info prev up next book cdrom email home

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.




© 1996-9 Eric W. Weisstein
1999-05-26