Numerical Examples - Mollweide Projection

# Numerical Examples for Mollweide Projection #

## SPHERE #

### Forward Equations #

Given

 Radius of sphere: $R=\;\;$ unit Central meridian: $\lambda_0=\;$° Point: $\phi=\;$° $\lambda=\;$°
Find $x, y$.

From equation (31-4), using $\phi$ or $-50^\circ$ as the first trial $\theta’$,

\eqalign{ \Delta\theta' =& -[(-50^\circ)+\sin(-50^\circ)-\pi\sin(-50^\circ)]/ \cr & [1+\cos(-50^\circ)]\times 180^\circ/\pi \cr =& -26.7818469^\circ }
The next trial $\theta' = -50^\circ-26.7818469^\circ = -76.7818469^\circ$ . Using this in place of $-50^\circ$ for $\theta’$ (not $\phi$) in equation (31-4), subsequent iterations produce the following:
\eqalign{\Delta\theta' &= -4.3367097^\circ\cr \theta' &= -81.1185566^\circ \cr\Delta\theta' &= -0.1391597^\circ\cr \theta' &= -81.2577163^\circ \cr\Delta\theta' &= -0.000145^\circ\cr \theta' &= -81.2578612^\circ \cr\Delta\theta' &= 0^\circ}

Since there is no change to seven decimal places, using (31-5),

\eqalign{ \theta &= -81.2578612^\circ/2 \cr &= -40.6289306^\circ }
Using (31-1) and (31-2),
\eqalign{ x =& (8^{1/2}/\pi)\times1.0\times[-75^\circ-(-90^\circ)]\cos(-40.6289306^\circ)\times\pi/180^\circ \cr =& 0.1788845\text{ units} }
\eqalign{ y &= 2^{1/2}\times1.0\times\sin(-40.6289306^\circ) \cr &= -0.9208758\text{ units} }

### Inverse Equations #

Inversing forward example:

Given: $R, \lambda_0$, for forward example

 $x=\;$ units $y=\;$ units
Find $\phi, \lambda$.

Using equations (31-6) through (31-8) in order,

\eqalign{ \theta &= \arcsin[-0.9208758/(2^{1/2}\times1.0)] \cr &= -40.6289311^\circ }
\eqalign{ \phi &= \arcsin\{[2\times(-40.6289311^\circ)\times\pi/180^\circ +\sin[2\times(-40.6289311^\circ)]]/\pi\} \cr &= -50.0000005^\circ }
\eqalign{ \lambda &= -90^\circ+\pi\times0.1788845/[8^{1/2}\cos(-40.6289311^\circ)]\times180^\circ/\pi \cr &= -74.999999^\circ }