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Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude (north/south) $ϕ$ , longitude (east/west) $λ$ , and ellipsoidal height $h$ (also known as geodetic height^[1]). The triad is also known as Earth ellipsoidal coordinates^[2] (not to be confused with ellipsoidal-harmonic coordinates).

Definitions

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, 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 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 is the distance between the point of interest and the ellipsoid surface, evaluated along the ellipsoidal normal direction; it is defined as a signed distance, such that points inside the ellipsoid have negative height.

Geodetic and geocentric latitudes

Geodetic latitude and geocentric latitude have slightly different definitions. Geodetic latitude is defined as the angle between the equatorial 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 coordinates is geodetic latitude. The standard notation for geodetic latitude is $φ$ . There is no standard notation for geocentric latitude; examples include $θ$ , $ψ$ , $φ'$ .

Similarly, geodetic altitude is defined as the height above the ellipsoid surface, normal to the ellipsoid; whereas geocentric altitude is defined as the height above the ellipsoid surface along a line to the center of the ellipsoid (the radius). When used without qualification, the term altitude refers to geodetic altitude; as is used in aviation. Geocentric altitude is typically used in orbital mechanics.

Conversion

Given geodetic coordinates, one can compute the geocentric Cartesian coordinates of the point as follows:^[3]

{\begin{aligned}X&={\big (}N(\phi )+h{\big )}\cos {\phi }\cos {\lambda }\\Y&={\big (}N(\phi )+h{\big )}\cos {\phi }\sin {\lambda }\\Z&=\left({\frac {b^{2}}{a^{2}}}N(\phi )+h\right)\sin {\phi }\end{aligned}}

where $a$ and $b$ are the equatorial radius (semi-major axis) and the polar radius (semi-minor axis), respectively. $N$ is the prime vertical radius of curvature:

N(\phi )={\frac {a^{2}}{\sqrt {a^{2}\cos ^{2}\phi +b^{2}\sin ^{2}\phi }}},

In contrast, extracting $ϕ$ , $λ$ and $h$ from the rectangular coordinates usually requires iteration as $ϕ$ and $h$ are mutually involved through $N$ :^[4]^[5]

\lambda =\operatorname {atan2} (Y,X)

.

h={\frac {p}{\cos \phi }}-N,

\phi =\arctan \left((Z/p)/(1-e^{2}N/(N+h))\right).

More sophisticated methods are available.

References

^ National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986). Geodetic Glossary. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107. Retrieved 2021-10-24.
^ Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010). Algebraic Geodesy and Geoinformatics. Springer Berlin Heidelberg. p. 156. ISBN 978-3-642-12124-1. Retrieved 2021-10-24.
^ B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins (1994). GPS - theory and practice. Section 10.2.1. p. 282. ISBN 3-211-82839-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ A guide to coordinate systems in Great Britain. This is available as a pdf document at ["Archived copy". Archived from the original on 2012-02-11. Retrieved 2012-01-11.{{cite web}}: CS1 maint: archived copy as title (link)]] Appendices B1, B2
^ Osborne, P (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Section 5.4

[National_Geodetic_Survey_(U.S.)._National_Geodetic_Survey_(U.S.)_1986_p._107-1] National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986). Geodetic Glossary. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107. Retrieved 2021-10-24.

[Awange_Grafarend_Paláncz_Zaletnyik_2010_p._156-2] Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010). Algebraic Geodesy and Geoinformatics. Springer Berlin Heidelberg. p. 156. ISBN 978-3-642-12124-1. Retrieved 2021-10-24.

[gps-chap10-3] B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins (1994). GPS - theory and practice. Section 10.2.1. p. 282. ISBN 3-211-82839-7.{{cite book}}: CS1 maint: multiple names: authors list (link)

[osgb-4] A guide to coordinate systems in Great Britain. This is available as a pdf document at ["Archived copy". Archived from the original on 2012-02-11. Retrieved 2012-01-11.{{cite web}}: CS1 maint: archived copy as title (link)]] Appendices B1, B2

[osborne-5] Osborne, P (2008). The Mercator Projections Archived 2012-01-18 at the Wayback Machine Section 5.4

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[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
+{{broader|Geographic coordinate system}}
-#redirect [[Reference ellipsoid#Coordinates]]
+[[File:Geodetic coordinates.svg|thumb|right|upright=0.9|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.br/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 | last=Awange | first=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.br/books?id=XrCBEVCwAewC&pg=PA156 | access-date=2021-10-24 | page=156}}</ref> (not to be confused with ''[[ellipsoidal-harmonic coordinates]]'').
+==Definitions==
+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 is the distance between the point of interest and the ellipsoid surface, evaluated along the ellipsoidal normal direction; it is defined as a [[signed distance]], such that points inside the ellipsoid have negative height.
+=== Geodetic and geocentric latitudes ===
+{{see also|Latitude#Geodetic and geocentric latitudes}}
+Geodetic latitude and ''[[geocentric latitude]]'' have slightly 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 height above the ellipsoid surface along a line to the center of the ellipsoid (the radius). When used without qualification, the term altitude refers to geodetic altitude; as is used in aviation. Geocentric altitude is typically used in [[orbital mechanics]].
+==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|author=B. Hofmann-Wellenhof, H. Lichtenegger, J. Collins|isbn=3-211-82839-7|page=282|others=Section 10.2.1|year=1994}}</ref>
+:<math>\begin{align}
+X &= \big( N(\phi) + h\big)\cos{\phi}\cos{\lambda} \\
+Y &= \big( N(\phi) + h\big)\cos{\phi}\sin{\lambda} \\
+Z &= \left( \frac{b^2}{a^2} N(\phi) + 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]]'':
+:<math>N(\phi) = \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>A guide to coordinate systems in Great Britain. This is available as a pdf document at
+[{{cite web |url=http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents |title=Archived copy |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 }}]] Appendices B1, B2</ref><ref name=osborne>Osborne, P (2008). [http://mercator.myzen.co.uk/mercator.pdf The Mercator Projections] {{webarchive|url=https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf |date=2012-01-18 }} 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>
+More sophisticated methods are [[Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates|available]].
+==See also==
+*[[Geodesics on an ellipsoid]]
+==References==
+{{reflist}}
+[[Category:Geodesy]]
+[[Category:Orthogonal coordinate systems]]