Numerical Examples for Eckert VI Projection #
SPHERE #
Forward Equations #
Given
| Radius of sphere: | $R=\;\;$ unit |
| Central meridian: | $\lambda_0=\;$° |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
From equation (32-8), using $\phi$ or or $ -50^\circ $ as the first trial $\theta$
$$
\eqalign{
\Delta\theta =& -[(-50^\circ)\times\pi/180^\circ+\sin(-50^\circ)-(1+\pi/2)\sin(-50^\circ)]/ \cr
& [1+\cos(-50^\circ)] \cr
=& -11.5316184^\circ
}
$$
The next trial
$ \theta = (-50^\circ)+(-11.5316184^\circ) = -61.5316184^\circ $
. Using this in place of
$ -50^\circ $
for $\theta$ in equation (32-8), subsequent iterations produce the following:$$
\eqalign{\Delta\theta' &= -0.6337921^\circ\cr
\theta &= -62.1654105^\circ \cr\Delta\theta' &= -0.0021049^\circ\cr
\theta &= -62.1675154^\circ \cr\Delta\theta' &= 0^\circ}
$$
Since there is no change to seven decimal places, $ \theta = -62.1675154^\circ $ . Using (32-5) and (32-6),
$$
\eqalign{
x &= 1.0\times(-75^\circ-(-90^\circ))\times(\pi/180^\circ)\times[1+\cos(-62.1675154^\circ)]/(2+\pi)^{1/2} \cr
&= 0.1693623\text{ units}
}
$$
$$
\eqalign{
y &= 2\times1.0\times(-62.1675154^\circ)\times\pi/180^\circ/(2+\pi)^{1/2} \cr
&= -0.9570223\text{ units}
}
$$
Inverse Equations #
Inversing forward example:
Given: $R, \lambda_0$, for forward example
| $x=\;$ units |
| $y=\;$ units |
Using equations (32-12), (32-13), and (32- 14) in order,
$$
\eqalign{
\theta &= (2+\pi)^{1/2}\times(-0.9570223)/(2\times1.0) \cr
&= -62.1675178^\circ
}
$$
$$
\eqalign{
\phi &= \arcsin[((-62.1675178^\circ)\times\pi/180^\circ+ \sin(-62.1675178^\circ))/(1+\pi/2)] \cr
&= -50.0000021^\circ
}
$$
$$
\eqalign{
\lambda &= -90^\circ +(2+\pi)^{1/2}\times0.1693623/[1.0\times(1+\cos(-62.1675178^\circ))] \cr
&= -75.0000008^\circ
}
$$