Rectifying Latitude

An Auxiliary Latitude which gives a sphere having correct distances along the meridians. It is denoted $\mu$ (or $\omega$) and is given by

$$\mu={\pi M\over 2M_p}.$$ (1)

$M_p$ is evaluated for $M$ at the north pole ($\phi=90^\circ$), and $M$ is given by

$$M = a(1-e^2)\int_0^\phi {d\phi\over (1-e^2\sin^2\phi)^{3/2}}$$

$$= a\left[{\int_0^\phi \sqrt{1-e^2\sin^2\phi}\,d\phi-{e^2\sin\phi\cos\phi\over\sqrt{1-e^2\sin^2\phi}}}\right].$$ (2)

A series for $M$ is

$$M = a[(1-{\textstyle{1\over 4}}e^2-{\textstyle{3\over 64}}e^4-{\textstyle{5\over 256}}e^6-\ldots)\phi$$

$$\phantom{=} -({\textstyle{3\over 8}}e^2+{\textstyle{3\over 32}}e^4+{\textstyle{45\over 1024}}e^6+\ldots)\sin(2\phi)$$

$$\phantom{=} +({\textstyle{15\over 256}}e^4+{\textstyle{45\over 1024}}e^6+\ldots)\sin(4\phi)$$

$$\phantom{=} -({\textstyle{35\over 3072}}e^6+\ldots)\sin(6\phi)+\ldots],$$ (3)

and a series for $\mu$ is

$$\mu = \phi-({\textstyle{3\over 2}}e_1-{\textstyle{9\over 16}}{e_1}^3+\ldots)\sin(2\phi)$$

$$\phantom{=} +({\textstyle{15\over 16}}{e_1}^2-{\textstyle{15\over 32}}{e_1}^4+\ldots)\sin(4\phi)$$

$$\phantom{=} -({\textstyle{35\over 48}}{e_1}^3-\ldots)\sin(6\phi)+({\textstyle{315\over 512}}{e_1}^4-\ldots)\sin(8\phi)+\ldots,$$ (4)

where

$$e_1\equiv {1-\sqrt{1-e^2}\over 1+\sqrt{1-e^2}}.$$ (5)

The inverse formula is

$$\phi = \mu+({\textstyle{3\over 2}}e_1-{\textstyle{27\over 32}}{e_1}^3+\ldots)\sin(2\mu)$$

$$\phantom{=} +({\textstyle{21\over 16}}{e_1}^2-{\textstyle{55\over 32}}{e_1}^4+\ldots)\sin(4\mu)$$

$$\phantom{=} +({\textstyle{151\over 96}}{e_1}^3-\ldots)\sin(6\mu)$$

$$\phantom{=} +({\textstyle{1097\over 512}}{e_1}^4-\ldots)\sin(8\mu)+\ldots.$$ (6)

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, pp. 125-128, 1921.

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 16-17, 1987.