Editing Latitude (section)

===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>