 |  |  |  |  | Map Projections |
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| Lambert's equal-area | Gall's orthographic | Behrmann's equal-area |
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| Gall's stereographic | Braun's stereographic | Central cylindrical |
| Cross-section diagrams of a few projections at same scale make clear differences in map size (infinite for the central cylindrical projection) and aspect ratio. Red light rays "paint" points at 0°N, 30°N, 45°N, 60°N and 90°N of a single meridian on a blue tube. | ||
Some cylindrical projections are defined by a geometric process based on perspective. It can roughly be imagined as a semitransparent spherical shell wrapped by a tube, secant or tangent. While both sphere and tube rotate around the latter's axis, a fixed source shoots light rays along a single meridian, projecting "shadows" of spherical features onto the tube. After a complete revolution, the tube is cut along a line parallel to its axis and unrolled.
Just by changing the source's position and tube's diameter, different maps result. The source may also be located infinitely away, making rays parallel.
In contrast, other cylindrical projections like the equidistant cylindrical, Miller and Mercator have conventional graticules defined arbitrarily, not by a light source analogy.
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| Lambert's equal-area cylindrical map |
The equal-area projection on a tangent cylinder - making the Equator a standard parallel in the normal aspect - was rigorously defined by Johann H. Lambert in both equatorial and transverse aspects, among several other projections (1772). It preserves areas by progressively compressing parallels away from the Equator in order to compensate horizontal scale exaggeration. Still, only the Equator is free of shape distortion. This projection is sometimes associated with Archimedes, but this is probably a confusion originated from his diagram of volumes of a sphere and a circumscribed cylinder.
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| Schematic development of Lambert's equal-area cylindrical projection. With a tangent cylinder, 0° is a standard parallel. |
This projection's perspective is easily visualized by rolling a flexible sheet around the globe and projecting each point horizontally onto the tube so formed. In other words, light rays shoot from the cylinder's axis towards its surface, which is afterwards cut along a meridian and unrolled.
Like most cylindrical projections, it is quite acceptable along the standard parallel, but practically useless at polar regions, which are rather compressed, resulting in a map much broader than tall. Again like in other cylindrical projections, deformation is uniform along the same parallel.
![Equal-area cylindrical map](javascript:showImage('Gall's orthographic projection, also known as the "Peters" projection: a special case of Lambert's equal-area cylindrical projection with standard latitudes 45°N and S', '2008 C.A.Furuti', 'Img/Z1/mp2_EqACyl-s75-p45.png', 674, 432))
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| Gall/Gall-Peters's orthographic projection |
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| Development of Gall's orthographic projection. If Lambert's cylinder becomes secant at 45°N and 45°S, the result is only 21/2/2 as wide; after the projection, it is vertically stretched by the inverse ratio. |
James Gall's orthographic equal-area projection (1855) is trivially similar to Lambert's version, but with standard parallels at 45°N and 45°S. Therefore, the projection cylinder is secant and narrower; the vertical amplitude must be proportionally expanded in order to preserve the total mapped area. Thus, the only real difference is the aspect ratio (i.e., width divided by height): Gall's orthographic is twice as tall.
Although area-preserving, this projection's unconventional pattern of shape distortion limits its usefulness.
In 1967, Arno Peters published a cylindrical projection essentially identical to Gall's orthographic version of 1855. Perhaps it was actually an independent creation; nevertheless, Peters persisted in claiming it as an original design and novelty even after heavy criticism. After 1973, the projection was vigorously promoted, gaining widespread press coverage. Its supposed virtues were mainly compared with Mercator's projection's shortcomings.
It was claimed Peters's projection presented no distance or area distortion, and no extreme shape distortion. Since it was equal-area, it was egalitarian: Third World nations, many of which are located in tropical areas, are presented in real size proportion, while the Mercator projection greatly exaggerates size in higher latitudes, including Europe, North America and the former Soviet Union.
Especially this last point, the supposed correction of a historical injustice, helped the design to be accepted as the main or only projection by several organizations (like UNESCO), a choice deplored by professional cartographers, who accused Peters of political propaganda and manipulation of uninformed media. In the episode, named by some the "Map Wars", Peters was often seen as David against the Goliath of established cartography.
| Equal-area cylindrical projections compared at identical scales. Standard parallels are outlined. | ||
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Regrettably, arguments in favor of Peters's projection are false, overstated, or fallacious. It does have severe shape distortion, and its distortion pattern greatly changes along a meridian (the Mercator projection has no local shape distortion). There is of course distance distortion (infinite at the poles, as in all cylindrical projections), and distance is only preserved along the two standard parallels. Area distortion is certainly absent; on the other hand, many other equal-area projections already existed, several with much lesser overall shape distortion (cartographers were especially aggravated by Peters's suggestions, at least initially, that his was the first equal-area projection). Finally, the Mercator projection was a false target for criticism, since it was designed as a navigation device, and never intended for world maps; the unfortunate fact that it was - misleadingly - adopted by many naïve magazines and textbooks meant only it should be replaced by a better candidate - and none of Peters's assertions proved his to be the best one.
The whole event demonstrated that projections must not be chosen due to a single feature or on the basis of publicity or political propaganda, no matter how sympathetic the cause. Today Gall's orthographic projection is still occasionally published under Peters's name. Ironically, for a projection advertised as free of Eurocentrism or any privilege towards developed nations, it shows both Europe and the United States/Canadian border with lower shape distortion, since they are near the northern standard parallel; on the other hand, most of Africa, Southeast Asia and Latin America lie on the most distorted areas.
Lambert's principle is employed by a few lesser-known equal-area cylindrical projections, changing only standard parallels and therefore general map proportions. Each of them can be converted to any other simply by rescaling both width and height by reciprocal factors.
Their patterns of shape distortion are similar and, like all cylindrical projections, independent of longitude: horizontal scale is more affected the farther from the standard parallels (compressed between them and exaggerated in the outer portions).
Some of these variants were explicitly designed in order to reduce maximum or average deformation (as conveniently defined by the author), as is the case of Behrmann's and Trystan Edwards's projections. Notably, for some reason the latter specified a deformation criterion whose standard parallel does not match the value actually chosen.
Several authors have suggested an equal-area cylindrical map on a square; more recently, Tobler and Chen mentioned it in the context of a geographic information system based on quadtrees. A quadtree is a hierarchical data structure which recursively divides a rectangular region, four smaller pieces per level: information may be efficiently stored and retrieved in coarse or fine detail as required. A square region is not a requirement, but eases implementation; area equivalence is much more important.
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| Gall's stereographic projection. |
The geometric construction for James Gall's preferred projection (1885) resembles the perspective for the azimuthal stereographic, with two differences:
Areas are not preserved and the map is not conformal. Scale is true only along the standard parallels 45°N and 45°S. There are no outstanding features except overall distortion.
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| Braun's stereographic cylindrical projection |
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| Central cylindrical projection, clipped at 70°N and S |
In the central cylindrical (also called centrographic cylindrical) projection, vertical scale increases very fast far from the map's centerline, even faster than in Mercator's projection; likewise, poles cannot be shown in the equatorial aspect.
Its origin is unknown, though it has an obvious analogue in the azimuthal gnomonic projection. With no favorable property, neither equal-area nor conformal, it is almost never used, either in equatorial or transverse (called the Wetch projection, after J.Wetch, 19th century) aspects.
| A Romanian translation of this page, courtesy of Alexander Ovsov, is available at Web Geek Science. | |
| A Danish translation by Excellent Worlds |
| | | | | www.progonos.com/furuti January 21, 2013 |
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: a special case of Lambert's equal-area cylindrical projection with standard latitudes approximately at 37°24'N and S', '2008 C.A.Furuti', 'Img/Z1/mp2_EqACyl-s75-p37.4.png', 756, 386))
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: a special case of Lambert's equal-area cylindrical projection with standard latitudes approximately at 50°52'N and S', '2008 C.A.Furuti', 'Img/Z1/mp2_EqACyl-s75-p50.86666666666.png', 602, 484))
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