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Eckert IV Projection

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

The equations are

$\displaystyle x$ $\textstyle =$ $\displaystyle {2\over\sqrt{\pi(4+\pi)}} (\lambda-\lambda_0)(1+\cos\theta)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle 2\sqrt{\pi\over 4+\pi} \sin\theta,$ (2)

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

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

The inverse Formulas are
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \sin^{-1}\left({\theta+\sin\theta\cos\theta+2\sin\theta\over 2+{\textstyle{1\over 2}}\pi}\right)$ (5)
$\displaystyle \lambda$ $\textstyle =$ $\displaystyle \lambda_0+{\pi\sqrt{4+\pi}\,x\over 1+\cos\theta},$ (6)

where
\begin{displaymath}
\theta=\sin^{-1}\left({{y\over 2}\sqrt{4+\pi\over\pi}}\right).
\end{displaymath} (7)


References

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
1999-05-25