Authalic Latitude

An Auxiliary Latitude which gives a Sphere equal Surface Area relative to an Ellipsoid. The authalic latitude is defined by

$$\beta=\sin^{-1}\left({q\over q_p}\right),$$ (1)

where

$$q\equiv (1-e^2)\left[{{\sin\phi\over 1-e^2\sin^2\phi}-{1\over 2e}\ln\left({1-e\sin\phi\over 1+e\sin\phi}\right)}\right],$$ (2)

and $q_p$ is $q$ evaluated at the north pole ($\phi=90^\circ$). Let $R_q$ be the Radius of the Sphere having the same Surface Area as the Ellipsoid, then

$$R_q=a\sqrt{q_p\over 2}\,.$$ (3)

The series for $\beta$ is

$$\beta = \phi-({\textstyle{1\over 3}}e^2+{\textstyle{31\over 180}}e^4+{\textstyle{59\over 560}}e^6+\ldots)\sin(2\phi)$$

$$\phantom{=} +({\textstyle{17\over 360}}e^4+{\textstyle{61\over 1260}}e^6+\ldots)\sin(4\phi)$$

$$\phantom{=} -({\textstyle{383\over 45360}}e^6+\ldots)\sin(6\phi)+\ldots.$$ (4)

The inverse Formula is found from

$$\Delta\phi={(1-e^2\sin^2\phi)^2\over 2\cos\phi}\left[{{q\ove... ...ver 2e}\ln\left({1-e\sin\phi\over 1+e\sin\phi}\right)}\right],$$ (5)

where

$$q=q_p\sin\beta$$ (6)

and $\phi_0=\sin^{-1}(q/2)$. This can be written in series form as

$$\phi = \beta+({\textstyle{1\over 3}}e^2+{\textstyle{31\over 180}}e^4+{\textstyle{517\over 5040}}e^6+\ldots)\sin(2\beta)$$

$$\phantom{=} +({\textstyle{23\over 360}}e^4+{\textstyle{251\over 3780}}e^6+\ldots)\sin(4\beta)$$

$$\phantom{=} +({\textstyle{761\over 45360}}e^6+\ldots)\sin(6\beta)+\ldots.$$ (7)

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. 16, 1987.