| The "polycylindrical" concept can be presented as a generalization of the cylindrical group of projections, applied to discrete latitude ranges. | |
| Nine strips with partial cylindrical equidistant maps (parts removed for clarity) cover the globe. Each strip fits a different standard parallel; therefore, strip lengths vary, but vertical scale is identical in all nine maps. |
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| The flattened discrete map. The continuous limiting case with infinitely many strips would be the pseudocylindrical sinusoidal projection; since the original strips would coincide with the globe surface, it is easy to see the final result is equal-area. | |
Pseudocylindrical projections could be called "polycylindricals", by analogy to polyconic projections being a generalization of the conic group. Indeed, any pseudocylindrical map can be conceptually created by juxtaposing a (possibly infinite) number of partial cylindrical maps.
Of infinitely many possible pseudocylindrical projections, several are useful didactical devices and popular choices for world maps.
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| Sinusoidal (Sanson-Flamsteed) map, graticule spacing 10° |
Both the Equator and central meridian are standard lines, thus the whole map is twice wide as tall.
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| Mollweide map |
Created in 1805 by Karl B. Mollweide from Germany, but actually popularized by Jacques Babinet from 1857 onwards, this classic equal-area projection was designed to inscribe the world into a 2:1 ellipse, keeping parallels as straight (but not standard) lines while still preserving areas. All meridians but the central one map to elliptical arcs. This projection is also called homalographic (Babinet's appellation), homolographic (from Greek homo, "same"), elliptical or Babinet, and an interrupted form was popularized by J.P. Goode.
In a sense, the sinusoidal and Mollweide projections handle polar regions in complementary ways: while the first overcrowds them, the latter creates much more spaced meridians, with correspondingly greater angular distortion. Those two designs were often combined in hybrid approaches like Goode's homolosine projection.
Although relatively seldom used in its original form, Mollweide's projection has been extremely influential. Besides the developments by Goode already mentioned, derived works include the interrupted Sinu-Mollweide projection by A.K. Philbrick's (1953), oblique maps like the Atlantis, and simple rescalings by reciprocal factors which preserve its features - e.g., making the Equator a standard parallel (Bromley, 1965), or making the whole map circular instead of elliptical (Tobler).
The Mollweide and Hammer projections are occasionally confused, since they are both equal-area and share the elliptical boundary; however, the latter design has curved parallels and is not pseudocylindrical.
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| Collignon map, symmetrical diamond form | |
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| Interrupted Collignon map | Common Collignon map |
The two most common arrangements for the worldwide map are either the isosceles triangle (with unbroken meridians and a flat South Pole for base) or the diamond. The two complementary options (upside-down triangle and hourglass-shaped) and the interrupted variant in two or more diamonds are equally valid. In spite of its simple construction, this projection is regarded as little more than a curiosity.
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| Quartic authalic map |
The quartic authalic is another equal-area (authalic from Greek autos ailos, "same area") projection, designed by Karl Siemon (1937) and independently by Adams (1944). Its meridians are fourth-order curves, hence the name.
Polelines avoid the crowded appearance of projections like the sinusoidal and Mollweide at the cost of scale distortion.
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| Flat polar quartic map |
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| In odd-numbered Eckert projections (Eckert I here), the graticule is nearly square at the center |
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| Eckert's projection II is similar to his first design, but nonconstant parallel spacing makes it equal-area |
Therefore, in all six proposals the poles are framed by a square, and the whole map by a rectangle twice as broad. The boundary meridians are simple curves.
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| Eckert III map |
Although none of the six is conformal, the odd-numbered projections present a better overall shape (there's no shape distortion at the very center); in order to preserve area, the even-numbered projections compress vertical scale near the poles and stretch it near the Equator.
Eckert's second design is equal-area and maps all meridians to straight lines broken at the Equator. The first projection is similar, but not equal-area since the parallels are equally spaced. Neither is more than a curiosity.
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| Eckert IV map |
For his third and fourth projections, Eckert made the outer meridians as half circles; all other meridians are regularly spaced elliptical arcs except the central which, like in all Eckert flat-polar maps, is straight and half as long as the Equator. The fourth design was moderately used for world maps; the third is sometimes mistaken for Ortelius's oval map, which has not constant scale along parallels.
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| Eckert V map |
The sixth and most popular of Eckert's flat-polar projections has boundary meridians shaped as half the period of a sinusoid. The superficially similar fifth design has regularly spaced parallels and is not equal-area.
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| Eckert VI map |
Several other pseudocylindrical projections, most notably by Siemon, McBryde and Thomas, and a series by Wagner, are also based on polelines and sinusoidal meridians.
Following a widespread controversy about the adequacy of cylindrical world maps for teaching, Rand McNally, the traditional atlas publisher, requested the distinguished cartographer and educator Arthur H. Robinson to develop a new map projection with reduced overall distortion and a simple, uninterrupted graticule.
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| Robinson map, calculated with 3rd degree polynomial interpolation |
Instead of using a perspective process such as that used for classic azimuthal projections, or simple mathematical functions like the sinusoidal or elliptical arcs of the Sanson-Flamsteed, Mollweide and Eckert's series, the resulting compromise projection had the boundary meridians in the equatorial aspect defined by conventional values, calculated by hand in order to yield a "right-looking" map (thus its common name, orthophanic). A table defines x, y coordinate values for 5° increments of latitude for those meridians; other points must be interpolated. Like in all pseudocylindrical projections, meridians are equally spaced along all horizontal, straight parallels (regularly spaced between 38°N and 38°S). The Equator is nearly twice as long as the central meridian; poles are flat.
Designed in 1963 and formally published in 1974, the Robinson
projection became truly popular only after praised by
the cartographic staff of the National Geographic Society; in
1988, it was published as an insert of the Society's
magazine and chosen as its reference world map, replacing
the van der
Grinten projection.