Numerical Examples - Orthographic Projection

# Numerical Examples for Orthographic Projection #

## SPHERE #

### Forward Equations #

Given

 Radius of sphere: $R=\;\;$ units Center: $\phi_1=\;$ ° $\lambda_0=\;$ ° Point: $\phi=\;$ ° $\lambda=\;$ °

Find $x, y$

In general calculations, to determine whether this point is beyond viewing, using equation (5-3),

\eqalign{ \cos c &= \sin 40^\circ\sin30^\circ+\cos 40^\circ\cos 30^\circ\cos(-110^\circ-(-100^\circ)) \cr &= 0.9747290 }

Since this is positive, the point is within view.

Using equations (20-3) and (20-4),

\eqalign{ x &= 1.0\cos30^\circ\sin(-110^\circ-(-100^\circ)) \cr &= -0.1503837 }
\eqalign{ y &= 1.0[\cos40^\circ\sin30^\circ - \sin40^\circ\cos30^\circ\cos(-110^\circ-(-100^\circ))] \cr &= -0.1651911 }
Examples of other forward equations are omitted, since the formulas for the oblique aspect apply generally.

### Inverse Equations #

Inversing forward example: Given $R, \phi_1, \lambda_0$ for forward example

 Point: $x=\;$ units $y=\;$ units
Find $\phi, \lambda$

Using equations (20-18) and (20-19),

\eqalign{ \rho &= [(-0.1503837)^2 + (-0.1651911)]^{1/2} \cr &= 0.2233906 }
\eqalign{ c &= \arcsin(0.2233906/1.0) \cr &= 12.9082572^\circ }
Using equations (20-14) and (20-15),
\eqalign{ \phi =& \arcsin[\cos12.9082572^\circ\sin40^\circ + (-0.1651911)\sin12.9082572^\circ \cr & \cos40^\circ/0.2233906] \cr =& 30.0000004^\circ }
\eqalign{ \lambda =& -100^\circ + \arctan[-0.1503837\sin12.9082572^\circ/(0.2233906 \cr &\cos40^\circ\cos12.9082572^\circ - (-0.1651911)\sin40^\circ \sin12.9082572^\circ)] \cr =& -109.9999978^\circ }