Numerical Examples for Cassini Projection #
SPHERE #
Forward Equations #
Given
| Radius of sphere: | $R=\;\;$ unit |
| Origin: | $\phi_0=\;$° |
| $\lambda_0=\;$° | |
| Point: | $\phi=\;$° |
| $\lambda=\;$° | |
Find $x, y$ Using equations (8-5), and (13-1) through (13-3) in order,
$$
\eqalign{
B &= \cos25^\circ\sin[(-90^\circ)-(-75^\circ)] \cr
&= -0.2345697
}
$$
$$
\eqalign{
x &= 1.0\times \arcsin(-0.2345697) \times \pi/180^\circ \cr
&= -0.2367759\;\text{units}
}
$$
$$
\eqalign{
y &= 1.0\times \{\arctan[\tan 25^\circ/\cos[(-90^\circ)-(-75^\circ)]]-(-20^\circ)\}\times\pi/180^\circ \cr
&= 1.0\times 45.7692621^\circ\times\pi/180^\circ = 0.7988243\;\text{units}
}
$$
$$
\eqalign{
h' &= 1/[1-(-0.2345697)^2]^{1/2} \cr
&= 1.0287015
}
$$
Inverse Equations #
Inversing forward example: Given: $R, \phi_0, \lambda_0$, for forward example
| $x=\;$ units |
| $y=\;$ units |
Using equations (13-6), (13-4), and (13-5) in order,
$$
\eqalign{
D &=(0.7988243/1.0)\times 180^\circ/\pi+(-20^\circ) \cr
&= 25.7692610^\circ
}
$$
$$
\eqalign{
\phi &=\arcsin\{\sin 25.7692610^\circ\cos[((-0.2367759)/1.0)\times 180^\circ/\pi] \} \cr
&=\arcsin 0.4226182 \cr
&= 24.9999989^\circ
}
$$
$$
\eqalign{
\lambda &= -75^\circ+\arctan\{\tan[(-0.2367759/1.0)\times 180^\circ/\pi]\} \cr
&= -75^\circ + \arctan (-0.2679492) \cr
&= -75^\circ + (-14.9999992^\circ) = -89.9999992^\circ
}
$$
ELLIPSOID #
Forward equations #
Given:
| ellipsoid | $a=6378206.4\,\text{m}$ |
| $e^2=0.00676866$ | |
| or | $e=0.0822719$ |
| Origin: | $\phi_0=\;$° |
| $\lambda_0=\;$° | |
| Point: | $\phi=$° | $\lambda=$° |
Using equations (4-20), (8-13), (8-15), (8-14), and (3-21) in order,
$$
\eqalign{
N &= 6378206.4/(1-0.0067687\times \sin^243^\circ)^{1/2} \cr
&= 6388270.27\;\text{m}
}
$$
$$
T = \tan^2 43^\circ = 0.8695844
$$
$$
\eqalign{
A &= [(-73^\circ)-(-75^\circ)]\times(\pi/180^\circ)\times\cos43^\circ\cr
&= 0.0255291
}
$$
$$
\eqalign{
C &= 0.0067687\times\cos^2 43^\circ/(1-0.0067687) \cr
&= 0.0036451
}
$$
$$
\begin{align}
M =& 6378206.4\times[(1-0.0067687/4-3\times 0.0067687^2/64 \cr
&- 5\times 0.0067687^3/256)\times 43^\circ\times\pi/180^\circ \cr
&-(3\times 0.0067687/8+3\times 0.0067687^2/32 +45\times0.0067687^3/1024)\sin(2\times43^\circ) \cr
&+(15\times 0.0067687^2/256 +45\times 0.0067687^3/1024)\sin(4\times 43^\circ) \cr
&-(35\times 0.0067687^3/3072)\sin(6\times43^\circ)] \cr
=& 4762504.81\;\text{m}
\end{align}
$$
Substituting $40^\circ$ for $43^\circ$ in equation (3-21),$$
M_0 = 4429318.90\;\text{m}
$$
Using equations (13-7) through (13-9) in order$$
\eqalign{
x =& 6388270.27\times[0.0255291-0.8695844\times0.0255291^3/6 \cr
& -(8-0.8695844+8\times0.0036451)\times0.8695844\cr
& \times0.0255291^5/120] \cr
=& 163071.13\;\text{m}
}
$$
$$
\eqalign{
y =& 4762504.81 - 4429318.90 + 6388270.27\times \tan 43^\circ \cr
& \times[0.0255291^2/2+(5-0.8695844+6\times0.0036451) \cr
& \times 0.0255291^4/24] \cr
=& 335127.59\;\text{m}
}
$$
$$
\eqalign{
s =& 1+ 163071.13^2\cos^230^\circ\times(1-0.0067687\times\sin^243^\circ)^2/ \cr
& [2\times 6378206.4^2 \times(1-0.0067687)] \cr
=& 1.0002452
}
$$
Inverse Equations #
Inversing forward example:
Given: $a, e, \phi_0, \lambda_0$ for forward example,
| $x=\;$m |
| $y=\;$m |
Calculating $M_0$ from equation (3-21) as in the forward example for $\phi_0 = 40^\circ$
$$
M_0 = 4429318.90\;\text{m}
$$
Using equations (13-12), (7-19), and (3-24) in order,$$
\eqalign{
M_1 &= 4429318.90 + 335127.59 \cr
&= 4764446.49\;\text{m}
}
$$
$$
\eqalign{
\mu_1 =& 4764446.49/[6378206.40\times (1-0.0067687/4 \cr
& -3\times0.0067687^2/64 -5\times0.0067687^3/256)] \cr
=& 0.7482562\;\text{radians} = 42.8719240^\circ
}
$$
$$
\eqalign{
e_1 &= [1-(1-0.0067687)^{1/2}]/[1-(1-0.0067687)^{1/2}] \cr
&= 0.001697916
}
$$
Using equations (3-26), (8-22), (8-23), (8-24), and (13-13) in order,$$
\eqalign{
\phi_1 =& 42.8719240^\circ + \cr
&[(3\times0.001697916/2-27\times0.001697916^3/32)\sin(2\times42.8719240^\circ) \cr
&+(21\times0.001697916^2/16-55\times0.001697916^4/32)\sin(4\times42.8719240^\circ) \cr
&+(151\times0.001697916^3/96)\sin(6\times42.8719240^\circ) \cr
&+(1097\times0.001697916^4/512)\sin(8\times42.8719240^\circ)]\times 180^\circ/\pi \cr
=& 43.0174782^\circ
}
$$
$$
\eqalign {
T_1 &= \tan^243.0174782^\circ \cr
&= 0.8706487
}
$$
$$
\eqalign{
N_1 &= 6378206.4/(1-0.0067687\sin^243.0174782^\circ)^{1/2} \cr
&= 6388276.87\;\text{m}
}
$$
$$
\eqalign{
R_1 &= 6378206.40(1-0.0067687)/(1-0.0067687\sin^243.0174782^\circ)^{3/2} \cr
&= 6365088.80\;\text{m}
}
$$
$$
\eqalign{
D &= 163071.13/6388276.87 \cr
&= 0.0255266
}
$$
Using equations (13-10) and (13-11) in order,$$
\eqalign{
\phi =& 43.0174782^\circ-(6388276.87\tan43.0174782^\circ/6365088.80) \cr
& \times [0.0255266^2/2-(1+3\times0.8706487)\times0.0255266^4/24)]\times 180^\circ/\pi \cr
=& 42.9999951^\circ
}
$$
$$
\eqalign{
\lambda =& -75^\circ + \{[0.0255266-0.8706487\times0.0255266^3 \cr
& + (1+3\times0.8706487)\times0.8706487\times0.0255266^5/15]/ \cr
& \cos 43.0174782^\circ\}\times180^\circ/\pi \cr
=& -73^\circ
}
$$