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Subsections

• One-standard conic projections
  • One-standard perspective conic projection
  • One-standard equidistant conic projection
  • One-standard equal area conic projection
  • One-standard orthomorphic conic projection
• Two-standard conic projections
  • Two-standard perspective conic projection
  • Two-standard equidistant conic projection
  • Two-standard equal area conic projection
  • Two-standard orthomorphic conic projection
• Poly-standard conic projections
  • Bonne's equal area projection
  • Polyconic projection

Conic projections

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

$$\theta \phi$$ x = (72)

$$\theta \phi$$ y = (73)

The requirement for conformality is

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

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

$${\frac{\partial A_\phi}{\partial \phi}}$$ (75)

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

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

$${\frac{\partial A_\phi}{\partial \phi}}$$ C = (77)

$${\frac{r_0 \cos \theta_1}{R_{\theta_1}}}$$ = (78)

$${\frac{r_0 \cos \theta_2}{R_{\theta_2}}}$$ = (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 $\theta_{1}^{}$ = 45^o.

One-standard perspective conic projection

$$\theta \theta_{1}^{} \theta \theta_{1}^{}$$ = (80)

$$\phi \phi \theta_{1}^{}$$ = (81)

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

One-standard equidistant conic projection

The spacing between parallels is true.

$$\theta \theta_{1}^{} \theta \theta_{1}^{} {\frac{\pi}{180}}$$ = (82)

$$\phi \phi \theta_{1}^{}$$ = (83)

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

One-standard equal area conic projection

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

$$\theta \left(\vphantom{ \cot^2 \theta_1 + 2 - 2 \frac{\sin \theta}{\sin \theta_1} }\right. \theta_{1}^{} {\frac{\sin \theta}{\sin \theta_1}} \left.\vphantom{ \cot^2 \theta_1 + 2 - 2 \frac{\sin \theta}{\sin \theta_1} }\right)^{\frac{1}{2}}_{}$$ = (84)

$$\phi \phi \theta_{1}^{}$$ = (85)

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

One-standard orthomorphic conic projection

$$\theta \theta_{1}^{} \left[\vphantom{ \frac{\tan \left ( \frac{90 - \theta}{2} \right )} {\tan \left ( \frac{90 - \theta_1}{2} \right )} }\right. {\frac{\tan \left ( \frac{90 - \theta}{2} \right )}{\tan \left ( \frac{90 - \theta_1}{2} \right )}} \left.\vphantom{ \frac{\tan \left ( \frac{90 - \theta}{2} \right )} {\tan \left ( \frac{90 - \theta_1}{2} \right )} }\right]^{\sin \theta_1}_{}$$ = (86)

$$\phi \phi \theta_{1}^{}$$ = (87)

Limits: diverges at latitude $\theta$ = - 90^o.
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 $\theta_{1}^{}$ = 30^o and $\theta_{2}^{}$ = 60^o.

Two-standard perspective conic projection

$$\theta \sqrt{1 + \alpha^2} {\frac{\sin \theta_1 + \alpha \cos \theta_1}{\alpha + \tan \theta}}$$ = (88)

$$\phi \phi \alpha^{2}_{} {\textstyle\frac{1}{2}}$$ = (89)

where

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

Note that

$$\theta_{1}^{} \alpha \theta_{1}^{} \theta_{2}^{} \alpha \theta_{2}^{}$$ = (91)

$${\frac{\sin (\theta_2 - \theta_1)}{\cos \theta_1 - \cos \theta_2}}$$ = (92)

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

Two-standard equidistant conic projection

The spacing between parallels is true.

$$\theta {\frac{\pi}{180}} \left[\vphantom{ \frac{1}{\alpha} ( \theta_2 \cos \theta_1 - \theta_1 \cos \theta_2 ) - \theta }\right. {\frac{1}{\alpha}} \theta_{2}^{} \theta_{1}^{} \theta_{1}^{} \theta_{2}^{} \theta \left.\vphantom{ \frac{1}{\alpha} ( \theta_2 \cos \theta_1 - \theta_1 \cos \theta_2 ) - \theta }\right]$$ = (93)

$$\phi \phi \alpha {\frac{\pi}{180}} \theta_{2}^{} \theta_{1}^{}$$ = (94)

where

$$\alpha \theta_{1}^{} \theta_{2}^{}$$ (95)

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

$$\theta \sqrt{1 + \sin \theta_1 \sin \theta_2 - \alpha \sin \theta} \alpha$$ = (96)

$$\phi \phi \alpha$$ = (97)

where

$$\alpha \theta_{1}^{} \theta_{2}^{}$$ (98)

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

Two-standard orthomorphic conic projection

$$\theta \alpha \left[\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right ) }\right. \left(\vphantom{ \frac{90 - \theta}{2} }\right. {\frac{90 - \theta}{2}} \left.\vphantom{ \frac{90 - \theta}{2} }\right) \left.\vphantom{ \tan \left ( \frac{90 - \theta}{2} \right ) }\right]^{C}_{}$$ = (99)

$$\phi \phi$$ = (100)

where

$$\alpha {\frac{\cos \theta_1}{C \left [ \tan \left ( \frac{90 - \theta_1}{2} \right ) \right ]^C}}$$ = (101)

$${\frac{\cos \theta_2}{C \left [ \tan \left ( \frac{90 - \theta_2}{2} \right ) \right ]^C}}$$ = (102)

$${\frac{\ln \left ( \frac{\cos \theta_2}{\cos \theta_1} \right )}{... ...heta_2}{2} \right )} {\tan \left ( \frac{90 - \theta_1}{2} \right )} \right ]}}$$ C = (103)

Limits: diverges at latitude $\theta$ = - 90^o.
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 $\theta_{1}^{}$ = 45^o.

$$\theta \theta_{1}^{} \theta \theta_{1}^{} \pi$$ = (104)

$$\phi \phi \theta \theta$$ = (105)

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.

$$\theta \phi \theta$$ x = (106)

$$\theta \phi \theta \theta \left(\vphantom{ \frac{\pi}{180} }\right. {\frac{\pi}{180}} \left.\vphantom{ \frac{\pi}{180} }\right)$$ y = (107)

Limits: none.
Conformal along the central meridian.