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Introduction

This document presents a concise mathematical description of uninterrupted spherical map projections in common use, particularly with reference to their application in astronomy. It is currently incomplete in four ways,

Two diagrams are missing.
The inverse formulae for the conics are missing.
Some important map projections are missing.
Many projections are known by several names and a complete list of aliases will be produced.

These will be added in good time.

Spherical projections can be classified as either perspective or non-perspective. Perspective projections may be constructed by ray-tracing of points on a generating sphere from a single point of projection onto a surface which can subsequently be flattened without further distortion. In zenithal projections, the surface is a plane; in cylindrical projections it is a cylinder which is conceptually cut along its length and unrolled; the surface in conic projections is a semi-cone which may likewise be cut and flattened. All perspective projections have as free parameters the distance of the point of projection from the centre of the generating sphere, and the distance of the surface of projection from the generating sphere.

Non-perspective projections are constructed mathematically so as to have particular useful properties. Conceptually, they are also based on a generating sphere, but the relationship between sphere and surface of projection cannot be defined in Euclidean terms - that is, with straight edge and compass. However, the surface of projection may still be described as planar, cylindrical, or conic.

Although certain perspective projections are important in astronomy, for example the orthographic and gnomonic projections, the general distinction between perspective and non-perspective projections is not particularly important - the days of ruler and compass are long past! A more important distinction is the mathematical one based on the plane of projection, and suggests the division into the following classes: zenithal, cylindrical, conic, and conventional. These classes may be further subdivided, for example conic projections can be either one-standard, two-standard, or poly-standard.

Conventional projections are pure-mathematical. Certain of them are closely related to particular projections from other classes, for example Aitov's projection is developed from the equatorial case of the zenithal equal area projection, and the Sanson-Flamsteed projection is actually the equatorial case of Bonne's projection, a poly-standard conic. From a mathematical viewpoint the conventional projections are closely related to the cylindricals, but really they belong in a class of their own.

When constructing a projection a distinction must be drawn between mapping the earth and mapping the celestial sphere. The two cases are related by a simple inversion, the difference between looking at a sphere from the outside (the earth), or from the inside (the sky). In the remainder of this document we will be concerned only with mapping the sky. To map the earth, simply apply the relation

(x, y) = (- x, y),

(1)

where (x, y) henceforth refers to the sky projection.

The relationship between the sphere and the surface of projection is a geometrical one, independent of any coordinate system that may be ascribed to the sphere. This is clearly the case for perspective projections, which may be constructed with straight edge and compass, but is also true for non-perspective projections. However, the mathematical description of a projection may only be made in terms of a spherical coordinate system. In a sense, this coordinate system is imposed on the sphere by the projection itself, and it may not bear any relationship to the coordinate systems usually associated with the celestial sphere, be they ecliptic, equatorial, or galactic. For example, zenithal projections are naturally described by a coordinate system in which the ``north'' pole is located at the point of tangency of the plane of projection.

Throughout the remainder of this document the longitudinal and latitudinal components of the native coordinate system will be denoted by the ordered pair ( $\phi$ , $\theta$ ).

An oblique projection is one in which the coordinate system of interest does not coincide with the coordinate system used to define the projection mathematically. The term ``oblique projection'' is somewhat of a misnomer, since obliquity is a property of a coordinate system and not of the spherical projection itself. It is best treated in terms of a spherical coordinate rotation from the ( $\phi{^\prime}$ , $\theta{^\prime}$ ) system to the ( $\phi$ , $\theta$ ) system defined by three Euler angles, ( $\Phi{^\prime}$ , $\Theta{^\prime}$ , $\Phi$ ) via the transformation equations:

$\displaystyle \phi$	=	$\displaystyle \Phi$ + arg(cos $\displaystyle \theta{^\prime}$ cos( $\displaystyle \phi{^\prime}$ - $\displaystyle \Phi{^\prime}$ ), sin $\displaystyle \theta{^\prime}$ sin $\displaystyle \Theta{^\prime}$ + cos $\displaystyle \theta{^\prime}$ cos $\displaystyle \Theta{^\prime}$ sin( $\displaystyle \phi{^\prime}$ - $\displaystyle \Phi{^\prime}$ )),	(2)
$\displaystyle \theta$	=	sin^-1(sin $\displaystyle \theta{^\prime}$ cos $\displaystyle \Theta{^\prime}$ - cos $\displaystyle \theta{^\prime}$ sin $\displaystyle \Theta{^\prime}$ sin( $\displaystyle \phi{^\prime}$ - $\displaystyle \Phi{^\prime}$ )).	(3)

The following equations derived from these are often found to be useful:

cos( $\displaystyle \phi$ - $\displaystyle \Phi$ )cos $\displaystyle \theta$	=	cos $\displaystyle \theta{^\prime}$ cos( $\displaystyle \phi{^\prime}$ - $\displaystyle \Phi{^\prime}$ ),	(4)
sin( $\displaystyle \phi$ - $\displaystyle \Phi$ )cos $\displaystyle \theta$	=	sin $\displaystyle \theta{^\prime}$ sin $\displaystyle \Theta{^\prime}$ + cos $\displaystyle \theta{^\prime}$ cos $\displaystyle \Theta{^\prime}$ sin( $\displaystyle \phi{^\prime}$ - $\displaystyle \Phi{^\prime}$ ).	(5)

Oblique coordinate projections arise in many places in astronomy even without being noticed. The ancient Arab astronomers faced with the task of constructing astrolabes, hit upon the expedient of using the stereographic projection which has the interesting and valuable property that great circles and small circles alike are projected as circles on the plane of projection. This makes oblique coordinate grids in stereographic projections easier to compute and draw than in most other projections. Any planisphere which contains an ecliptic or galactic grid in addition to the equatorial grid contains an oblique coordinate projection. The so called SIN projection used in radio astronomy is really an oblique orthographic projection whose natural pole is at the field centre of the observation, offset from the north celestial pole. On the other hand, the so called NCP projection is, in fact, a non-oblique orthographic projection with rescaled y-coordinate.

There are a number of important properties which a spherical projection can have:

Equidistance: where angles are to be measured directly from a map ease of measurement may be an important criterion. This is facilitated in equidistant projections such as Plate Carreé, in which the meridians and parallels of the native coordinate system are uniformly divided.
Equal area: an equal area projection has the valuable property that two regions on the generating sphere which enclose equal areas also enclose equal areas in the plane of projection. Such projections are referred to as being authalic. This property is particularly useful when mapping the density of objects in different regions of the sky, or for unbiassed political maps of the earth.
All of the equal area projections depicted in this document enclose the same area (16 $\pi$ cm²).
Conformality (orthomorphism): It can be proved mathematically that no spherical projection exists which can represent a finite portion of the sphere without distortion. Conformality is a property which refers to the faithful representation of regions of vanishingly small size. Conformality exists at points on a projection where the meridians and parallels are orthogonal and equally scaled, so that a tiny circle at the corresponding point of the generating sphere is projected as a circle in the plane of projection. Most projections are conformal within a restricted set of points, but certain projections are conformal at all points.
Although conformality is a local property, distortions are usually minimized within the immediate vicinity of the conformal points. However, the importance of conformality can be overemphasised. For example, while the Sanson-Flamsteed projection is conformal along the equator and central meridian, it doesn't minimize distortions across the whole of the sphere as well as does Aitov's projection which is conformal only at the centre.
Boundaries: Some projections are incapable of representing the whole sphere and diverge at some latitude in their native coordinate system. Other projections, particularly the poly-standard conics and conventional projections, are particularly suited to representing the whole sphere.
A number of projections have properties shared by no other:
- gnomonic: great circle arcs are projected as straight line intervals, but with non-uniform scale.
- stereographic: small circles are projected as circles.
- orthographic: represents the visual appearance of a sphere when seen from infinity.
- Mercator's projection: lines of constant bearing (rhumb lines) are projected as straight lines.

Certain projections are particularly suited for representing ``continent'' sized regions of the sphere. In particular, the two-standard and poly-standard conic projections are much favoured in atlases of the world.

The following sections present mathematical formulae which may be used to construct the spherical projections. It is usually easier to derive formulae for computing coordinates in the plane of projection, (x, y), from the spherical coordinates in the native coordinate system of the projection, ( $\phi$ , $\theta$ ), than vica versa. However, the inverse formulae are also very useful, and are provided for all projections.

In some circumstances it may be necessary to do a hybrid calculation, that is, compute x and $\theta$ from y and $\phi$ , compute x and $\phi$ from y and $\theta$ , compute y and $\theta$ from x and $\phi$ , or compute y and $\phi$ from x and $\theta$ . However, the hybrid formulae often cannot be specified analytically and iterative methods must be used. We will not attempt to develop any hybrid formulae here but instead rely on iterative solutions in all such cases. Formulae of the following forms will be presented for each projection (or class of projections):

x	=	x( $\displaystyle \phi$ , $\displaystyle \theta$ ),	(6)
y	=	y( $\displaystyle \phi$ , $\displaystyle \theta$ ),	(7)
$\displaystyle \phi$	=	$\displaystyle \phi$ (x, y),	(8)
$\displaystyle \theta$	=	$\displaystyle \theta$ (x, y).	(9)

Then for any mixed pair of coordinates, say x and $\phi$ , each of the unknown coordinates, y and $\theta$ , can be obtained by iterative solution of a single formula, for example y can be obtained from the equation $\phi$ = $\phi$ (x, y) and $\theta$ from x = x( $\phi$ , $\theta$ ).

Throughout the rest of this paper the radius of the generating sphere is denoted by r₀, and latitudes of significance in defining the projection as $\theta_{x}^{}$ . The angles $\phi$ and $\theta$ are always specified in degrees rather than radians, and the formulae explicitly include the $\pi$ /180 conversion factor to radians whenever required. It is assumed that all trigonometric and inverse trigonometric functions accept and return angles in degrees. The arg() function referred to in some of the formulae is such that if

(x, y) = (rcos $\displaystyle \alpha$ , rsin $\displaystyle \alpha$ ),

(10)

then

arg(x, y) = $\displaystyle \alpha$ .

(11)

In other words, arg(x, y) is effectively the same as tan^-1(y/x) except that it returns the angle in the correct quadrant.

Since the mathematical formulae which define the projections are hard to interpret by themselves, each projection is illustrated with a diagram of its native coordinate grid. The grid interval is 15^o, and all projections are based on a generating sphere of radius r₀ = 2cm. The diagrams themselves were encoded directly in POSTSCRIPT.

Next: Zenithal (azimuthal) projections Up: No Title Previous: No Title