Editing Geographic coordinate conversion (section)

==== 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>.
-->