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.