Albers Equal-Area Conic Projection

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Let $\phi_0$ be the Latitude for the origin of the Cartesian Coordinates and $\lambda_0$ its Longitude. Let $\phi_1$ and $\phi_2$ be the standard parallels. Then

$\displaystyle x$	$\textstyle =$	$\displaystyle \rho\sin\theta$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle \rho_0-\rho\cos\theta,$	(2)

where

$\displaystyle \rho$	$\textstyle =$	$\displaystyle {\sqrt{C-2n\sin\phi}\over n}$	(3)
$\displaystyle \theta$	$\textstyle =$	$\displaystyle n(\lambda-\lambda_0)$	(4)
$\displaystyle \rho_0$	$\textstyle =$	$\displaystyle {\sqrt{C-2n\sin\phi_0}\over n}$	(5)
$\displaystyle C$	$\textstyle =$	$\displaystyle \cos^2\phi_1+2n\sin\phi_1$	(6)
$\displaystyle n$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}(\sin\phi_1+\sin\phi_2).$	(7)

The inverse Formulas are

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}\left({C-\rho^2 n^2\over 2n}\right)$	(8)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+{\theta\over n},$	(9)

where

$\displaystyle \rho$	$\textstyle =$	$\displaystyle \sqrt{x^2+(\rho_0-y)^2}$	(10)
$\displaystyle \theta$	$\textstyle =$	$\displaystyle \tan^{-1}\left({x\over\rho_0-y}\right).$	(11)

References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 98-103, 1987.