Editing Latitude (section)

===Parametric latitude (or reduced latitude){{anchor|Parametric latitude|Reduced latitude}}===
[[File:Ellipsoid reduced angle definition.svg|thumb|upright=0.9|right|Definition of the parametric latitude ({{mvar|β}}) on the ellipsoid]] 
The '''parametric latitude''' or '''reduced latitude''', {{mvar|β}}, is defined by the radius drawn from the centre of the ellipsoid to that point {{math|Q}} on the surrounding sphere (of radius {{mvar|a}}) which is the projection parallel to the Earth's axis of a point {{math|P}} on the ellipsoid at latitude {{mvar|ϕ}}. It was introduced by Legendre<ref name=legendre>{{cite journal|first=A. M. |last=Legendre |date=1806 |title=Analyse des triangles tracés sur la surface d'un sphéroïde |journal=Mém. Inst. Nat. Fr. |pages=130–161 |volume=1st semester}}</ref> and Bessel<ref name=bessel>{{cite journal|first=F. W. |last=Bessel |date=1825 |title=Über die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen |journal=Astron. Nachr. |volume=4|issue=86 |pages=241–254 |doi=10.1002/asna.201011352|arxiv=0908.1824 |bibcode=2010AN....331..852K |s2cid=118760590 }}<br>'''Translation:''' {{cite journal|first1=C. F. F. |last1=Karney |first2=R. E. |last2=Deakin |title=The calculation of longitude and latitude from geodesic measurements |journal=Astron. Nachr. |volume=331|issue=8 |pages=852–861 |date=2010 |arxiv=0908.1824 |bibcode=1825AN......4..241B|doi=10.1002/asna.18260041601 |s2cid=118630614 }}</ref> who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, {{math|''u''(''ϕ'')}}, is also used in the current literature. The parametric latitude is related to the geodetic latitude by:<ref name=osborne/><ref name=rapp/>
:<math>\beta(\phi) = \tan^{-1}\left(\sqrt{1 - e^2}\tan\phi\right) = \tan^{-1}\left((1 - f)\tan\phi\right)</math>

The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates {{mvar|p}}, the distance from the minor axis, and {{mvar|z}}, the distance above the equatorial plane, the equation of the [[ellipse]] is:
:<math> \frac{p^2}{a^2} + \frac{z^2}{b^2} =1\, .</math>

The Cartesian coordinates of the point are parameterized by
:<math> p = a\cos\beta\,, \qquad z = b\sin\beta\,; </math>

Cayley suggested the term ''parametric latitude'' because of the form of these equations.<ref name=cayley>{{cite journal|first=A. |last=Cayley |date=1870 |title=On the geodesic lines on an oblate spheroid |journal=Phil. Mag. |volume=40 |issue=4th ser |pages=329–340|doi=10.1080/14786447008640411 }}</ref>

The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, ([[Vincenty's formulae|Vincenty]], Karney<ref name=Karney>{{cite journal|first=C. F. F. |last=Karney |date=2013 |title=Algorithms for geodesics |journal=Journal of Geodesy |volume=87|issue=1 |pages=43–55 |doi= 10.1007/s00190-012-0578-z|arxiv=1109.4448 |bibcode=2013JGeod..87...43K |s2cid=119310141 }}</ref>).