Numerical Examples - Eckert VI Projection

# Numerical Examples for Eckert VI Projection #

## SPHERE #

### Forward Equations #

Given

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

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
Find $\phi, \lambda$.

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 }