Unlike the “classic” 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.
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, and often restricted to insets for polar caps.
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, the radial scale is not constant: in the polar aspect, parallels get closer together towards the border, just enough to preserve areas.
Relatively simple in construction, this projection is frequently used in all aspects.
The polar aspect of Lambert's azimuthal projection was independently devised by Anton-Mario Lorgna (1789), and during a short period named after him.
In 1949, Russian Georgiy A.Ginzburg proposed two azimuthal projections for hemispheres in school maps. Since Lambert's equal-area projection compresses distances from the center of the map, causing considerable shape distortion near the borders, Ginzburg added a power term to Lambert's equations, slightly expanding the map. The result is neither conformal nor equivalent.