Mollweide Projection

$\begin{figure}\begin{center}\BoxedEPSF{maps/moll.epsf scaled 700}\end{center}\end{figure}$

A Map Projection also called the Elliptical Projection or Homolographic Equal Area Projection. The forward transformation is

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

where $\theta$ is given by

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

(3)

Newton's Method can then be used to compute $\theta'$ iteratively from

$\begin{displaymath} \Delta\theta'=-{\theta'+\sin\theta'-\pi\sin\phi\over 1+\cos\theta'}, \end{displaymath}$

(4)

where

$\begin{displaymath} \theta'={\textstyle{1\over 2}}\theta' \end{displaymath}$

(5)

or, better yet,

$\begin{displaymath} \theta'=2\sin^{-1}\left({2\phi\over\pi}\right) \end{displaymath}$

(6)

can be used as a first guess.

The inverse Formulas are

$\displaystyle \phi$	$\textstyle =$	$\displaystyle \sin^{-1}\left[{2\theta+\sin(2\theta)\over\pi}\right]$	(7)
$\displaystyle \lambda$	$\textstyle =$	$\displaystyle \lambda_0+{\pi x\over 2\sqrt{2}\,\cos\theta},$	(8)

where

$\begin{displaymath} \theta=\sin^{-1}\left({y\over\sqrt{2}}\right). \end{displaymath}$

(9)

References

Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, pp. 249-252, 1987.