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Modified Azimuthal Projections

In true azimuthal projections, all directions are preserved from the reference point, usually tangent at the center of the map. The three classic perspective azimuthal projections can show no more than one hemisphere at a time; others (like the azimuthal equidistant) are defined by arbitrary constraints instead of purely geometrical models.

Some projections are inspired by azimuthal principles or modifications of mentioned projections; the result is not, usually, entirely azimuthal itself. E.g., most star-like projections are based on an azimuthal hemisphere.

Map in Wiechel projection Map in Wiechel projection
Map in Wiechel projection Map in Wiechel projection
Wiechel's projection in hemispheric (top) and whole-world (above) polar versions; the latter two make apparent its limitations near the circular pole. Non-polar aspects (right) are visually intriguing novelties but of little practical value.
In polar aspects, Wiechel maps combine the areal preservation of Lambert's azimuthal design with meridian lengths of the azimuthal equidistant projection.
Map in Wiechel projection

Wiechel's Projection

H.Wiechel's projection of 1879 shares some features of both the azimuthal equidistant and Lambert's azimuthal equal-area projections. In its polar aspect, one pole is a point and the other, if the whole world is shown, the map's circular border.

A map in Wiechel's projection can be imagined as an azimuthal equal-area map sliced in infinitely many concentric rings which are rotated all in the same direction, either arbitrarily clockwise or counterclockwise; each ring's rotation angle is proportional to its radius. The total mapped area does not change, therefore the projection remains equal-area, but no more azimuthal. In the polar aspect, meridians become circular arcs, and like in the azimuthal equidistant projection, standard lines; in full world maps, meridians are semicircles: they approach nearly parallel to the circular pole, creating large shape distortion. Therefore, maps are usually limited to a single hemisphere.

This rarely used projection is of some interest only in polar aspects; sometimes it is classified as pseudoazimuthal, by analogy with pseudocylindrical and pseudoconic projections, which share some attributes with cylindrical and conic designs, respectively, but have curved meridians in the normal aspects.

Aitoff Projection

Map in Aitoff projection
Map in Aitoff's projection

In 1889, David Aitoff announced a very simple modification of the equatorial aspect of an azimuthal equidistant map. Doubling longitudinal values enabled the whole world to fit in the inner disc of the map; the horizontal scale was then doubled, creating a 2 : 1 ellipse. As a result, the map is neither azimuthal nor equidistant, except along the Equator and central meridian. Neither it is equivalent or conformal.

The Aitoff projection is a very interesting compromise between shape and scale distortion. It clearly suggests the Earth's shape with less polar shearing than Mollweide's elliptical projection. However, this influential design was quickly superseded by Hammer's work.

Hammer Projection

Hammer's projection
Hammer map

Properly crediting Aitoff's previous work, in 1892 Ernst Hammer applied exactly the same principle to Lambert's azimuthal equal-area projection.

The resulting 2 : 1 elliptical equal-area design, called by the author Aitoff-Hammer, by others at first Hammer-Aitoff and then simply the Hammer projection, soon became popular and is used even today for world maps. It was itself the base for several modified projections, like the oblique contribution by Briesemeister.

The strong superficial resemblance of Aitoff's and Hammer's projections led to considerable confusion, even in technical literature.

Schjerning II map
Partial reconstruction of Schjerning's second projection, minus the Asian enlargement
Schjerning III map's base projection Schjerning III map centered on the Equator
Above left, the base projection for Schjerning's projection III, centered on a pole. Above right, a transverse aspect of the base. Below, Schjerning's final map, centered on London.
Schjerning III map

Schjerning's Second and Third Projections

In a sense, all projections published by W.Schjerning in 1904 are related to the azimuthal equidistant, including the three variations of Werner's pseudoconic cordiform projection. Two others are summarized here.

Schjerning's second proposal is more easily described using a polar base projection which, like many star projections, has an inner hemisphere identical to a polar azimuthal equidistant's. The outer hemisphere is split in two appendages, and the whole map fits a 2:1 ellipse. In each appendage, parallels comprise circular arcs which retain the inner hemisphere's spacing. Meridians are marked equidistantly along each parallel. The Schjerning II projection is a transverse aspect of this base projection, moving the poles to the end of the ellipse's minor axis. Unfortunately there are conspicuous breaks between the inner and outer hemispheres; Schjerning applied an enlargement to improve the appearance of Asia.

The Schjerning III projection can also be described with the help of a base projection, in this case comprising two circles connected at a central pole; the other pole is split between the two points farthest from the center. Parallels are drawn at correct distances from the central pole, then meridians are marked equidistantly along each parallel. The final projection is an oblique aspect, centered on London.

In both the second and third proposals by Schjerning, every point has correct distances from the center of the map. Azimuths from the center are correct only in the inner hemisphere of the second design. Neither projection is conformal or equal-area.

Wagner IX Projection

Wagner's projection IX
Wagner IX map

Part of a series by Karlheinz (Karl Heinrich) Wagner, his ninth proposal (1949) is a rescaling of Aitoff's projection. Parallels are projected as in Aitoff's, but at 7/9 of their actual value; as a result, the poles are mapped as curved lines along the parallels 70°N and 70°S of Aitoff's projection. Therefore, polar angular distortion is lesser than usual in pseudocylindrical projections with polelines. Conversely, meridians are mapped at 5/18 of the actual value. The projected coordinates are then stretched horizontally and vertically at the reciprocal rates, thus keeping the original aspect ratio (the Equator is twice as long as the central meridian).

The projection is neither equal-area nor conformal. Scale is constant and the same along only the Equator and central meridian.

Eckert-Greifendorff Projection

Eckert-Greifendorff's projection
Eckert-Greifendorff map
Much like Hammer's projection horizontally stretched part of an equatorial equal-area azimuthal map, the projection announced in 1935 by Max Eckert-Greifendorff (previously known as Max Eckert) stretched the corresponding portion of a Hammer map. In other words, exactly the same idea as Hammer's, but with longitude compressed four times and horizontal scale multiplied fourfold. Before rescaling it uses only a narrow region near the central meridian of the original azimuthal map; as a consequence, parallels are almost straight lines.

Winkel Tripel Projection

Winkel's Tripel projection
Winkel Tripel map using 50°28'N/S as reference, but not standard, parallels
Winkel's Tripel projection
Winkel Tripel map using 40°N/S as reference parallels

The third and best known of Oswald Winkel's hybrid projections was called tripel (from the German for triple, possibly referring to a triple compromise of reduced shape, area and distance distortion). Like his two other proposals published in 1921, it is defined by a simple arithmetic mean including the equidistant cylindrical projection, using an arbitrary value for standard parallels (the author preferred approximately 50°28'N/S; another common value is 40°N/S); these are not standard in the final result. However, the other projection is Aitoff's, therefore, the result is not pseudocylindrical.

Winkel's Tripel projection is peculiarly irregular: it is neither equal-area nor conformal; parallels are straight at Equator and poles, curved elsewhere; scales are constant, but not equal, only at the Equator and central meridian.
Nevertheless, it manages to present a pleasant and balanced view of the world, which led to its choice by several popular atlases. In 1998, it was selected by the prestigious National Geographic Society for its new reference world map, in place of the Robinson projection.


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