Editing Latitude (section)

===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).