Editing Geographic coordinate conversion (section)

=== From ECEF to geodetic coordinates ===
====Conversion for the longitude====
The conversion of ECEF coordinates to longitude is:
: <math>\lambda = \operatorname{atan2}(Y,X)</math>. 
where [[atan2]] is the quadrant-resolving arc-tangent function.
The geocentric longitude and geodetic longitude have the same value; this is true for Earth and other similar shaped planets because they have a large amount of rotational symmetry around their spin axis (see [[triaxial ellipsoidal longitude]] for a generalization).

==== Simple iterative conversion for latitude and height ====
The conversion for the latitude and height involves a circular relationship involving ''N'', which is a function of latitude:
:<math>\frac{Z}{p} \cot \phi = 1 - \frac{e^2 N}{N + h}</math>,
:<math>h=\frac{p}{\cos\phi} - N</math>.
It can be solved iteratively,<ref name=osgb>A guide to coordinate systems in Great Britain. This is available as a pdf document at
{{cite web|url=http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents |title=ordnancesurvey.co.uk |access-date=2012-01-11 |url-status=dead |archive-url=https://web.archive.org/web/20120211075826/http://www.ordnancesurvey.co.uk/oswebsite/gps/information/coordinatesystemsinfo/guidecontents/ |archive-date=2012-02-11 }} Appendices B1, B2</ref><ref name=osborne>Osborne, P (2008). [http://mercator.myzen.co.uk/mercator.pdf The Mercator Projections] {{webarchive|url=https://web.archive.org/web/20120118224152/http://mercator.myzen.co.uk/mercator.pdf |date=2012-01-18 }} Section 5.4</ref> for example, starting with a first guess ''h''≈0 then updating ''N''.
More elaborate methods are shown below. 
The procedure is, however, sensitive to small accuracy due to <math>N</math> and <math>h</math> being maybe 10{{sup|6}} apart.<ref>[https://web.archive.org/web/20080920155754/http://www.ferris.edu/faculty/burtchr/papers/cartesian_to_geodetic.pdf R. Burtch, A Comparison of Methods Used in Rectangular to Geodetic Coordinate Transformations.]</ref><ref>{{cite journal |last1=Featherstone |first1=W. E. |last2=Claessens |first2=S. J. |title=Closed-Form Transformation between Geodetic and Ellipsoidal Coordinates |journal=Stud. Geophys. Geod. |volume=52 |issue=1 |pages=1–18 |year=2008 |doi=10.1007/s11200-008-0002-6 |bibcode=2008StGG...52....1F |hdl=20.500.11937/11589 |s2cid=59401014 |hdl-access=free }}</ref>
<!-- There are several methods that solve the equation; two are shown. -->

==== Newton–Raphson method ====
The following Bowring's irrational geodetic-latitude equation,<ref>{{cite journal |last=Bowring |first=B. R. |title=Transformation from Spatial to Geographical Coordinates |journal=Surv. Rev. |volume=23 |issue=181 |pages=323–327 |year=1976 |doi=10.1179/003962676791280626 }}</ref> derived simply from the above properties, is efficient to be solved by [[Newton–Raphson]] iteration method:<ref>{{cite journal |last=Fukushima |first=T. |title=Fast Transform from Geocentric to Geodetic Coordinates |journal=J. Geod. |volume=73 |issue=11 |pages=603–610 |year=1999 |doi=10.1007/s001900050271 |bibcode=1999JGeod..73..603F |s2cid=121816294 }} (Appendix B)</ref><ref>{{cite book|first1=J. J. |title=Proceedings of the IEEE 1997 National Aerospace and Electronics Conference. NAECON 1997|volume=2|pages=646–650|last1=Sudano|doi=10.1109/NAECON.1997.622711|chapter=An exact conversion from an earth-centered coordinate system to latitude, longitude and altitude|year=1997|isbn=0-7803-3725-5|s2cid=111028929 }}</ref>
: <math>\kappa - 1 - \frac{e^2 a\kappa}{\sqrt{p^2 + \left(1 - e^2\right) Z^2 \kappa^2}} = 0,</math>

where <math>\kappa = \frac{p}{Z} \tan \phi</math> and <math>p = \sqrt{X^2 + Y^2}</math> as before. The height is calculated as:
: <math>\begin{align}
         h &= e^{-2} \left(\kappa^{-1} - {\kappa_0}^{-1}\right) \sqrt{p^2 + Z^2 \kappa^2}, \\
  \kappa_0 &\triangleq \left(1 - e^2\right)^{-1}.
\end{align}</math>

The iteration can be transformed into the following calculation:
: <math>\kappa_{i+1} = \frac{c_i + \left(1 - e^2\right) Z^2 \kappa_i^3}{c_i - p^2} = 1 + \frac{p^2 + \left(1 - e^2\right) Z^2 \kappa_i^3}{c_i - p^2},</math>

where <math>c_i = \frac{\left(p^2 + \left(1 - e^2\right) Z^2 \kappa_i ^2\right)^\frac{3}{2}}{ae^2} .</math>

The constant <math>\,\kappa_0</math> is a good starter value for the iteration when <math>h \approx 0</math>. Bowring showed that the single iteration produces a sufficiently accurate solution. He used extra trigonometric functions in his original formulation.
<!--
: <math>\kappa \approx \kappa_1 = \left(c + \frac{z^2}{1 - e^2 }\right)/\left(c - \left(1 - e^2\right)\left(x^2 + y^2\right)\right),</math>

where
: <math>c = \frac{\left(\left(1 - e^2\right)\left(x^2 + y^2\right) + z^2\right)^\frac{3}{2}}{ae^2 \sqrt{1 - e^2}}.</math> --><!--
For <math>h = 0</math>, <math>\kappa = \frac{1}{1 - e^2}</math>, which is a good starter for the iteration. Bowring showed that the single iteration produces the sufficiently accurate solution under the condition of <math>h \approx 0</math>.
-->

==== Ferrari's solution ====
The quartic equation of <math>\kappa</math>, derived from the above, <!--for this transformation--> can be solved by [[Quartic equation#Ferrari.27s solution|Ferrari's solution]]<ref>{{cite journal|first1=H. |last1=Vermeille, H.|title=Direct Transformation from Geocentric to Geodetic Coordinates|journal= J. Geod.|volume=76|number=8|pages=451–454
|year= 2002|doi=10.1007/s00190-002-0273-6|s2cid=120075409 }}</ref><ref>{{cite journal|first1=Laureano|last1=Gonzalez-Vega|first2=Irene|last2=PoloBlanco|title=A symbolic analysis of Vermeille and Borkowski polynomials for transforming 3D Cartesian to geodetic coordinates|journal=J. Geod.|volume=83|number=11|pages=1071–1081|doi=10.1007/s00190-009-0325-2|year=2009|bibcode=2009JGeod..83.1071G |s2cid=120864969 }}</ref> to yield:
: <math>
\begin{align}
   \zeta &= \left(1 - e^2\right)\frac{z^2}{a^2} ,\\[4pt]
    \rho &= \frac{1}{6}\left(\frac{p^2}{a^2} + \zeta - e^4\right) ,\\[4pt]
       s &= \frac{e^4 \zeta p^2}{4\rho^3 a^2} ,\\[4pt]
       t &= \sqrt[3]{1 + s + \sqrt{s(s + 2)}} ,\\[4pt]
       u &= \rho \left(t + 1 + \frac{1}{t}\right) ,\\[4pt]
       v &= \sqrt{u^2 + e^4 \zeta} ,\\[4pt]
       w &= e^2 \frac{u + v - \zeta}{2v} ,\\[4pt]
  \kappa &= 1 + e^2 \frac{\sqrt{u + v + w^2} + w}{u + v}.
\end{align}
</math>

===== The application of Ferrari's solution =====
A number of techniques and algorithms are available but the most accurate, according to Zhu,<ref>{{cite journal|first1=J.|last1=Zhu|title=Conversion of Earth-centered Earth-fixed coordinates to geodetic coordinates|journal=IEEE Transactions on Aerospace and Electronic Systems|volume=30|issue=3|year=1994|pages=957–961|doi=10.1109/7.303772|bibcode=1994ITAES..30..957Z }}</ref> is the following procedure established by Heikkinen,<ref>{{cite journal|first1=M.|last1=Heikkinen|title=Geschlossene formeln zur berechnung räumlicher geodätischer koordinaten aus rechtwinkligen koordinaten.|journal=Z. Vermess.|volume=107|year=1982|pages=207–211|language=de}}</ref> as cited by Zhu. This overlaps with above. It is assumed that geodetic parameters <math>\{a,\, b,\, e\}</math> are known

: <math>\begin{align}
        a &= 6378137.0 \text{     m. Earth Equatorial Radius} \\[3pt]
        b &= 6356752.3142 \text{      m. Earth Polar Radius} \\[3pt]
      e^2 &= \frac{a^2-b^2}{a^2} \\[3pt]
     e'^2 &= \frac{a^2 - b^2}{b^2} \\[3pt]
        p &= \sqrt{X^2 + Y^2} \\[3pt]
        F &= 54b^2 Z^2 \\[3pt]
        G &= p^2 + \left(1 - e^2\right)Z^2 - e^2\left(a^2 - b^2\right) \\[3pt]
        c &= \frac{e^4 Fp^2}{G^3} \\[3pt]
        s &= \sqrt[3]{1 + c + \sqrt{c^2 + 2c}} \\[3pt]
        k &= s + 1 + \frac{1}{s}\\[3pt]
        P &= \frac{F}{3 k^2 G^2} \\[3pt]
        Q &= \sqrt{1 + 2e^4 P} \\[3pt]
      r_0 &= \frac{-Pe^2 p}{1 + Q} + \sqrt{\frac{1}{2} a^2\left(1 + \frac{1}{Q}\right) - \frac{P\left(1 - e^2\right)Z^2}{Q(1 + Q)} - \frac{1}{2}Pp^2} \\[3pt]
        U &= \sqrt{\left(p - e^2 r_0\right)^2 + Z^2} \\[3pt]
        V &= \sqrt{\left(p - e^2 r_0\right)^2 + \left(1 - e^2\right)Z^2} \\[3pt]
      z_0 &= \frac{b^2 Z}{aV} \\[3pt]
        h &= U\left(1 - \frac{b^2}{aV}\right) \\[3pt]
     \phi &= \arctan\left[\frac{Z + e'^2 z_0}{p}\right] \\[3pt]
  \lambda &= \operatorname{arctan2}[Y,\, X]
\end{align}</math>

Note: [[atan2|arctan2]][Y, X] is the four-quadrant inverse tangent function.

==== Power series ====
For small {{math|e<sup>2</sup>}} the power series
:<math>\kappa = \sum_{i\ge 0} \alpha_i e^{2i}</math>

starts with
:<math>\begin{align}
  \alpha_0 &= 1; \\
  \alpha_1 &= \frac{a}{\sqrt{Z^2 + p^2}}; \\
  \alpha_2 &= \frac{aZ^2\sqrt{Z^2 + p^2} + 2a^2 p^2}{2\left(Z^2 + p^2\right)^2}.
\end{align}</math>