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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.



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© 1996-9 Eric W. Weisstein
1999-05-25