Editing Latitude (section)

==Auxiliary latitudes==

There are six '''auxiliary latitudes''' that have applications to special problems in geodesy, geophysics and the theory of map projections:
* [[Geocentric latitude]]
* Parametric (or reduced) latitude
* Rectifying latitude
* [[Authalic latitude]]
* Conformal latitude
* Isometric latitude
The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional [[geographic coordinate system]] as discussed [[#Latitude and coordinate systems|below]]. The remaining latitudes are not used in this way; they are used ''only'' as intermediate constructs in map projections of the reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower.

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, {{mvar|a}}, and the eccentricity, {{mvar|e}}. (For inverses see [[#Inverse formulae and series|below]].) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder.<ref name=snyder>{{Cite book| last=Snyder| first=John P.| title=Map Projections: A Working Manual| series=U.S. Geological Survey Professional Paper 1395| publisher=United States Government Printing Office| location=Washington, DC| year=1987| url=https://pubs.er.usgs.gov/pubs/pp/pp1395| access-date=2017-09-02| archive-url=https://web.archive.org/web/20080516070706/http://pubs.er.usgs.gov/pubs/pp/pp1395| archive-date=2008-05-16| url-status=dead}}</ref> Derivations of these expressions may be found in Adams<ref name=adams1921>{{cite book|last=Adams |first=Oscar S. |author-link=Oscar S. Adams |date=1921 |title=Latitude Developments Connected With Geodesy and Cartography (with tables, including a table for Lambert equal area meridional projection |series=Special Publication No. 67|publisher=US Coast and Geodetic Survey|url=https://geodesy.noaa.gov/library/pdfs/Special_Publication_No_67.pdf}} (''Note'': Adams uses the nomenclature isometric latitude for the conformal latitude of this article (and throughout the modern literature).)</ref> and online publications by Osborne<ref name=osborne/> and Rapp.<ref name=rapp/>

===Geocentric latitude===
{{see also|#Geodetic and geocentric latitudes}}
[[File:Geocentric coords 03.svg|right|thumb|upright=1.15|The definition of geodetic latitude ({{mvar|ϕ}}) and geocentric latitude ({{mvar|θ}})]]
The '''geocentric latitude''' is the angle between the equatorial plane and the radius from the centre to a point of interest.

When the point is on the surface of the ellipsoid, the relation between the geocentric latitude ({{mvar|θ}}) and the geodetic latitude ({{mvar|ϕ}}) is:
:<math>\theta(\phi) = \tan^{-1}\left(\left(1 - e^2\right)\tan\phi\right) = \tan^{-1}\left((1 - f)^2\tan\phi\right)\,.</math>
For points not on the surface of the ellipsoid, the relationship involves additionally the [[ellipsoidal height]] ''h'':
: <math>\theta(\phi,h) = \tan^{-1}\left( \frac{N(1 - f)^2 + h}{ N + h}\tan\phi \right)</math>

where {{mvar|N}} is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of <math>\phi{-}\theta</math> may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45°&nbsp;6′.{{efn|An elementary calculation involves differentiation to find the maximum difference of the geodetic and geocentric latitudes.
}}

===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>).

===Rectifying latitude===
{{see also|Rectifying radius}}
The '''rectifying latitude''', {{mvar|μ}}, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or {{sfrac|{{pi}}|2}} radians:
:<math>\mu(\phi) = \frac{\pi}{2}\frac{m(\phi)}{m_\mathrm{p}}</math>

where the meridian distance from the equator to a latitude {{mvar|ϕ}} is (see [[Meridian arc]])
:<math>m(\phi) = a\left(1 - e^2\right)\int_0^\phi \left(1 - e^2 \sin^2 \phi'\right)^{-\frac{3}{2}}\, d\phi'\,,</math>

and the length of the meridian quadrant from the equator to the pole (the [[Meridian arc#Polar distance|polar distance]]) is
:<math>m_\mathrm{p} = m\left(\frac{\pi}{2}\right)\,.</math>

Using the rectifying latitude to define a latitude on a sphere of radius
:<math>R = \frac{2m_\mathrm{p}}{\pi}</math>

defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale. The sphere may then be projected to the plane with an [[equirectangular projection]] to give a double projection from the ellipsoid to the plane such that all meridians have true length and uniform meridian scale. An example of the use of the rectifying latitude is the [[equidistant conic projection]]. (Snyder, Section 16).<ref name=snyder/> The rectifying latitude is also of great importance in the construction of the [[Transverse Mercator projection]].

===Authalic latitude===
{{see also|Authalic radius}}
The '''authalic latitude''' (after the Greek for "[[wiktionary:authalic|same area]]"), {{mvar|ξ}}, gives an [[equal-area projection]] to a sphere.
:<math>\xi(\phi) = \sin^{-1}\left(\frac{q(\phi)}{q_\mathrm{p}}\right)</math>

where
:<math>\begin{align}
  q(\phi) &= \frac{\left(1 - e^2\right)\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) \\[2pt]
          &= \frac{\left(1 - e^2\right)\sin\phi}{1 - e^2 \sin^2 \phi} + \frac{1 - e^2}{e}\tanh^{-1}(e\sin\phi)
\end{align}</math>

and
:<math>\begin{align}
  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{align}</math>

and the radius of the sphere is taken as
:<math>R_q = a\sqrt{\frac{q_\mathrm{p}}{2}}\,.</math>

An example of the use of the authalic latitude is the [[Albers projection|Albers equal-area conic projection]].<ref name=snyder/>{{rp|§14}}

===Conformal latitude===
The '''conformal latitude''', {{mvar|χ}}, gives an angle-preserving ([[Conformal map|conformal]]) transformation to the sphere.
<ref>{{cite book |first=Joseph-Louis |last=Lagrange |author-link= Joseph-Louis Lagrange |year=1779 |title=Oevres |volume=IV |chapter=Sur la Construction des Cartes Géographiques |page=667 |language=fr |chapter-url=https://archive.org/details/oeuvresdelagrang04lagr/page/663 }}</ref>

:<math>\begin{align}
  \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} \\[2pt]
             &= 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} \\[2pt]
             &= \tan^{-1}\left[\sinh\left(\sinh^{-1}(\tan\phi) - e\tanh^{-1}(e\sin\phi)\right)\right] \\
             &= \operatorname{gd}\left[\operatorname{gd}^{-1}(\phi) - e\tanh^{-1}(e\sin\phi)\right]
\end{align}</math>

where {{math|gd(''x'')}} is the [[Gudermannian function]]. (See also [[Mercator projection#Alternative expressions|Mercator projection]].)

The conformal latitude defines a transformation from the ellipsoid to a sphere of ''arbitrary'' radius such that the angle of intersection between any two lines on the ellipsoid is the same as the corresponding angle on the sphere (so that the shape of ''small'' elements is well preserved). A further conformal transformation from the sphere to the plane gives a conformal double projection from the ellipsoid to the plane. This is not the only way of generating such a conformal projection. For example, the 'exact' version of the [[Transverse Mercator projection]] on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).

===Isometric latitude===
The '''isometric latitude''', {{mvar|ψ}}, is used in the development of the ellipsoidal versions of the normal [[Mercator projection]] and the [[Transverse Mercator projection]]. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of {{mvar|ψ}} and longitude {{mvar|λ}} give rise to equal distance displacements along the meridians and parallels respectively. The [[Geographic coordinate system|graticule]] defined by the lines of constant {{mvar|ψ}} and constant {{mvar|λ}}, divides the surface of the ellipsoid into a mesh of squares (of varying size). The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude, tending to infinity at the poles. The conventional notation is given in Snyder (page 15):<ref name=snyder/>
:<math>\begin{align}
  \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] \\
             &= \sinh^{-1}(\tan\phi) -e\tanh^{-1}(e\sin\phi) \\
             &= \operatorname{gd}^{-1}(\phi)-e\tanh^{-1}(e\sin\phi).
\end{align}</math>

For the ''normal'' Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is {{mvar|E}} (units of length or pixels) then the distance, {{mvar|y}}, of a parallel of latitude {{mvar|ϕ}} from the equator is
:<math>y(\phi) = \frac{E}{2\pi}\psi(\phi)\,.</math>

The isometric latitude {{mvar|ψ}} is closely related to the conformal latitude {{mvar|χ}}:
:<math>\psi(\phi) = \operatorname{gd}^{-1} \chi(\phi)\,.</math>

===Inverse formulae and series===
The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and parametric latitudes may be inverted directly but this is impossible in the four remaining cases: the rectifying, authalic, conformal, and isometric latitudes. There are two methods of proceeding.
* The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are [[fixed-point iteration]] and [[Newton's method|Newton–Raphson]] root finding.
** When converting from isometric or conformal to geodetic, two iterations of Newton-Raphson gives [[double precision]] accuracy.<ref>{{cite journal |last1=Karney |first1=Charles F. F. |title=Transverse Mercator with an accuracy of a few nanometers |journal=Journal of Geodesy |date=August 2011 |volume=85 |issue=8 |pages=475–485 |doi=10.1007/s00190-011-0445-3 |arxiv=1002.1417|bibcode=2011JGeod..85..475K |s2cid=118619524 }}</ref>
* The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of [[Lagrange reversion]]. Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity.<ref name=adams1921/>  Orihuela<ref>{{cite web |last = Orihuela |first = Sebastián |date = 2013 |title = Funciones de Latitud |url = https://www.academia.edu/7580468}}</ref> gives series for the conversions between all pairs of auxiliary latitudes in terms of the third flattening, {{math|''n'' {{=}} (''a'' - ''b'')/(''a'' + ''b'')}}. Karney<ref>{{cite journal |last = Karney |first = Charles F. F. |date = 2023 |title = On auxiliary latitudes |journal = Survey Review |volume = 56 |issue = 395 |pages = 165–180 |doi = 10.1080/00396265.2023.2217604 |arxiv = 2212.05818}}</ref> establishes that the truncation errors for such series are consistently smaller that the equivalent series in terms of the eccentricity. The series method is not applicable to the isometric latitude and one must find the conformal latitude in an intermediate step.<ref name=osborne/>

===Numerical comparison of auxiliary latitudes===
[[File:Auxiliary Latitudes Difference.svg|inline|right|frameless]]
The plot to the right shows the difference between the geodetic latitude and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles) for the case of the WGS84 ellipsoid. The differences shown on the plot are in arc minutes. 
In the Northern hemisphere (positive latitudes), ''θ'' ≤ ''χ'' ≤ ''μ'' ≤ ''ξ'' ≤ ''β'' ≤ ''ϕ''; in the Southern hemisphere (negative latitudes), the inequalities are reversed, with equality at the equator and the poles. Although the graph appears symmetric about 45°, the minima of the curves actually lie between 45° 2′ and 45° 6′. Some representative data points are given in the table below. The conformal and geocentric latitudes are nearly indistinguishable, a fact that was exploited in the days of hand calculators to expedite the construction of map projections.<ref name=snyder/>{{rp|108}}

To first order in the flattening ''f'', the auxiliary latitudes can be expressed as
''ζ'' = ''ϕ'' − ''Cf'' sin 2''ϕ''
where the constant ''C'' takes on the values
[{{frac|2}}, {{frac|2|3}}, {{frac|3|4}}, 1, 1]
for
''ζ'' = [''β'', ''ξ'', ''μ'', ''χ'', ''θ''].

{| class="wikitable" style="margin: 1em auto 1em auto; text-align:right;" 
|+ Approximate difference from geodetic latitude ({{mvar|ϕ}})
! {{mvar|ϕ}}
! Parametric<br />{{math|''β'' − ''ϕ''}}
! Authalic<br />{{math|''ξ'' − ''ϕ''}}
! Rectifying<br />{{math|''μ'' − ''ϕ''}}
! Conformal<br />{{math|''χ'' − ''ϕ''}}
! Geocentric<br />{{math|''θ'' − ''ϕ''}}
|-
|  0°||  0.00′ ||  0.00′ ||  0.00′ ||   0.00′ ||   0.00′
|-
| 15°|| −2.88′ || −3.84′ || −4.32′ ||  −5.76′ ||  −5.76′
|-
| 30°|| −5.00′ || −6.66′ || −7.49′ ||  −9.98′ ||  −9.98′
|-
| 45°|| −5.77′ || −7.70′ || −8.66′ || −11.54′ || −11.55′
|-
| 60°|| −5.00′ || −6.67′ || −7.51′ || −10.01′ || −10.02′
|-
| 75°|| −2.89′ || −3.86′ || −4.34′ ||  −5.78′ ||  −5.79′
|-
| 90°||  0.00′ ||  0.00′ ||  0.00′ ||   0.00′ ||   0.00′
|}
{{clear}}