Numerical Examples for Mollweide Projection #
SPHERE #
Forward Equations #
Given
| Radius of sphere: | $R=\;\;$ unit |
| Central meridian: | $\lambda_0=\;$° |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
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 |
Using equations (31-6) through (31-8) in order,
$$
\eqalign{
\theta &= \arcsin[-0.9208758/(2^{1/2}\times1.0)] \cr
&= -40.6289311^\circ
}
$$
$$
\eqalign{
\phi &= \arcsin\{[2\times(-40.6289311^\circ)\times\pi/180^\circ +\sin[2\times(-40.6289311^\circ)]]/\pi\} \cr
&= -50.0000005^\circ
}
$$
$$
\eqalign{
\lambda &= -90^\circ+\pi\times0.1788845/[8^{1/2}\cos(-40.6289311^\circ)]\times180^\circ/\pi \cr
&= -74.999999^\circ
}
$$