Bipolar Oblique Conic Conformal Projection

# 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).

From equations (17-16) through (17-22),

\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
Find $\phi, \lambda$.

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).