# Geodetic coordinates

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Geographic coordinate system

For broader coverage of this topic, see [Geographic coordinate system](/source/Geographic_coordinate_system).

Geodetic coordinates P(*ɸ*,*λ*,*h*)

**Geodetic coordinates** are a type of [curvilinear](/source/Curvilinear_coordinate_system) [orthogonal coordinate system](/source/Orthogonal_coordinate_system) used in [geodesy](/source/Geodesy) based on a *[reference ellipsoid](/source/Reference_ellipsoid)*. They include **geodetic latitude** (north/south) ϕ, *[longitude](/source/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](/source/Ellipsoidal-harmonic_coordinates)*).

## Definitions

Further information: [Longitude](/source/Longitude), [Latitude](/source/Latitude), and [Vertical position](/source/Vertical_position)

Longitude measures the rotational [angle](/source/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](/source/Meridian_(geography))*, which for Earth is usually the [Prime Meridian](/source/Prime_Meridian). For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the crater [Airy-0](/source/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](/source/Surface_normal) to the reference ellipsoid. Depending on the flattening, it may be slightly different from the *[geocentric latitude](/source/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](/source/Planetographic_latitude)* and *[planetocentric latitude](/source/Planetocentric_latitude)* are used instead.

Ellipsoidal height (or ellipsoidal [altitude](/source/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](/source/Ellipsoidal_normal_vector); it is defined as a [signed distance](/source/Signed_distance) such that points inside the ellipsoid have negative height.

## Geodetic vs. geocentric coordinates

See also: [Latitude § Geodetic and geocentric latitudes](/source/Latitude#Geodetic_and_geocentric_latitudes)

Geodetic latitude and *[geocentric latitude](/source/Geocentric_latitude)* have different definitions. Geodetic latitude is defined as the angle between the [equatorial](/source/Equator) plane and the [surface normal](/source/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](/source/Geographic_coordinate_system) 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](/source/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](/source/Altitude) refers to geodetic altitude (possibly with further refinements, such as in [orthometric heights](/source/Orthometric_height)). Geocentric altitude is typically used in [orbital mechanics](/source/Orbital_mechanics) (see [orbital altitude](/source/Orbital_altitude)).

If the impact of Earth's [equatorial bulge](/source/Equatorial_bulge) is not significant for a given application (e.g., [interplanetary spaceflight](/source/Interplanetary_spaceflight)), the [Earth ellipsoid](/source/Earth_ellipsoid) may be simplified as a [spherical Earth](/source/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](/source/Earth's_radius) (see also: [spherical coordinate system](/source/Spherical_coordinate_system)).

## Conversion

Main article: [Geographic coordinate conversion](/source/Geographic_coordinate_conversion)

Given geodetic coordinates, one can compute the *[geocentric Cartesian coordinates](/source/Geocentric_Cartesian_coordinates)* of the point as follows:[3]

- X = ( N + h ) cos ⁡ ϕ cos ⁡ λ Y = ( N + h ) cos ⁡ ϕ sin ⁡ λ Z = ( b 2 a 2 N + h ) sin ⁡ ϕ {\displaystyle {\begin{aligned}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{aligned}}}

where a and b are the equatorial radius ([semi-major axis](/source/Semi-major_axis)) and the polar radius ([semi-minor axis](/source/Semi-minor_axis)), respectively. N is the *[prime vertical radius of curvature](/source/Prime_vertical_radius_of_curvature)*, function of latitude ϕ:

- N = a 2 a 2 cos 2 ⁡ ϕ + b 2 sin 2 ⁡ ϕ , {\displaystyle N={\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](/source/Iterative_method) as ϕ and h are mutually involved through N:[4][5]

- λ = atan2 ⁡ ( Y , X ) {\displaystyle \lambda =\operatorname {atan2} (Y,X)} .

- h = p cos ⁡ ϕ − N , {\displaystyle h={\frac {p}{\cos \phi }}-N,}

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

where p = X 2 + Y 2 {\displaystyle p={\sqrt {X^{2}+Y^{2}}}} . More sophisticated methods are [available](/source/Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates).

## See also

- [Local geodetic coordinates](/source/Local_geodetic_coordinates)

- [Geodetic datum](/source/Geodetic_datum)

- [Geodesics on an ellipsoid](/source/Geodesics_on_an_ellipsoid)

- [Planetary coordinate system](/source/Planetary_coordinate_system)

## References

1. **[^](#cite_ref-National_Geodetic_Survey_(U.S.)._National_Geodetic_Survey_(U.S.)_1986_p._107_1-0)** National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986). [*Geodetic Glossary*](https://books.google.com/books?id=sBlyBIfdHL8C&pg=PA107). 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.

1. **[^](#cite_ref-Awange_Grafarend_Paláncz_Zaletnyik_2010_p._156_2-0)** Awange, J.L.; Grafarend, E.W.; Paláncz, B.; Zaletnyik, P. (2010). [*Algebraic Geodesy and Geoinformatics*](https://books.google.com/books?id=XrCBEVCwAewC&pg=PA156). Springer Berlin Heidelberg. p. 156. [ISBN](/source/ISBN_(identifier)) [978-3-642-12124-1](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-12124-1). Retrieved 2021-10-24.

1. **[^](#cite_ref-gps-chap10_3-0)** Hofmann-Wellenhof, B.; Lichtenegger, H.; Collins, J. (1994). *GPS – theory and practice*. Section 10.2.1. Springer. p. 282. [ISBN](/source/ISBN_(identifier)) [3-211-82839-7](https://en.wikipedia.org/wiki/Special:BookSources/3-211-82839-7).

1. **[^](#cite_ref-osgb_4-0)** ["A guide to coordinate systems in Great Britain"](https://web.archive.org/web/20120211075826/http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/). *Ordnance Survey*. Appendices B1, B2. Archived from [the original](http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents) on 2012-02-11. Retrieved 2012-01-11.

1. **[^](#cite_ref-osborne_5-0)** Osborne, P (2008). ["The Mercator Projections"](https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf) (PDF). Section 5.4. Archived from [the original](http://mercator.myzen.co.uk/mercator.pdf) (PDF) on 2012-01-18.

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