Question

What is a zenithal map projection?
Present its complete classification
scheme.

Answer 1

Explanation:

Zenithal (azimuthal) projections

Zenithal projections (also known as azimuthal projections) are a class of projections in which the surface of projection is a plane. The native coordinate system is such that the polar axis is orthogonal to the plane of projection, whence the meridians are projected as equispaced rays emanating from a central point, and the parallels are mapped as concentric circles centered on the same point. The projection is therefore defined by R$\scriptstyle \theta$ and A$\scriptstyle \phi$ as in the following diagram:

\epsffile{figure1.eps}

All zenithal projections have

A$\scriptstyle \phi$ = $\displaystyle \phi$, (12)

whence

x = - R$\scriptstyle \theta$sin$\displaystyle \phi$, (13)

y = - R$\scriptstyle \theta$cos$\displaystyle \phi$. (14)

These equations may be inverted as follows

R$\scriptstyle \theta$ = $\displaystyle \sqrt{x^2 + y^2}$, (15)

$\displaystyle \phi$ = arg(- y, - x). (16)

Since

$\displaystyle {\frac{\partial A_\phi}{\partial \phi}}$ = 1, (17)

the requirement for conformality of zenithal projections is

$\displaystyle \left\vert\vphantom{ \frac{\partial R_\theta}{\partial \theta} }\right.$$\displaystyle {\frac{\partial R_\theta}{\partial \theta}}$ $\displaystyle \left.\vphantom{ \frac{\partial R_\theta}{\partial \theta} }\right\vert$ = $\displaystyle {\frac{R_\theta}{\cos \theta}}$. (18)

This differential equation has the general solution

R$\scriptstyle \theta$ $\displaystyle \propto$ tan$\displaystyle \left(\vphantom{ \frac{90 - \theta}{2} }\right.$$\displaystyle {\frac{90 - \theta}{2}}$ $\displaystyle \left.\vphantom{ \frac{90 - \theta}{2} }\right)$, (19)

and this is the form of R$\scriptstyle \theta$ for the stereographic projection.

Let an oblique coordinate system be denoted by ($ \phi{^\prime}$,$ \theta{^\prime}$), and let the coordinates of the pole of the native coordinate system in the oblique system be ($ \phi{^\prime}_{0}$,$ \theta{^\prime}_{0}$). The meridian of the oblique system defined by $ \phi{^\prime}$ = $ \phi{^\prime}_{0}$ will be projected as a straight line segment; suppose it overlies the native meridian of $ \phi$ = $ \phi_{0}^{}$ in the same sense of increasing or decreasing latitude (to distinguish it from the native meridian on the opposite side of the pole), then the Euler angles for the transformation from ($ \phi{^\prime}$,$ \theta{^\prime}$) to ($ \phi$,$ \theta$) are

($\displaystyle \Phi{^\prime}$,$\displaystyle \Theta{^\prime}$,$\displaystyle \Phi$) = ($\displaystyle \phi{^\prime}_{0}$ + 90o, 90o - $\displaystyle \theta{^\prime}_{0}$,$\displaystyle \phi_{0}^{}$ + 90o). (20)

Perspective zenithal projections

Let r0 be the radius of the generating sphere and let the distance of the origin of the projection from the centre of the generating sphere be $ \mu$r0. If the plane of projection intersects the generating sphere at latitude $ \theta_{x}^{}$ in the native coordinate system of the projection as in the following diagram

...diagram...

then it is straightforward to show that

R$\scriptstyle \theta$ = r0cos$\displaystyle \theta$$\displaystyle \left(\vphantom{

\frac{\mu + \sin \theta_x}{\mu + \sin \theta} }\right.$$\displaystyle {\frac{\mu + \sin \theta_x}{\mu + \sin \theta}}$ $\displaystyle \left.\vphantom{

\frac{\mu + \sin \theta_x}{\mu + \sin \theta} }\right)$. (21)

From this equation it can be seen that the effect of $ \theta_{x}^{}$ is to rescale r0 by ($ \mu$ + sin$ \theta_{x}^{}$)/$ \mu$, thereby uniformly scaling the projection as a whole. Consequently the projections presented in this section only consider $ \theta_{x}^{}$ = 90o.

The equation for R$\scriptstyle \theta$ is invertible as follows:

$\displaystyle \theta$ = arg($\displaystyle \rho$, 1) - sin-1$\displaystyle \left(\vphantom{

\frac{\rho \mu}{\sqrt{\rho^2 +1}} }\right.$$\displaystyle {\frac{\rho \mu}{\sqrt{\rho^2 +1}}}$ $\displaystyle \left.\vphantom{

\frac{\rho \mu}{\sqrt{\rho^2 +1}} }\right)$, (22)

where

$\displaystyle \rho$ = $\displaystyle {\frac{R_\theta}{r_0 ( \mu + \sin \theta_x )}}$. (23)

For |$ \mu$| $ \leq$ 1 the perspective zenithal projections diverge at latitude $ \theta$ = sin-1(- $ \mu$), while for |$ \mu$| > 1 the projection of the near and far sides of the generating sphere are superposed, with the overlap beginning at latitude $ \theta$ = sin-1(- 1/$ \mu$).

All perspective zenithal projections are conformal at latitude ( $ \theta$ = 90o). The projection with $ \mu$ = 1 (stereographic) is conformal at all points.