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 | |
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),$$
\eqalign{
\phi =& \arcsin[\cos12.9082593^\circ\sin40^\circ+(-0.1694739)\cr
& \sin12.9082593^\circ\cos40^\circ/0.2291823]\cr
=& 29.9999988^\circ
}
$$
$$
\eqalign{
\lambda =& -100^\circ +\arctan[ -0.1542826\sin12.9082593^\circ/ \cr
& (0.2291823\cos40^\circ\cos12.9082593^\circ - (-0.1694739) \cr
& \sin40^\circ\sin12.9082593^\circ)] \cr
=& -100^\circ +\arctan(-0.0344653/0.1954624) \cr
=& -109.9999993^\circ
}
$$