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Eckert VI Projection

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

The equations are

$\displaystyle x$ $\textstyle =$ $\displaystyle {(\lambda-\lambda_0)(1+\cos\theta)\over\sqrt{2+\pi}}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {2\theta\over\sqrt{2+\pi}},$ (2)

where $\theta$ is the solution to
\theta+\sin\theta=(1+{\textstyle{1\over 2}}\pi)\sin\phi.
\end{displaymath} (3)

This can be solved iteratively using Newton's Method with $\theta_0=\phi$ to obtain
\Delta\theta=-{\theta+\sin\theta-(1+{\textstyle{1\over 2}}\pi)\sin\phi\over 1+\cos\theta}.
\end{displaymath} (4)

The inverse Formulas are
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \sin^{-1}\left({\theta+\sin\theta\over 1+{\textstyle{1\over 2}}\pi}\right)$ (5)
$\displaystyle \lambda$ $\textstyle =$ $\displaystyle \lambda_0+{\sqrt{2+\pi}\,x\over 1+\cos\theta},$ (6)

\theta={\textstyle{1\over 2}}\sqrt{2+\pi}\,y.
\end{displaymath} (7)


Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 253-258, 1987.

© 1996-9 Eric W. Weisstein