 |  |  |  |  | Map Projections |
| | | | | Map 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.
| Wiechel's projection in hemispheric and whole-world polar maps; the latter make apparent its limitations near the circular pole. Non-polar aspects are visually intriguing but of little practical value. | |
H.Wiechel's projection of 1879 shares some features of both Lambert's azimuthal equal-area and azimuthal equidistant projections. In the polar aspect, one pole is a point and the other, if the whole world is shown, a circle at the border of the map.
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 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.
| 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 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.
| 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 map |
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| Winkel Tripel map using 50°28'N/S as reference, but not standard, parallels |
| 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.
| | | | | www.progonos.com/furuti December 22, 2008 |