Map Projections: Azimuthal Projections

Map Projections

Given reference point A, the azimuth remains unchanged from points B and C on the sphere to corresponding B' and C' on an azimuthal map.

Azimuthal Projections

Introduction

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.


Orthographic	Stereographic

Vertical perspective	Gnomonic
Schematic cross-section of development of some perspective azimuthal projections in a polar aspect. Red light rays shoot from latitudes 0°, 30°, 45°, 60° and 90° to the blue projection plane. By construction, not every point can be represented.

Compared polar aspects of five azimuthal projections with parallels spaced 10° apart. Orthographic and stereographic stop at Equator, gnomonic is arbitrarily clipped at 20°. Equatorial zone is red, polar caps blue.

The polar aspect is easily built for azimuthal projections; one of the poles is the central point, making the graticule trivial:

meridians are straight lines, radiating regularly spaced from the central point
parallels are complete concentric circles
projections are only distinguished by parallel spacing

In any aspect, all straight lines touching the central point are geodesics, and distortion depends only on distance from the center.

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.

Classic Azimuthal Projections

Among the oldest projections, three classic azimuthal designs are defined by pure geometric perspective constructions.

Azimuthal Orthographic Projection


Orthographic map centered on Campinas, Brazil (Progonos's home)	South polar orthographic map

Equatorial map, central meridian 110°W	Equatorial map, central meridian 70°E

Used by the Greek Hipparchus (2nd century B.C.), but probably known earlier, this projection was called analemma by Ptolemy and renamed orthographic by d'Aiguillon (1613).

The orthographic projection is mainly used for (sometimes dramatic) 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. Severe shape and area distortion near the map borders prevents its general use for world maps.

Its construction can be easily explained and compared to other azimuthal projections's.

Azimuthal Stereographic Projection


Stereographic map, central meridian 110°W	Stereographic map, central meridian 70°E

Probably the most widely used azimuthal and known since Classical eras, this projection usually attributed to Hipparchus was called Planisphaerum by Ptolemy and stereographic by d'Aiguillon (1613). It is a 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, the loxodrome is plotted as a logarithmic spiral.

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 antipode. The result is the map backface, which covers the entire plane (regions near the source point lie at infinity, and that point itself cannot be mapped).

Because - in contrast to the azimuthal orthographic - scale is greatly stretched away from the center of the map, azimuthal stereographic maps are usually constrained to the hemisphere opposite the source point, or an even smaller region.

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 part of the UTM grid.

Gnomonic Projection


Equatorial gnomonic map, arbitrarily clipped at 70° from the center	Oblique gnomonic map: meridians and Equator are still straight lines

The gnomonic (also called central) 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 distortion is 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 (but not the direction to follow).

Nonperspective Azimuthal Projections

Unlike the "classical" (orthographic, stereographic and gnomonic) designs, azimuthal projections like the equidistant and equal-area were derived mathematically without a real perspective process. Both can map a full sphere, with an "inner" hemisphere surrounded by a ringlike "outer" one. However, for lesser overall distortion the latter may be presented in a separate map centered on the antipodal point.

Azimuthal Equidistant Projection


North polar and equatorial aspects of azimuthal equidistant projection

Able to present the whole Earth in a single map and with constant radial scale (distances increase linearly from the center of projection), the azimuthal equidistant projection is further discussed elsewhere due to its important features.

In the north polar aspect, the azimuthal equidistant is familiar as part of both flag and emblem of the United Nations Organization, with olive branches replacing Antarctica. The austral continent, here turned "inside-out", illustrates this projection's extreme distortion of shapes and areas far from the center.

Simple in construction, this projection is sometimes clipped to a single hemisphere, or even restricted to insets for polar caps.

Lambert's Azimuthal Equal-area Projection


North polar aspect	Equatorial aspect, central meridian 5°E

Western hemisphere, central meridian 110°W	Eastern hemisphere, central meridian 70°E
Azimuthal equal-area maps

Like the superficially similar azimuthal equidistant, the azimuthal projection published by Johann H. Lambert in 1772 strongly distorts shapes in the boundary of a worldwide map. However, radial scale is not constant: in the polar aspects, parallels get closer together towards the border. As a result, the map preserves areas.

Relatively simple in construction, this projection is frequently used in all aspects.