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 |
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
}
$$
ELLIPSOID #
Forward Equations #
Given:
| ellipsoid | $a=$ | 6378206.4 m |
| $e^2=$ | 0.00676866 | |
| Standard parallels: | $\phi_1=$ | ° |
| $\phi_2=$ | ° | |
| Origin: | $\phi_0=$ | ° |
| $\lambda_0=$ | ° | |
| Point: | $\phi=$ | ° | $\lambda=$ | ° |
From equation (14-15) and (3-21),
$$
\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
}
$$
Inverse Equations #
Inversing forward example:
Given
| $x=\;$m |
| $y=\;$m |
$$
\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
}
$$