Cylindrical Equal-Area Projection

$\begin{figure}\begin{center}\BoxedEPSF{maps/cyea.epsf scaled 600}\end{center}\end{figure}$

The Map Projection having transformation equations,

$\displaystyle x$	$\textstyle =$	$\displaystyle (\lambda-\lambda_0)\cos\phi_s$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {\sin\phi\over\cos\phi_s}$	(2)

for the normal aspect, and inverse transformation equations

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}(y\cos\phi_s)$	(3)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle {x\over\cos\phi_s}+\lambda_0.$	(4)

$\begin{figure}\begin{center}\BoxedEPSF{maps/cyob.epsf scaled 600}\end{center}\end{figure}$

An oblique form of the cylindrical equal-area projection is given by the equations

$\displaystyle \lambda_p$	$\textstyle =$	$\displaystyle \tan^{-1}\left({\cos\phi_1\sin\phi_2\cos\lambda_1-\sin\phi_1\cos\... ...ver \sin\phi_1\cos\phi_2\sin\lambda_2-\cos\phi_1\sin\phi_2\sin\lambda_1}\right)$
			(5)
$\displaystyle \phi_p$	$\textstyle =$	$\displaystyle \tan^{-1}\left[{-{\cos(\lambda_p-\lambda_1)\over\tan\phi_1}}\right],$	(6)

and the inverse Formulas are

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}(y\sin\phi_p+\sqrt{1-y^2}\,\cos\phi_p\sin x)$	(7)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+\tan^{-1}\left({\sqrt{1-y^2}\,\sin\phi_p\sin x-y\cos\phi_p\over\sqrt{1-y^2}\,\cos x}\right).$
			(8)

$\begin{figure}\begin{center}\BoxedEPSF{maps/cytr.epsf scaled 600}\end{center}\end{figure}$

A transverse form of the cylindrical equal-area projection is given by the equations

$\displaystyle x$	$\textstyle =$	$\displaystyle \cos\phi\sin(\lambda-\lambda_0)$	(9)
$\displaystyle y$	$\textstyle =$	$\displaystyle \tan^{-1}\left[{\tan\phi\over\cos(\lambda-\lambda_0)}\right]-\phi_0,$	(10)

and the inverse Formulas are

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}[\sqrt{1-x^2}\sin (y+\phi_0)]$	(11)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+\tan^{-1}\left[{{x\over\sqrt{1-x^2}}\cos(y+\phi_0)}\right].$	(12)

References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 76-85, 1987.