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Next: Conventional projections Up: No Title Previous: Cylindrical projections

Subsections


Conic projections

Parallels are projected as arcs of circles. Projections for which the parallels are concentric may be described by R$\scriptstyle \theta$ and A$\scriptstyle \phi$, where R$\scriptstyle \theta$ is the radius of the arc for latitude $ \theta$. Then,


x = - R$\scriptstyle \theta$sinA$\scriptstyle \phi$, (72)
y = - R$\scriptstyle \theta$cosA$\scriptstyle \phi$. (73)

The requirement for conformality is

$\displaystyle \left\vert\vphantom{ \frac{\partial R_\theta}{\partial \theta} }\right.$$\displaystyle {\frac{\partial R_\theta}{\partial \theta}}$ $\displaystyle \left.\vphantom{ \frac{\partial R_\theta}{\partial \theta} }\right\vert$ = $\displaystyle {\frac{R_\theta}{\cos \theta}}$$\displaystyle \left\vert\vphantom{ \frac{\partial A_\phi}{\partial \phi} }\right.$$\displaystyle {\frac{\partial A_\phi}{\partial \phi}}$ $\displaystyle \left.\vphantom{ \frac{\partial A_\phi}{\partial \phi} }\right\vert$. (74)

In the special case satisfied by one-, and two-standard conic projections

$\displaystyle {\frac{\partial A_\phi}{\partial \phi}}$ = C, (75)

a constant known as the constant of the cone, this differential equation has the general solution

R$\scriptstyle \theta$ = $\displaystyle \kappa$$\displaystyle \left[\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right )
}\right.$tan$\displaystyle \left(\vphantom{ \frac{90 - \theta}{2} }\right.$$\displaystyle {\frac{90 - \theta}{2}}$ $\displaystyle \left.\vphantom{ \frac{90 - \theta}{2} }\right)$ $\displaystyle \left.\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right )
}\right]^{C}_{}$ (76)

where $ \kappa$ is a constant. This solution is used to construct orthomorphic conic projections. The apical angle of the projected cone is 2$ \pi$C where


C = $\displaystyle {\frac{\partial A_\phi}{\partial \phi}}$ (77)
  = $\displaystyle {\frac{r_0 \cos \theta_1}{R_{\theta_1}}}$ (78)
  = $\displaystyle {\frac{r_0 \cos \theta_2}{R_{\theta_2}}}$ (79)

where $ \theta_{1}^{}$ (and $ \theta_{2}^{}$) is the latitude of the standard parallel(s).

One-standard conic projections

The standard parallel is at latitude $ \theta_{1}^{}$. All of the diagrams presented here have been computed for $ \theta_{1}^{}$ = 45o.

One-standard perspective conic projection


R$\scriptstyle \theta$ = r0[cot$\displaystyle \theta_{1}^{}$ - tan($\displaystyle \theta$ - $\displaystyle \theta_{1}^{}$)] (80)
A$\scriptstyle \phi$ = $\displaystyle \phi$sin$\displaystyle \theta_{1}^{}$ (81)

\epsffile{OneConicPersp.eps}

Limits: diverges at latitude $ \theta$ = $ \theta_{1}^{}$ - 90o.
Conformal at latitude $ \theta_{1}^{}$.
Constant of the cone: C = sin$ \theta_{1}^{}$.

One-standard equidistant conic projection

The spacing between parallels is true.


R$\scriptstyle \theta$ = r0[cot$\displaystyle \theta_{1}^{}$ - ($\displaystyle \theta$ - $\displaystyle \theta_{1}^{}$)$\displaystyle {\frac{\pi}{180}}$] (82)
A$\scriptstyle \phi$ = $\displaystyle \phi$sin$\displaystyle \theta_{1}^{}$ (83)

\epsffile{OneConicEquid.eps}

Limits: none.
Conformal at latitude $ \theta_{1}^{}$.
Constant of the cone: C = sin$ \theta_{1}^{}$.

One-standard equal area conic projection

R$\scriptstyle \theta$ is defined so that the area between any two parallels on the projection is true.


R$\scriptstyle \theta$ = r0$\displaystyle \left(\vphantom{ \cot^2 \theta_1 + 2 -
2 \frac{\sin \theta}{\sin \theta_1} }\right.$cot2$\displaystyle \theta_{1}^{}$ + 2 - 2$\displaystyle {\frac{\sin \theta}{\sin \theta_1}}$ $\displaystyle \left.\vphantom{ \cot^2 \theta_1 + 2 -
2 \frac{\sin \theta}{\sin \theta_1} }\right)^{\frac{1}{2}}_{}$ (84)
A$\scriptstyle \phi$ = $\displaystyle \phi$sin$\displaystyle \theta_{1}^{}$ (85)

\epsffile{OneConicEqArea.eps}

Limits: none.
Conformal at latitude $ \theta_{1}^{}$.
Constant of the cone: C = sin$ \theta_{1}^{}$.

One-standard orthomorphic conic projection


R$\scriptstyle \theta$ = r0cot$\displaystyle \theta_{1}^{}$$\displaystyle \left[\vphantom{
\frac{\tan \left ( \frac{90 - \theta}{2} \right )}
{\tan \left ( \frac{90 - \theta_1}{2} \right )}
}\right.$$\displaystyle {\frac{\tan \left ( \frac{90 - \theta}{2} \right )}{\tan \left ( \frac{90 - \theta_1}{2} \right )}}$ $\displaystyle \left.\vphantom{
\frac{\tan \left ( \frac{90 - \theta}{2} \right )}
{\tan \left ( \frac{90 - \theta_1}{2} \right )}
}\right]^{\sin \theta_1}_{}$ (86)
A$\scriptstyle \phi$ = $\displaystyle \phi$sin$\displaystyle \theta_{1}^{}$ (87)

\epsffile{OneConicOrtho.eps}

Limits: diverges at latitude $ \theta$ = - 90o.
Conformal at all points.
Constant of the cone: C = sin$ \theta_{1}^{}$.

Two-standard conic projections

The standard parallels are at latitudes $ \theta_{1}^{}$ and $ \theta_{2}^{}$, $ \theta_{1}^{}$ < $ \theta_{2}^{}$. All of the diagrams presented here have been computed for $ \theta_{1}^{}$ = 30o and $ \theta_{2}^{}$ = 60o.

Two-standard perspective conic projection


R$\scriptstyle \theta$ = r0$\displaystyle \sqrt{1 + \alpha^2}$$\displaystyle {\frac{\sin \theta_1 + \alpha \cos \theta_1}{\alpha + \tan \theta}}$ (88)
A$\scriptstyle \phi$ = $\displaystyle \phi$(1 + $\displaystyle \alpha^{2}_{}$)- $\scriptstyle {\textstyle\frac{1}{2}}$ (89)

where

$\displaystyle \alpha$ = $\displaystyle {\frac{\sin \theta_2 - \sin \theta_1}{\cos \theta_1 - \cos \theta_2}}$ (90)

Note that


sin$\displaystyle \theta_{1}^{}$ + $\displaystyle \alpha$cos$\displaystyle \theta_{1}^{}$ = sin$\displaystyle \theta_{2}^{}$ + $\displaystyle \alpha$cos$\displaystyle \theta_{2}^{}$ (91)
  = $\displaystyle {\frac{\sin (\theta_2 - \theta_1)}{\cos \theta_1 - \cos \theta_2}}$ (92)

\epsffile{TwoConicPersp.eps}

Limits: diverges at latitude $ \theta$ = tan-1(- $ \alpha$).
Conformal at latitude $ \theta$ = tan-1(1/$ \alpha$).
Constant of the cone: C = (1 + $ \alpha^{2}_{}$)- $\scriptstyle {\frac{1}{2}}$.

Two-standard equidistant conic projection

The spacing between parallels is true.


R$\scriptstyle \theta$ = r0$\displaystyle {\frac{\pi}{180}}$$\displaystyle \left[\vphantom{ \frac{1}{\alpha}
( \theta_2 \cos \theta_1 - \theta_1 \cos \theta_2 )
- \theta }\right.$$\displaystyle {\frac{1}{\alpha}}$($\displaystyle \theta_{2}^{}$cos$\displaystyle \theta_{1}^{}$ - $\displaystyle \theta_{1}^{}$cos$\displaystyle \theta_{2}^{}$) - $\displaystyle \theta$ $\displaystyle \left.\vphantom{ \frac{1}{\alpha}
( \theta_2 \cos \theta_1 - \theta_1 \cos \theta_2 )
- \theta }\right]$ (93)
A$\scriptstyle \phi$ = $\displaystyle \phi$$\displaystyle \alpha$/[$\displaystyle {\frac{\pi}{180}}$($\displaystyle \theta_{2}^{}$ - $\displaystyle \theta_{1}^{}$)] (94)

where

$\displaystyle \alpha$ = cos$\displaystyle \theta_{1}^{}$ - cos$\displaystyle \theta_{2}^{}$ (95)

\epsffile{TwoConicEquid.eps}

Limits: none.
Conformal at latitudes $ \theta_{1}^{}$ and $ \theta_{2}^{}$.
Constant of the cone: $ \alpha$/[$ {\frac{\pi}{180}}$($ \theta_{2}^{}$ - $ \theta_{1}^{}$)].

Two-standard equal area conic projection


R$\scriptstyle \theta$ = 2r0$\displaystyle \sqrt{1 + \sin \theta_1 \sin \theta_2 -
\alpha \sin \theta}$/$\displaystyle \alpha$ (96)
A$\scriptstyle \phi$ = $\displaystyle \phi$$\displaystyle \alpha$/2 (97)

where

$\displaystyle \alpha$ = sin$\displaystyle \theta_{1}^{}$ + sin$\displaystyle \theta_{2}^{}$ (98)

\epsffile{TwoConicEqArea.eps}

Limits: none.
Conformal at latitudes $ \theta_{1}^{}$ and $ \theta_{2}^{}$.
Constant of the cone: $ \alpha$/2.

Two-standard orthomorphic conic projection


R$\scriptstyle \theta$ = r0$\displaystyle \alpha$$\displaystyle \left[\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right )
}\right.$tan$\displaystyle \left(\vphantom{ \frac{90 - \theta}{2} }\right.$$\displaystyle {\frac{90 - \theta}{2}}$ $\displaystyle \left.\vphantom{ \frac{90 - \theta}{2} }\right)$ $\displaystyle \left.\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right )
}\right]^{C}_{}$ (99)
A$\scriptstyle \phi$ = $\displaystyle \phi$C (100)

where


$\displaystyle \alpha$ = $\displaystyle {\frac{\cos \theta_1}{C \left [
\tan \left ( \frac{90 - \theta_1}{2} \right )
\right ]^C}}$ (101)
  = $\displaystyle {\frac{\cos \theta_2}{C \left [
\tan \left ( \frac{90 - \theta_2}{2} \right )
\right ]^C}}$ (102)
C = $\displaystyle {\frac{\ln \left ( \frac{\cos \theta_2}{\cos \theta_1} \right )}{...
...heta_2}{2} \right )}
{\tan \left ( \frac{90 - \theta_1}{2} \right )}
\right ]}}$ (103)

\epsffile{TwoConicOrtho.eps}

Limits: diverges at latitude $ \theta$ = - 90o.
Conformal at all points.

Poly-standard conic projections

Bonne's equal area projection

Parallels are concentric equidistant arcs of circles of true length. The diagram presented here has been computed for $ \theta_{1}^{}$ = 45o.


R$\scriptstyle \theta$ = r0(cot$\displaystyle \theta_{1}^{}$ - ($\displaystyle \theta$ - $\displaystyle \theta_{1}^{}$)$\displaystyle \pi$/180) (104)
A$\scriptstyle \phi$ = r0$\displaystyle \phi$cos$\displaystyle \theta$/R$\scriptstyle \theta$ (105)

\epsffile{Bonne.eps}

Limits: none.
Conformal at latitude $ \theta_{1}^{}$ and along the central meridian.

Polyconic projection

Every parallel is projected as standard, i.e. as arcs of circles of radius rcot$ \theta$ at their true length, 2$ \pi$rcos$ \theta$, and correctly divided. The scale along the central meridian is true, consequently the parallels are not concentric.


x = r0cot$\displaystyle \theta$sin($\displaystyle \phi$sin$\displaystyle \theta$) (106)
y = r0{cot$\displaystyle \theta$[1 - cos($\displaystyle \phi$sin$\displaystyle \theta$)] + $\displaystyle \theta$$\displaystyle \left(\vphantom{ \frac{\pi}{180} }\right.$$\displaystyle {\frac{\pi}{180}}$ $\displaystyle \left.\vphantom{ \frac{\pi}{180} }\right)$} (107)

\epsffile{Polyconic.eps}

Limits: none.
Conformal along the central meridian.


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Next: Conventional projections Up: No Title Previous: Cylindrical projections
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2006-10-15