Editing Latitude (section)

===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/>