Numerical Examples - Gnomonic Projection

Numerical Examples for Gnomonic Projection #

SPHERE #

Forward Equations #

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

Find $x, y$

Using equation (5-3),

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

Since $\cos c$ is positive (not zero or negative), the point is in view and may be plotted. Using equations (22-3) through (22-5) in order,

\eqalign{ k' &= 1/0.9747290 \cr &= 1.0259262 }
\eqalign{ x &= 1.0\times1.0259262\cos 30^\circ\sin[-110^\circ-(-100^\circ)] \cr &= -0.1542826\text{ units} }
\eqalign{ y =& 1.0\times1.0259262\times\{\cos40^\circ\sin30^\circ - \sin40^\circ \cr & \cos30^\circ\cos[-110^\circ-(-100^\circ)] \} \cr =& -0.1694739\text{ units} }

Examples of other forward equations are omitted, since the above equations are general.

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 (22-16),

\eqalign{ \rho &= [( -0.1542826)^2 + (-0.1694739)^2]^{1/2} \cr &= 0.2291823 \text{ units} }
\eqalign{ c &= \arctan(0.2291823/1.0) \cr &= 12.9082593^\circ }
Using equations (20-14) and (20-15),
\eqalign{ \phi =& \arcsin[\cos12.9082593^\circ\sin40^\circ+(-0.1694739)\cr & \sin12.9082593^\circ\cos40^\circ/0.2291823]\cr =& 29.9999988^\circ }
\eqalign{ \lambda =& -100^\circ +\arctan[ -0.1542826\sin12.9082593^\circ/ \cr & (0.2291823\cos40^\circ\cos12.9082593^\circ - (-0.1694739) \cr & \sin40^\circ\sin12.9082593^\circ)] \cr =& -100^\circ +\arctan(-0.0344653/0.1954624) \cr =& -109.9999993^\circ }