Numerical Examples - Cassini Projection

# 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
Find $\phi, \lambda$.

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:

 Clarke 1866WGS-84 ellipsoid $a=6378206.4\,\text{m}$ $e^2=0.00676866$ or $e=0.0822719$ Origin: $\phi_0=\;$° $\lambda_0=\;$° Point: $\phi=$° $\lambda=$°
Find: $x, y, s$ at $Az =$ °

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
Find $\phi, \lambda$.

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 }