [[ellipsoidal normal vector]]; it is defined as a [[signed distance]] such that points inside the ellipsoid have negative height.
== Geodetic vs. geocentric coordinates == {{see also|Latitude#Geodetic and geocentric latitudes}}
Geodetic latitude and ''[[geocentric latitude]]'' have different definitions. Geodetic latitude is defined as the angle between the [[equator]]ial plane and the [[surface normal]] at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure). When used without qualification, the term latitude refers to geodetic latitude. For example, the latitude used in [[Geographic_coordinate_system|geographic coordinates]] is geodetic latitude. The standard notation for geodetic latitude is {{mvar|φ}}. There is no standard notation for geocentric latitude; examples include {{mvar|θ}}, {{mvar|ψ}}, {{mvar|φ′}}.
Similarly, geodetic altitude is defined as the height above the ellipsoid surface, normal to the ellipsoid; whereas ''[[geocentric altitude]]'' is defined as the distance to the reference ellipsoid along a radial line to the geocenter. When used without qualification, as in aviation, the term [[altitude]] refers to geodetic altitude (possibly with further refinements, such as in [[orthometric height]]s). Geocentric altitude is typically used in [[orbital mechanics]] (see [[orbital altitude]]).
If the impact of Earth's [[equatorial bulge]] is not significant for a given application (e.g., [[interplanetary spaceflight]]), the [[Earth ellipsoid]] may be simplified as a [[spherical Earth]], in which case the geocentric and geodetic latitudes are equal and the latitude-dependent geocentric radius simplifies to a global mean [[Earth's radius]] (see also: [[spherical coordinate system]]).
==Conversion== {{main|Geographic coordinate conversion}}
Given geodetic coordinates, one can compute the ''[[geocentric Cartesian coordinates]]'' of the point as follows:<ref name="gps-chap10">{{cite book|title=GPS – theory and practice|first1=B. |last1=Hofmann-Wellenhof |first2=H. |last2=Lichtenegger |first3=J. |last3=Collins|isbn=3-211-82839-7|page=282|others=Section 10.2.1|year=1994|publisher=Springer }}</ref>
:<math>\begin{align} X &= \big( N + h\big)\cos{\phi}\cos{\lambda} \\ Y &= \big( N + h\big)\cos{\phi}\sin{\lambda} \\ Z &= \left( \frac{b^2}{a^2} N + h\right)\sin{\phi} \end{align}</math>
where {{mvar|a}} and {{mvar|b}} are the equatorial radius ([[semi-major axis]]) and the polar radius ([[semi-minor axis]]), respectively. {{mvar|N}} is the ''[[prime vertical radius of curvature]]'', function of latitude {{mvar|ϕ}}: :<math>N = \frac{a^2}{\sqrt{a^2\cos^2\phi + b^2\sin^2 \phi }},</math>
In contrast, extracting {{mvar|ϕ}}, {{mvar|λ}} and {{mvar|h}} from the rectangular coordinates usually requires [[Iterative method|iteration]] as {{mvar|ϕ}} and {{mvar|h}} are mutually involved through {{mvar|N}}:<ref name=osgb>{{cite web |url=http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents |title=A guide to coordinate systems in Great Britain |access-date=2012-01-11 |url-status=dead |archive-url=https://web.archive.org/web/20120211075826/http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/ |archive-date=2012-02-11 |at=Appendices B1, B2 |website=Ordnance Survey }}</ref><ref name=osborne>{{cite web |last1=Osborne |first1=P |date=2008 |url=http://mercator.myzen.co.uk/mercator.pdf |title=The Mercator Projections |archive-url=https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf |archive-date=2012-01-18 |at=Section 5.4}}</ref> :<math>\lambda = \operatorname{atan2}(Y,X)</math>. :<math>h=\frac{p}{\cos\phi} - N,</math> :<math>\phi = \arctan\left( (Z / p)/(1 - e^2 N / (N + h)) \right).</math> where <math>p = \sqrt{X^2 + Y^2}</math>. More sophisticated methods are [[Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates|available]].
==See also== *[[Local geodetic coordinates]] *[[Geodetic datum]] *[[Geodesics on an ellipsoid]] *[[Planetary coordinate system]]
==References== {{reflist}}
[[Category:Geodesy]] [[Category:Orthogonal coordinate systems]] [[Category:Geographic coordinate systems]] [[Category:Ellipsoids]]