Numerical Examples for Bipolar Oblique Conic Conformal Projection #
SPHERE #
Forward Equations #
This example will illustrate equations (17-14) through (17-23), assuming prior calculation of the constants from equations (17-1) through (17-13). Given
| Radius of sphere: | $R=\;\;$ m |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
Find $x, y, k$
$$ $$ From equations (17-14) and (17-15)
$$
\eqalign{
z_B &= \arccos\{\sin 45^\circ\sin 40^\circ+\cos 45^\circ\cos 40^\circ\cos[(-19.9933333^\circ)-(-90^\circ)]\} \cr
&= 50.2287516^\circ
}
$$
$$
\eqalign{
Az_B &= \arctan\lbrace\sin[(-19.9933333^\circ)-(-90^\circ)]/[\cos 45^\circ\tan40^\circ-\sin 45^\circ\cos((-19.9933333^\circ)-(-90^\circ))]\rbrace \cr
&= 69.4885512^\circ
}
$$
Since $69.48856^\circ$ is less than $104.42834^\circ$, proceed to equation (17-16).
$$
\eqalign{
\rho_B &= 1.8972474\times6370997.0\tan^{0.6305584}(½\times 50.2287516^\circ) \cr
&= 7496092.0\;\text{m}
}
$$
$$
\eqalign{
k &= 7496092.0\times0.6305584/(6370997.0\sin 50.2287516^\circ) \cr
&= 0.9652723
}
$$
$$
\eqalign{
\alpha &= \arccos\{ [\tan^{0.6305584}(½\times 50.2287516^\circ) + \tan^{0.6305584}½(104^\circ - 50.2287516^\circ)]/1.2724658 \} \cr
&= 1.8750582^\circ
}
$$
$$
n(Az_BA-Az_B) = 0.6305584\times(104.4283332^\circ-69.4885512^\circ) = 22.0315747^\circ
$$
This is greater than $\alpha$, so $\rho’_B = \rho_B$.
$$
\eqalign{
x' &= 7496092.0\sin[0.6305584(104.4283332^\circ-69.4885512^\circ)] \cr
&= 2811915.2\;\text{m}
}
$$
$$
\eqalign{
y' =& 7496092.0\cos[0.6305584(104.4283332^\circ-69.4885512^\circ)] \cr
&- 1.2070912 \times 6370997.0 \cr
=& -741667.6\;\text{m}
}
$$
From equations (17-32) and (17-33),
$$
\eqalign{
x &= -2811915.2\cos 45.8199707^\circ -(-741667.6)\sin 45.8199707^\circ \cr
&= -1427776.8\;\text{m}
}
$$
$$
\eqalign{
y &= -(-741667.6)\cos 45.8199707^\circ + 2811915.2\sin 45.8199707^\circ \cr
&= 2533454.4\;\text{m}
}
$$
Inverse Equations #
Inversing forward example:
Given: $R$, for forward example
| $x=\;$ m |
| $y=\;$ m |
From equations (17-34) and (17-35),
$$
$$
$$
$$
Since $x’$ is positive, go to equations (17-36) through (17-44) in order:$$
$$
$$
$$
$$
$$
$$
$$
$$
$$
Since $Az’_B$, is greater than $\alpha$, go to equation (17-42).$$
$$
$$
$$
$$
$$