Given a reference point A and two other points B and C on a surface, the azimuth from B to C is the angle formed by the minimum-distance lines AB and AC (which, on a sphere, are geodesic or great circle arcs). In other words, it represents the angle one sitting on A and looking at B must turn in order to look at C. The bearing from A to C is the azimuth considering a pole as reference B.
All azimuthal projections preserve the azimuth from a reference point (the conceptual center of the map), thus presenting true direction (but not necessarily distance) to any other points. They are also called planar since several of them are obtained straightforwardly by direct perspective projection to a plane surface. This terminology is somewhat misleading because all map projections of interest result in a flat map, azimuthal or not, perspective or not.
The polar aspect is easily built for azimuthal projections; one of the poles is the central point, making the graticule trivial:
Schematic cross-sections of selected perspective azimuthal projections. Red rays (tangent at 15° latitude intervals) “paint” the visible pale blue surface of the Earth onto a blue projection plane
In any aspect, all straight lines touching the central point are geodesics, and distortion depends only on distance from that point, where it is zero.
In a few two-point azimuthal projections, correct angles are presented from two specific locations instead of one.
While azimuthal maps quickly tell the direction to anywhere from the central point, retroazimuthal projections have the opposite property, showing the correct direction to turn from any place to the central point.
Among the oldest projections, three azimuthal designs are defined by pure geometric perspective constructions. None can present the whole Earth, being usually clipped to a hemisphere or less.
Mentioned by the Greek Hipparchus in the 2nd century B.C., but probably known earlier, the azimuthal orthographic (usually simply referred to as the orthographic) projection was called analemma by Ptolemy and got its modern name from d'Aiguillon (1613).
The orthographic projection is mainly used — sometimes strikingly — for illustration purposes, since it clearly shows the Earth as seen from space infinitely far away, thus closely matching a student's view of a globe (an even better match is provided by near-side general vertical perspective projections). Severe shape and area distortion near the map borders prevents its general use for world maps: radial scale decreases sharply making features unrecognizable. On the other hand, scale is exactly correct along any circle whose center coincides with the projection center, like parallels in the polar aspect.
Usage of orthographic maps in modern atlases is mostly restricted to insets. However, the Space Age, which made available high-quality imagery of the Moon, other planets and the Earth as seen from orbit, has caused a recent revival of interest for this projection.
The geometric construction of orthographic maps can be easily explained and compared to other azimuthal projections's.
What is probably the most widely used azimuthal projection was known in the polar aspect since Classical eras and usually attributed to Hipparchus; it was named Planisphaerum by Ptolemy and stereographic by d'Aiguillon (1613). Still, its usage was long limited to star maps; world maps are known starting in the Modern Age. The carthographic stereographic projection must not of course be confused with “stereographic” projection techniques designed for stereoscopic imagery, i.e., compositing two pictures in order to simulate the depth of three-dimensional vision — an important feature related to mapping and analysis of high-altitude photographs.
Different aspects of the azimuthal stereographic projection demonstrate its conformality: as aspect changes “move” continents along, portions far from the center grow, but local angles remain the same: graticule lines are always circles crossing one another at right angles.
Among azimuthal designs, the stereographic is the unique conformal projection: over a small area, angles in the map are the same as the corresponding angles on Earth's surface. It also preserves circles, no matter how large (great circles passing on the central point are mapped into straight lines), although concentric circles on the sphere will not generally remain concentric on the map. On the other hand, loxodromes are plotted as logarithmic spirals.
An azimuthal stereographic map has a simple geometric interpretation: rays emanating from one point pierce the Earth's surface hitting a plane tangent at the point's antipode. The result is the map backface, which covers the entire plane; regions near the source point lie near infinity, and that point itself cannot be mapped.
Because — in contrast to the azimuthal orthographic — scale is greatly stretched far from the center of the map, azimuthal stereographic maps are commonly constrained to the hemisphere opposite the source point, or an even smaller region.
One of the “classic” conformal projections, the azimuthal stereographic was also modified for the ellipsoidal case; conformality is maintained, but the result is no more exactly azimuthal or circle-preserving. In this form, it is also an auxiliary part of the UTM grid.
The gnomonic (also called central, azimuthal centrographic or rarely gnomic) projection is constructed much like the azimuthal stereographic, but the ray source is located exactly on the sphere's center; therefore it can present even less than one hemisphere at a time. Distance and shape distortions are pronounced except very near the tangent point.
This unique projection's most important property is that every geodesic, including the Equator and all meridians, is mapped to a straight line, making easy finding the shortest route between any two points, although not the direction to follow.
The general vertical perspective projection is an azimuthal projection which can be defined by straight lines converging at an arbitrary zenithal point on a line passing through the center of the Earth and perpendicular to the projection plane, which is usually tangent at the planet's surface. Because it is a perspective projection, for each point on Earth the line passing through it defines the former's projection where it intersects the plane. The projection is parameterized by the distance between the convergence point and the center of the Earth; it is the general case of the azimuthal orthographic, stereographic and gnomonic projections, and is itself a limiting case of the oblique (or tilted) perspective projection, which does not require the projection plane to be perpendicular to the convergence line and is not necessarily azimuthal.
The curvature of the Earth and the height of an observer V above the surface towards the zenith Z determine the visible angular range A and the circle of the horizon H, beyond which the surface is hidden from view. The vertical near-side perspective pictures below were shot by a rear-looking camera aboard an imaginary rocket taking off from a hypothetical base on the Equator, near the mouth of the Amazon River. In this cloudless day, somehow the entire trajectory remained vertical over the launch point. Original unprojected satellite imagery from NASA's The Blue Marble New Generation project.
As normally happens with azimuthal projections, shape and area distortion grow large far from the central point and, for a tangent projection plane, are zero only at the center. Maps are neither conformal (except the special stereographic case) nor equal-area.
The general perspective projection is found in two varieties, “near-sided” when the convergence point is "above" the mapped surface, and “far-sided” when “below” it. The first kind reproduces a view from air or space directly downwards, bounded by a circular horizon, which is limited by the curvature of the globe; the visible angular range grows to a maximum of 90° (a whole hemisphere) of the zenith at infinity, which is the classic orthographic projection.
In contrast, the far-side kind normally shows more than one hemisphere. The visible angular range shrinks with distance; the limiting case at infinity is again the orthographic. And, like the stereographic case, the projecting lines first “see” the inner face of the globe.
Although near-side general vertical perspective maps have been limited to mimicking views from space, the far-side variant was adopted by several authors who chose different projection distances in order to minimize global distortion according to arbitrary criteria.
The best known far-side perspective projections were proposed by Philippe de La Hire (1701), who put the convergence point at 1 + 0.50.5 (about 1.707) times the Earth radius, Henry James (1857) with 1.5, and Alexander R. Clarke (1862, jointly with James) with 1.368 and later (1879) 1.4, for his more famous “Twilight” projection.
In La Hire's polar maps scale is nonlinear along meridians, but the 45°N parallel has exactly half the radius of the Equator. Shortly after (1702) Antoine Parent suggested three different distances for minimizing either distance or areal error.
Both James and Clarke preferred secant projection planes and oblique aspects presenting most of the continents. Both also clipped the angular range to less than can actually be shown: 113°30' for James and 108° for the “Twilight”; 113°30' adds to 90° (from the zenith to the visible horizon) 23°30', the ecliptic angle; 108° adds 18°, the angle below the visible horizon which defines the astronomical twilight.