Latitude

• In geography, latitude is a geographic coordinate that specifies the northsouth position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the Equator to 90° (North or South) at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. Without qualification the term latitude should be taken to be the geodetic latitude as defined in the following sections. Also defined are six auxiliary latitudes which are used in special applications. There is a separate article on the History of latitude measurements.

Two levels of abstraction are employed in the definition of latitude and longitude. In the first step the physical surface is modelled by the geoid, a surface which approximates the mean sea level over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a sphere, but the geoid is more accurately modelled by an ellipsoid. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a graticule on the reference surface. The latitude of a point on the actual surface is that of the corresponding point on the reference surface, the correspondence being along the normal to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a geographic coordinate system as defined in the specification of the ISO 19111 standard.

Since there are many different reference ellipsoids, the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.

 Arctic Circle 66° 34′ (66.57°) N Tropic of Cancer 23° 26′ (23.43°) N Tropic of Capricorn 23° 26′ (23.43°) S Antarctic Circle 66° 34′ (66.57°) S
${\displaystyle \phi }$ Δ1
lat
Δ1
long
110.574 km 111.320 km
15° 110.649 km 107.550 km
30° 110.852 km 96.486 km
45° 111.132 km 78.847 km
60° 111.412 km 55.800 km
75° 111.618 km 28.902 km
90° 111.694 km 0.000 km
Approximate difference from geodetic latitude (φ)
φ Reduced
φβ
Authalic
φξ
Rectifying
φμ
Conformal
φχ
Geocentric
φψ
0.00′ 0.00′ 0.00′ 0.00′ 0.00′
15° 2.91′ 3.89′ 4.37′ 5.82′ 5.82′
30° 5.05′ 6.73′ 7.57′ 10.09′ 10.09′
45° 5.84′ 7.78′ 8.76′ 11.67′ 11.67′
60° 5.06′ 6.75′ 7.59′ 10.12′ 10.13′
75° 2.92′ 3.90′ 4.39′ 5.85′ 5.85′
90° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′

 Arctic Circle 66° 34′ (66.57°) N Tropic of Cancer 23° 26′ (23.43°) N Tropic of Capricorn 23° 26′ (23.43°) S Antarctic Circle 66° 34′ (66.57°) S
${\displaystyle m(\phi )={\frac {\pi }{180^{\circ }}}R\phi _{\mathrm {degrees} }=R\phi _{\mathrm {radians} }}$
${\displaystyle f={\frac {a-b}{a}},\qquad e^{2}=2f-f^{2},\qquad b=a(1-f)=a{\sqrt {1-e^{2}}}\,.}$
${\displaystyle m(\phi )=\int _{0}^{\phi }M(\phi ')\,d\phi '=a(1-e^{2})\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '}$
${\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,}$
${\displaystyle \delta m(\phi )=M(\phi )\,\delta \phi =a(1-e^{2})\left(1-e^{2}\sin ^{2}\phi \right)^{-{\frac {3}{2}}}\,\delta \phi }$
${\displaystyle \Delta _{\mathrm {lat} }^{1}={\frac {\pi a\left(1-e^{2}\right)}{180^{\circ }\left(1-e^{2}\sin ^{2}\phi \right)^{\frac {3}{2}}}}}$
${\displaystyle \Delta _{\mathrm {lat} }^{1}=111\,132.954-559.822\cos 2\phi +1.175\cos 4\phi }$
${\displaystyle \Delta _{\mathrm {long} }^{1}={\frac {\pi a\cos \phi }{180^{\circ }{\sqrt {1-e^{2}\sin ^{2}\phi }}}}\,}$
${\displaystyle \psi (\phi )=\tan ^{-1}\left((1-e^{2})\tan \phi \right)\,.}$
${\displaystyle \beta (\phi )=\tan ^{-1}\left({\sqrt {1-e^{2}}}\tan \phi \right)}$
${\displaystyle {\frac {p^{2}}{a^{2}}}+{\frac {z^{2}}{b^{2}}}=1\,.}$
${\displaystyle p=a\cos \beta \,,\qquad z=b\sin \beta \,;}$
${\displaystyle \mu (\phi )={\frac {\pi }{2}}{\frac {m(\phi )}{m_{\mathrm {p} }}}}$
${\displaystyle m(\phi )=a(1-e^{2})\int _{0}^{\phi }\left(1-e^{2}\sin ^{2}\phi '\right)^{-{\frac {3}{2}}}\,d\phi '\,,}$
${\displaystyle m_{\mathrm {p} }=m\left({\frac {\pi }{2}}\right)\,.}$
${\displaystyle R={\frac {2m_{\mathrm {p} }}{\pi }}}$
${\displaystyle \xi (\phi )=\sin ^{-1}\left({\frac {q(\phi )}{q_{\mathrm {p} }}}\right)}$
{\displaystyle {\begin{aligned}q(\phi )&={\frac {(1-e^{2})\sin \phi }{1-e^{2}\sin ^{2}\phi }}-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)\\&={\frac {(1-e^{2})\sin \phi }{1-e^{2}\sin ^{2}\phi }}+{\frac {1-e^{2}}{e}}\tanh ^{-1}(e\sin \phi )\end{aligned}}}
{\displaystyle {\begin{aligned}q_{\mathrm {p} }=q\left({\frac {\pi }{2}}\right)&=1-{\frac {1-e^{2}}{2e}}\ln \left({\frac {1-e}{1+e}}\right)&=1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\end{aligned}}}
${\displaystyle R_{q}=a{\sqrt {\frac {q_{\mathrm {p} }}{2}}}\,.}$
{\displaystyle {\begin{aligned}\chi (\phi )&=2\tan ^{-1}\left[\left({\frac {1+\sin \phi }{1-\sin \phi }}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{e}\right]^{\frac {1}{2}}-{\frac {\pi }{2}}\\[2ex]&=2\tan ^{-1}\left[\tan \left({\frac {\phi }{2}}+{\frac {\pi }{4}}\right)\left({\frac {1-e\sin \phi }{1+e\sin \phi }}\right)^{\frac {e}{2}}\right]-{\frac {\pi }{2}}\\&=\sin ^{-1}\left[\tanh \left(\tanh ^{-1}(\sin \phi )-e\tanh ^{-1}(e\sin \phi )\right)\right]\\&=\operatorname {gd} \left[\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi )\right]\end{aligned}}}
{\displaystyle {\begin{aligned}\psi (\phi )&=\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\phi }{2}}\right)\right]+{\frac {e}{2}}\ln \left[{\frac {1-e\sin \phi }{1+e\sin \phi }}\right]\\&=\tanh ^{-1}(\sin \phi )-e\tanh ^{-1}(e\sin \phi )\\&=\operatorname {gd} ^{-1}(\phi )-e\tanh ^{-1}(e\sin \phi ).\end{aligned}}}
${\displaystyle y(\phi )={\frac {E}{2\pi }}\psi (\phi )\,.}$
${\displaystyle \psi (\phi )=\operatorname {gd} ^{-1}\chi (\phi )\,.}$
• a (equatorial radius): 6378137.0 m exactly
• 1/f (inverse flattening): 298.257223563 exactly
• b (polar radius): 6356752.3142 m
• e2 (eccentricity squared): 0.00669437999014
• Geodetic latitude: the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.
• Geocentric latitude: the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure below). There is no standard notation: examples from various texts include ψ, q, φ′, φc, φg. This article uses ψ.
• Spherical latitude: the angle between the normal to a spherical reference surface and the equatorial plane.
• Geographic latitude must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude.
• Latitude (unqualified) should normally refer to the geodetic latitude.
• Geocentric latitude
• Reduced (or parametric) latitude
• Rectifying latitude
• Authalic latitude
• Conformal latitude
• Isometric latitude
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