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 and A as in the following diagram:

R	=	$\displaystyle \sqrt{x^2 + y^2}$ ,	(15)
$\displaystyle \phi$	=	arg(- y, - x).	(16)

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

(18)

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

(19)

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

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

R = 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 $\theta_{x}^{}$ = 90^o.

$\displaystyle \theta$ = 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)

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

(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 ( $\theta$ = 90^o). The projection with $\mu$ = 1 (stereographic) is conformal at all points.

The gnomonic projection

The perspective zenithal projection with $\mu$ = 0 is known as the gnomonic projection. Since the projection is made from the centre of the generating sphere great circles are projected as straight lines. Thus the shortest distance between two points on the sphere is represented as a straight line interval which, however, is not uniformly divided.

R	=	r₀cot $\displaystyle \theta$ ,	(24)
$\displaystyle \theta$	=	tan^-1 $\displaystyle \left(\vphantom{ \frac{r_0}{R_\theta} }\right.$ $\displaystyle {\frac{r_0}{R_\theta}}$ $\displaystyle \left.\vphantom{ \frac{r_0}{R_\theta} }\right)$ .	(25)

Limits: diverges at latitude $\theta$ = 0^o.
Conformal at latitude $\theta$ = 90^o.

The stereographic projection

The perspective zenithal projection with $\mu$ = 1 is known as the stereographic projection. It is conformal at all points and also has the property that all circles on the generating sphere are projected as circles. The relative ease of constructing oblique stereographic projections led to its use by Arab astronomers for constructing astrolabes.

R	=	2r₀tan $\displaystyle \left(\vphantom{ \frac{90 - \theta}{2} }\right.$ $\displaystyle {\frac{90 - \theta}{2}}$ $\displaystyle \left.\vphantom{ \frac{90 - \theta}{2} }\right)$ ,	(26)
$\displaystyle \theta$	=	90^o - 2tan^-1 $\displaystyle \left(\vphantom{ \frac{R_\theta}{2 r_0} }\right.$ $\displaystyle {\frac{R_\theta}{2 r_0}}$ $\displaystyle \left.\vphantom{ \frac{R_\theta}{2 r_0} }\right)$ .	(27)

The orthographic projection

The perspective zenithal projection with $\mu$ = $\infty$ is known as the orthographic projection. It gives a representation of the visual appearance of a sphere (for example a planet) when seen from a great distance.

R	=	r₀cos $\displaystyle \theta$ ,	(28)
$\displaystyle \theta$	=	cos^-1 $\displaystyle \left(\vphantom{ \frac{R_\theta}{r_0} }\right.$ $\displaystyle {\frac{R_\theta}{r_0}}$ $\displaystyle \left.\vphantom{ \frac{R_\theta}{r_0} }\right)$ .	(29)

Limits: the front and rear sides of the generating sphere begin to overlap at latitude $\theta$ = 0^o.
Conformal at latitude $\theta$ = 90^o.

Approximate equidistant perspective zenithal projection

The perspective zenithal projection with $\mu$ = ( $\pi$ /2 - 1)^-1 = 1.7519 has approximately equidistant parallels, the length of the meridian from $\theta$ = 90^o to $\theta$ = 0^o being true. See also zenithal equidistant projection.

Approximate equal area perspective zenithal projection

The perspective zenithal projection with $\mu$ = $\sqrt{2}$ + 1 = 2.4142 is approximately equal area, the area of the hemisphere being true. See also zenithal equal area projection.

Non-perspective zenithal projections

This group contains mathematically defined projections which are designed to have certain useful properties.

Zenithal equidistant projection

The meridians are uniformly divided so as to give uniformly spaced parallels. This projection is useful for mapping the polar regions of a spherical coordinate system when ease of construction and measurement are required.

R	=	r₀(90 - $\displaystyle \theta$ ) $\displaystyle {\frac{\pi}{180}}$ ,	(30)
$\displaystyle \theta$	=	90^o - $\displaystyle {\frac{180 R_\theta}{\pi r_0}}$ .	(31)

Zenithal equal area projection

The spacing between parallels is computed to make this an equal area projection.

R	=	r₀ $\displaystyle \sqrt{2(1 - \sin \theta)}$ ,	(32)
$\displaystyle \theta$	=	sin^-1 $\displaystyle \left(\vphantom{ 1 - \frac{R_\theta^2}{2 r_0^2} }\right.$ 1 - $\displaystyle {\frac{R_\theta^2}{2 r_0^2}}$ $\displaystyle \left.\vphantom{ 1 - \frac{R_\theta^2}{2 r_0^2} }\right)$ .	(33)

x	=	- Rsin $\displaystyle \phi$ ,	(13)
y	=	- Rcos $\displaystyle \phi$ .	(14)