Isometric Latitude

An Auxiliary Latitude which is directly proportional to the spacing of parallels of Latitude from the equator on an ellipsoidal Mercator Projection. It is defined by

$$\psi=\ln\left\vert{\tan({\textstyle{1\over 4}}\pi+{\textstyl... ...\left({1-e\sin\phi\over 1+e\sin\phi}\right)^{e/2}}\right\vert,$$ (1)

where the symbol $\tau$ is sometimes used instead of $\psi$. The isometric latitude is related to the Conformal Latitude by

$$\psi=\ln\tan({\textstyle{1\over 4}}\pi+{\textstyle{1\over 2}}\chi).$$ (2)

The inverse is found by iterating

$$\phi=2\tan^{-1}\left[{\mathop{\rm exp}\nolimits (\psi)\left(... ...r 1-e\sin\phi}\right)^{e/2}}\right]-{\textstyle{1\over 2}}\pi,$$ (3)

with the first trial as

$$\phi_0=2\tan^{-1}(e^\psi)-{\textstyle{1\over 2}}\pi.$$ (4)

See also Latitude

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. 15, 1987.