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

$\begin{displaymath} \theta+\sin\theta=(1+{\textstyle{1\over 2}}\pi)\sin\phi. \end{displaymath}$

(3)

This can be solved iteratively using Newton's Method with $\theta_0=\phi$ to obtain

$\begin{displaymath} \Delta\theta=-{\theta+\sin\theta-(1+{\textstyle{1\over 2}}\pi)\sin\phi\over 1+\cos\theta}. \end{displaymath}$

(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)

where

$\begin{displaymath} \theta={\textstyle{1\over 2}}\sqrt{2+\pi}\,y. \end{displaymath}$

(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.