Conformal Latitude

An Auxiliary Latitude defined by

$$\chi \equiv 2\tan^{-1}\left\{{\tan({\textstyle{1\over 4}}\pi+{\textstyle{1\ov... ...[{1-e\sin\phi\over 1+e\sin\phi}\right]^{e/2}}\right\}-{\textstyle{1\over 2}}\pi$$

$$= 2\tan^{-1}\left[{{1+\sin\phi\over 1-\sin\phi}\left({1-e\sin\phi\over 1+e\sin\phi}\right)^e}\right]^{1/2}-{\textstyle{1\over 2}}\pi$$

$$= \phi-({\textstyle{1\over 2}}e^2+{\textstyle{5\over 24}}e^4+{\textstyle{3\over 32}}e^6+{\textstyle{281\over 5760}}e^8+\ldots)\sin(2\phi)$$

$$\phantom{=} +({\textstyle{5\over 48}}e^4+{\textstyle{7\over 80}}e^6+{\textstyle{697\over 11520}}e^8+\ldots)\sin(4\phi)$$

$$\phantom{=} -({\textstyle{13\over 480}}e^6+{\textstyle{461\over 13440}}+\ldots)\sin(6\phi)$$

$$\phantom{=} +({\textstyle{1237\over 161280}}e^8+\ldots)\sin(8\phi)+\ldots.$$

The inverse is obtained by iterating the equation

$$\phi=2\tan^{-1}\left[{\tan({\textstyle{1\over 4}}\pi+{\texts... ...er 1-e\sin\phi}\right)^{e/2}}\right]-{\textstyle{1\over 2}}\pi$$

using $\phi=\chi$ as the first trial. A series form is

$$\phi = \chi+({\textstyle{1\over 2}}e^2+{\textstyle{5\over 24}}e^4+{\textstyle{1\over 12}}e^6+{\textstyle{13\over 360}}e^8+\ldots)\sin(2\chi)$$

$$\phantom{=} +({\textstyle{7\over 48}}e^4+{\textstyle{29\over 240}}e^6+{\textstyle{811\over 11520}}e^9+\ldots)\sin(4\chi)$$

$$\phantom{=} +({\textstyle{7\over 120}}e^6+{\textstyle{81\over 1120}}e^8+\ldots)\sin(6\chi)$$

$$\phantom{=} +({\textstyle{4279\over 161280}}e^8+\ldots)\sin(8\chi)+\ldots$$

The conformal latitude was called the Isometric Latitude by Adams (1921), but this term is now used to refer to a different quantity.

See also Auxiliary Latitude, Latitude

References

Adams, O. S. ``Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridianal Projections.'' Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, pp. 18 and 84-85, 1921.

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 15-16, 1987.