Parametric Latitude

An Auxiliary Latitude also called the Reduced Latitude and denoted $\eta$ or $\theta$ . It gives the Latitude on a Sphere of Radius for which the parallel has the same radius as the parallel of geodetic latitude $\phi$ and the Ellipsoid through a given point. It is given by

$\begin{displaymath} \eta=\tan^{-1}(\sqrt{1-e^2}\,\tan\phi). \end{displaymath}$

In series form,

$\begin{displaymath} \eta=\phi-e_1\sin(2\phi)+{\textstyle{1\over 2}}{e_1}^2\sin(4\phi)-{\textstyle{1\over 3}}{e_1}^3\sin(6\phi)+\ldots, \end{displaymath}$

where

$\begin{displaymath} e_1\equiv{1-\sqrt{1-e^2}\over 1+\sqrt{1-e^2}}. \end{displaymath}$

See also Auxiliary Latitude, Ellipsoid, Latitude, Sphere

References

Adams, O. S. ``Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections.'' Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921.

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 18, 1987.