Numerical Examples - Eckert IV Projection

# Numerical Examples for Eckert IV Projection #

## SPHERE #

### Forward Equations #

Given

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

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
Find $\phi, \lambda$.

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 }