Editing Latitude (section)

==Latitude and coordinate systems==
The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional [[coordinate system]] on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and parametric latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.

===Geodetic coordinates===
{{main|Geodetic coordinates}}
[[File:Geodetic coordinates.svg|thumb|right|upright=0.9| Geodetic coordinates {{math|P(''ɸ'',''λ'',''h'')}}]]
At an arbitrary point {{math|P}} consider the line {{math|PN}} which is normal to the reference ellipsoid. The geodetic coordinates {{math|P(''ɸ'',''λ'',''h'')}} are the latitude and longitude of the point {{math|N}} on the ellipsoid and the distance {{math|PN}}. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of {{math|PN}} will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth.

===Spherical polar coordinates===

[[File:Geocentric coords 02.svg|thumb|right|upright=0.9| Geocentric coordinate related to spherical polar coordinates {{math|P(''r'',''θ''′,''λ'')}}]]
The geocentric latitude {{mvar|θ}} is the complement of the ''polar angle'' or ''[[colatitude]]'' {{mvar|θ′}} in conventional [[spherical polar coordinates]] in which the coordinates of a point are {{math|P(''r'',''θ''′,''λ'')}} where {{mvar|r}} is the distance of {{math|P}} from the centre {{math|O}}, {{mvar|θ′}} is the angle between the radius vector and the polar axis and {{mvar|λ}} is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points {{math|P'}} on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

===Ellipsoidal-harmonic coordinates===
[[File:Ellipsoidal coordinates.svg|thumb|right|upright=0.9|Ellipsoidal coordinates {{math|P(''u'',''β'',''λ'')}}]]

The parametric latitude can also be extended to a three-dimensional coordinate system. For a point {{math|P}} not on the reference ellipsoid (semi-axes {{math|OA}} and {{math|OB}}) construct an auxiliary ellipsoid which is confocal (same foci {{math|F}}, {{math|F′}}) with the reference ellipsoid: the necessary condition is that the product {{mvar|ae}} of semi-major axis and eccentricity is the same for both ellipsoids. Let {{mvar|u}} be the semi-minor axis ({{math|OD}}) of the auxiliary ellipsoid. Further let {{mvar|β}} be the parametric latitude of {{math|P}} on the auxiliary ellipsoid. The set {{math|(''u'',''β'',''λ'')}} define the '''ellipsoidal-harmonic coordinates'''<ref>Holfmann-Wellenfor & Moritz (2006) ''Physical Geodesy'', p.240, eq. (6-6) to (6-10).</ref> or simply ''ellipsoidal coordinates''<ref name=torge/>{{rp|§4.2.2}} (although that term is also used to refer to geodetic coordinate). These coordinates are the natural choice in models of the gravity field for a rotating ellipsoidal body.
The above applies to a biaxial ellipsoid (a spheroid, as in [[oblate spheroidal coordinates]]); for a generalization, see [[triaxial ellipsoidal coordinates]].

===Coordinate conversions===
The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in [[geographic coordinate conversion]]. The relation of Cartesian and spherical polars is given in [[spherical coordinate system]]. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.<ref name=torge/>