Numerical Examples - Equidistant Conic Projection

# Numerical Examples for Equidistant Conic Projection #

## SPHERE #

### Forward Equations #

Given

 Radius of sphere: $R=\;\;$ unit Standard parallels: $\phi_1=\;$° $\phi_2=\;$° Origin: $\phi_0=\;$° $\lambda_0=\;$° Point: $\phi=\;$° $\lambda=\;$°

Find $\rho, \theta, x, y, k$

From equations (16-4), (16-3), (16-2), (16-1), and (14-4) in order

\eqalign{ n &= (\cos29.5^\circ-\cos45.5^\circ)/[(45.5^\circ-29.5^\circ)\times\pi/180^\circ] \cr &= 0.6067853 }
\eqalign{ G &= (\cos29.5^\circ)/0.6067853 + 29.5^\circ\times\pi/180^\circ \cr &= 1.9492438 }
\eqalign{ \rho_0 &= 1.0\times(1.9492438-23^\circ\times\pi/180^\circ) \cr &= 1.5478181\;\text{units} }
\eqalign{ \rho &= 1.0\times(1.9492438-35^\circ\times\pi/180^\circ) \cr &= 1.3383786\;\text{units} }
\eqalign{ \theta &= 0.6067853\times[-75^\circ - (-96^\circ)] \cr &= 12.7424921^\circ }
Using equations (14-1), (14-2), and (16-5) in order,
\eqalign{ x &= 1.3383786\sin12.7424921^\circ \cr &= 0.2952057\;\text{units} }
\eqalign{ y &= 1.5478181-1.3383786\cos12.7424921^\circ \cr &= 0.2424021\;\text{units} }
\eqalign{ k &= (1.9492438-35^\circ\times\pi/180^\circ)/\cos35^\circ \cr &= 0.9914014 }

### Inverse Equations #

Inversing forward example:

Given: $R, \phi1, \phi2, \phi_0, \lambda_0$, for forward example

 $x=\;$ units $y=\;$ units
Find $\rho, \theta, \phi, \lambda$.

Calculating $n, G,$ and $\rho_0$ as in the forward example,

\eqalign{ n &= 0.6067853 \cr G &= 1.9492438 \cr \rho_0 &= 1.5478181\;\text{units} }
Using equations (14-10) and (14-11) in order,
\eqalign{ \rho &= +[0.2952057^2+(1.5478181-0.2424021)^2]^{1/2} \cr &= 1.3383786\;\text{units,}\; \text{positive because $n$ is positive} }
\eqalign{ \theta &= \arctan[0.2952057/(1.5478181-0.2424021)] \cr &= 12.7424936^\circ }
Using equations (16-6) and (14-9) in order,
\eqalign{ \phi &= [1.9492438 - 1.3383786/1.0]\times 180^\circ/\pi \cr &= 34.9999981^\circ }
\eqalign{ \lambda &= (-96^\circ) + 12.7424936^\circ/0.6067853 \cr &= -74.9999975^\circ }

## ELLIPSOID #

### Forward Equations #

Given:

 Clarke 1866WGS-84 ellipsoid $a=$ 6378206.4 m $e^2=$ 0.00676866 Standard parallels: $\phi_1=$ ° $\phi_2=$ ° Origin: $\phi_0=$ ° $\lambda_0=$ ° Point: $\phi=$ ° $\lambda=$ °
Find $\rho, \theta, x, y, k$.

From equation (14-15) and (3-21),

\eqalign{ m &= \cos35^\circ/(1-0.0067687\sin^235^\circ)^{1/2} \cr &= 0.8200656 }
\eqalign{ M =&6378206.4[(1-0.0067687/4-3\times 0.0067687^2/64 - 5\times 0.0067687^3/256)\times 35^\circ\times\pi/180^\circ \\ &-(3\times 0.0067687/8+3\times 0.0067687^2/32 +45\times0.0067687^3/1024)\sin(2\times35^\circ) \\ &+(15\times 0.0067687^2/256 +45\times 0.0067687^3/1024)\sin(4\times 35^\circ) \\ &-(35\times 0.0067687^3/3072)\sin(6\times35^\circ)] \\ =& 3874395.25\;\text{m} }

Using the same equations, but with $\phi_1 = 29.5^\circ$ in place of $35^\circ$,

$$m_1=0.8710708$$
$$M_1=3264511.19\;\text{m}$$
Similarly, with $\phi_2 = 45.5^\circ$ in place of $35^\circ$,
$$m_2=0.7021191$$
$$M_2=5040295.01\;\text{m}$$
and with $\phi_0 = 23^\circ$ in place of $35^\circ$
$$M_0=2544389.74\;\text{m}$$

Using equations (16-10), (16-11), (16-9), (16-8), and (14-4) in order,

\eqalign{ n &= 6378206.4\times(0.8710708-0.7021191)/(5040295.01-3264511.19) \cr &= 0.6068355 }
\eqalign{ G &= 0.8710708/0.6068355+ 3264511.19/6378206.4 \cr &= 1.9472543 }
\eqalign{ \rho_0 &= 6378206.4\times 1.9472543 - 2544389.74 \cr &= 9875600.03\;\text{m} }
\eqalign{ \rho &= 6378206.4\times 1.9472543 - 3874395.25 \cr &= 8545594.52\;\text{m} }
\eqalign{ \theta &= 0.6068355\times[-75^\circ-(-96^\circ)] \cr &= 12.7435456^\circ }

Constants $n, G$, and $\rho_0$ apply to the entire map. Using equations (14-1), (14-2), and (16-7) in order,

\eqalign{ x &= 8545594.52\times \sin 12.7435456^\circ \cr &= 1885051.86\;\text{m} }
\eqalign{ y &= 9875600.03 - 8545594.52\times\cos12.7435456^\circ \cr &= 1540507.64\;\text{m} }
\eqalign{ k =& (1.9472543-3874395.25/6378206.4) \times 0.6068355/0.8200656 \cr =& 0.9914392 }

### Inverse Equations #

Inversing forward example:

Given

 $x=\;$m $y=\;$m
Calculating $n, G$, and $\rho_0$ as in the forward example,
\eqalign{ n &= 0.6068355 \cr G &= 1.9472543 \cr \rho_0 &= 9875600.03\;\text{m} }
Using equations (14-10), (14-11), (16-12), (7-19), (3-24), and (3-26) in order,
\eqalign{ \rho &= +[1885051.86^2+(9875600.03-1540507.64)^2]^{1/2} \cr &= 8545594.52\;\text{m}\; \text{positive because $n$ is positive} }
\eqalign{ \theta &= \arctan[1885051.86/(9875600.03-1540507.64)] \cr &= 12.7435457^\circ }
\eqalign{ M &= 6378206.4\times 1.9472543 - 8545594.52 \cr &= 3874395.25\;\text{m} }
\eqalign{ \mu =& 3874395.25 /[6378206.4\times(1-0.0067687/4 \cr & -3\times0.0067687^2/64 - 5\times0.0067687^3/256)] \cr =& 0.6084737 \;\text{radians} = 34.8629750^\circ }
\eqalign{ e_1 &= [1-(1-0.0067687)^{1/2}]/[1+(1-0.0067687)^{1/2}] \cr &= 0.001697916 }
\eqalign{ \phi =& 34.8629750^\circ+[(3\times0.0016979/2-27\times0.0016979^3/32)\sin(2\times34.8629750^\circ) \cr &+(21\times0.0016979^2/16-55\times0.0016979^4/32)\sin(4\times34.8629750^\circ) \cr &+(151\times0.0016979^3/96)\sin(6\times34.8629750^\circ) \cr &+(1097\times0.0016979^4/512)\sin(8\times34.8629750^\circ)]\times180^\circ/\pi \cr =&35.0000000^\circ }

Using equation (14-9),

\eqalign{ \lambda &= -96^\circ + 12.7435457^\circ/0.6068355 \cr &= -75.0000000^\circ }