Numerical Examples for Eckert IV Projection #
SPHERE #
Forward Equations #
Given
| Radius of sphere: | $R=\;\;$ unit |
| Central meridian: | $\lambda_0=\;$° |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
From equation (32-4), using $(\phi/2)$ or $ -25^\circ $ as the first trial $\theta$
$$
\eqalign{
\Delta\theta =& -[(-25^\circ)\times\pi/180^\circ+\sin(-25^\circ)\cos(-25^\circ)+2\sin(-25^\circ) \cr
& -(2+\pi/2)\sin(-50^\circ)]/[2\cos(-25^\circ)\times(1-\cos(-25^\circ))] \cr
=& -17.7554344^\circ
}
$$
The next trial
$ \theta = (-25^\circ)+(-17.7554344^\circ) = -42.7554344^\circ $
. Using this in place of
$ -25^\circ $
for $\theta$ in equation (32-4), subsequent iterations produce the following:$$
\eqalign{\Delta\theta' &= -2.9912099^\circ\cr
\theta &= -45.7466443^\circ \cr\Delta\theta' &= -0.1113894^\circ\cr
\theta &= -45.8580337^\circ \cr\Delta\theta' &= -0.0001573^\circ\cr
\theta &= -45.858191^\circ \cr\Delta\theta' &= 0^\circ}
$$
Since there is no change to seven decimal places, $ \theta = -45.858191^\circ $ . Using (32-1a) and (32-2a),
$$
\eqalign{
x &= 0.4222382\times1.0\times[-75^\circ-(-90^\circ)]\times(\pi/180^\circ)\times[1+\cos(-45.858191^\circ)]\cr
&= 0.1875270\text{ units}
}
$$
$$
\eqalign{
y &= 1.3265004\times1.0\times\sin(-45.858191^\circ) \cr
&= -0.9519210\text{ units}
}
$$
Inverse Equations #
Inversing forward example:
Given: $R, \lambda_0$, for forward example
| $x=\;$ units |
| $y=\;$ units |
Using equations (32-9a), (32-10), and (32-11a) in order,
$$
\eqalign{
\theta &= \arcsin[-0.9519210/(1.3265004\times1.0)] \cr
&= -45.8581925^\circ
}
$$
$$
\eqalign{
\phi =& \arcsin\{[-45.8581925^\circ\times\pi/180^\circ+\sin(-45.8581925^\circ)\cos(-45.8581925^\circ)\cr
& +2\sin(-45.8581925^\circ)]/(2+\pi/2)\} \cr
=& -50.0000015^\circ
}
$$
$$
\eqalign{
\lambda =& -90^\circ+0.1875270/\{0.4222382\times1.0\times[1+\cos(-45.8581925^\circ)]\}\cr
=& -74.9999993^\circ
}
$$