Geodetic coordinates

Geodetic coordinates are a type of

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^{ellipsoidal-harmonic coordinates or ellipsoidal coordinates
).

Definitions
Further information: Longitude, Latitude, and 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
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
planetocentric latitude are used instead.
Ellipsoidal height (or ellipsoidal
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
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
orbital altitude
).
If the impact of Earth's Earth's radius (see also: spherical coordinate system
).

Conversion
Main article: Geographic coordinate conversion
Given geodetic coordinates, one can compute the
geocentric Cartesian coordinates of the point as follows:^[3]}

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

where $a$ and $b$ are the equatorial radius (
prime vertical radius of curvature
, function of latitude $ϕ$ :

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

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

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

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

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

where $p={\sqrt {X^{2}+Y^{2}}}$ . More sophisticated methods are available.

See also

Local geodetic coordinates

Geodetic datum

Geodesics on an ellipsoid

Planetary coordinate system

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.

ISBN 978-3-642-12124-1
. Retrieved 2021-10-24.

ISBN 3-211-82839-7
.

^ "A guide to coordinate systems in Great Britain". Ordnance Survey. Appendices B1, B2. Archived from the original on 2012-02-11. Retrieved 2012-01-11.

^ Osborne, P (2008). "The Mercator Projections" (PDF). Section 5.4. Archived from the original (PDF) on 2012-01-18.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Geodetic_coordinates&oldid=1215495247"

[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] ISBN 978-3-642-12124-1
. Retrieved 2021-10-24.

[gps-chap10-3] ISBN 3-211-82839-7
.

[osgb-4] "A guide to coordinate systems in Great Britain". Ordnance Survey. Appendices B1, B2. Archived from the original on 2012-02-11. Retrieved 2012-01-11.

[osborne-5] Osborne, P (2008). "The Mercator Projections" (PDF). Section 5.4. Archived from the original (PDF) on 2012-01-18.

[1]

ellipsoidal coordinates

Longitude

Latitude

Vertical position

angle

Airy-0

Latitude § Geodetic and geocentric latitudes

geographic coordinates

spherical coordinate system

Geographic coordinate conversion

[3]

[4]

[5]