Map Projections

### Perspective Cylindrical Projections

 Schematic cross-section diagrams of selected perspective cylindrical projections at the same scaling factor show differences in the map's size (infinite in one case) and aspect ratio. Red light rays "paint" points at 0°, 30°, 60° and 90° N and S of a single meridian onto a blue tube. Mathematical development follows immediately.

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.

#### Lambert's Cylindrical Equal-Area Projection

 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.

 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 any single parallel.

#### Gall's Orthographic Projection

 Gall/Gall-Peters's orthographic projection 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.

#### "Peters" (Gall-Peters) Projection

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.

Equal-area cylindrical projections compared at identical scaling factors. Standard parallels are outlined.
 Lambert (1772);standard latitude 0°, aspect ratio 3.141:1
 W.Behrmann (1910);standard latitudes 30°, aspect ratio 2.356:1
 Trystan Edwards (theoretical constraint, 1953);approximate standard latitudes 37°24', aspect ratio 1.983:1
 Hobo-Dyer (2002); standard latitudes 37°30', aspect ratio 1.977:1
 Gall (1855), Peters (1967); standard latitudes 45°, aspect ratio 1.571:1
 Trystan Edwards (actual, 1953);approximate standard latitudes 50°52', aspect ratio 1.251:1
 Tobler and Chen (1986); approximate standard latitudes 55°39', aspect ratio 1:1

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.

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 has been — misguidedly — 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.

#### Other Equal-area Cylindrical Projections

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).

 Gall's stereographic projection.

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, Waldo 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; areal equivalence is much more important.

#### Stereographic Cylindrical Projections: Gall's, BSAM cylindrical and Braun's

 Gall's stereographic cylindrical projection with 30° standard parallels (on the B.S.A.M., the central meridian is 10°E).

The geometric construction for James Gall's preferred projection (1885) resembles the perspective for the azimuthal stereographic, with two differences:

• the projection surface is a secant cylinder 45°N and 45°S, like in his orthographic
• the ray source for every projected point sits at the Equator on the opposite meridian

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.

 Braun's stereographic cylindrical projection

Carl Braun's stereographic cylindrical (1867) is a very similar projection, but developed from a tangent cylinder.

Two other variations, credited as special cases of Gall's stereographic, were published in the Soviet Union: a map by Kamenetskiy (1929) using 55° as the standard parallel, and several maps in a volume of the B.S.A.M. (Bol'shoy sovetskij atlas mira, Great Soviet World Atlas, 1937) using 30°. The latter is known as the BSAM cylindrical projection.

Yet another variation was briefly mentioned by Braun: moving the light source from the Equator to a point on the equatorial plane, but at 40% of the distance from the polar axis to the Equator. Between 80°N and 80°S, the result closely matches Mercator's, with a maximum error of 2.39% (Mercator's is hardly useful beyond 80° of latitude anyway), and 2.29% between 60°N and 60°S. Even better approximations can be achieved with distances like 40.7% (2.05% of error between 80°N and 80°S) and 46.71% (0.24% between 60° N and S). Despite the ability to roughly recreate the genuine conformal projection with trivial geometry, this is just a curiosity, whose best merit is emphasizing that Mercator's design is not a perspective process.

#### Central Cylindrical Projection

 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 obvious analogues in the azimuthal gnomonic and centrographic perspective conic projections. 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. An Indonesian translation by ChameleonJohn, courtesy of Jordan Silaen.

 www.progonos.com/furuti    July 31, 2017