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Pseudocylindrical Projections

From cylindrical to pseudocylindrical
Pseudocylindrical maps generalize the "polycylindrical" concept, which can be presented as a generalization of the cylindrical group of projections, applied to discrete latitude ranges.
Polycylindrical concept
Here, nine strips with partial cylindrical equidistant maps cover the globe (parts were removed for clarity). Each strip, tangent to and fitting a different standard parallel, has one half buried inside the globe (the equatorial strip is the exception); lengths vary, but the vertical scale is identical in all nine maps.
Polycylindrical concept
Generalizing the flattened discrete map above, the continuous limiting case with infinitely many strips would be the pseudocylindrical sinusoidal projection; since the strips would exactly cover the globe surface, it is easy to see the final result is equal-area.
Slanted sinusoidal map Slanted sinusoidal map
Aligning the central meridians of each strip is not mandatory: other arrangements (examples on the right) are possible but maybe not so useful.


In the normal aspect at higher latitudes, in all cylindrical projections there is strong shape distortion, and usually area is also greatly exaggerated. In particular, the poles are infinitely stretched to lines, or can not even be included, as in Mercator's projection. The pseudocylindrical class of projections attempts another trade-off of shape vs. area; in the normal equatorial aspect, they are defined by: Some properties, inherited from cylindrical maps, are noteworthy: Since their parallels and meridians do not always cross at right angles, conformality is denied to pseudocylindrical maps; in fact, most suffer from strong shape distortion at polar regions. Therefore, many were designed for equivalence.

The concept of "polycylindrical" maps, comprising many thin cylindrical maps with correct parallel length but arbitrary height, was suggested by W.Tobler by analogy to polyconic projections being a generalization of the conic group. More generally, any pseudocylindrical map can be conceptually created by juxtaposing a (possibly infinite) number of partial cylindrical maps, not necessarily using the same scale for parallels.

Even though their geometric constraints are much more flexible than for cylindrical designs, a few decisions are still common to all pseudocylindrical projections: the shape of meridians (straight lines, conic sections, sinusoidals, arbitrary), distance distribution of parallels, linear or pointed poles. Often identical combinations of choices have been adopted by different authors, and several pseudocylindrical "projections" are actually synonyms. Given those alternatives, projections in this large group are classified somewhat arbitrarily; many designs could easily be assigned to more than one section. Of infinitely many possible pseudocylindrical projections, several are useful didactical devices and popular choices for world maps.

Classic Pseudocylindrical Projections

Sinusoidal (Sanson-Flamsteed) Projection

Sinusoidal (Sanson-Flamsteed) map
Sinusoidal (Sanson-Flamsteed) map, graticule spacing 10°

Despite the common name, the "Sanson-Flamsteed" projection was not first studied by either Nicholas Sanson (ca. 1650) or John Flamsteed (1729, published posthumously), but possibly by Mercator — at least it was included for maps of South America in later editions (1606) of Mercator's atlas, and has been referred to as the Mercator equal-area projection, or Mercator-Sanson. Since in the normal aspect all meridians are sinusoids, it is also called sinusoidal, and can be easily deduced.

The sinusoidal projection is equal-area and preserves distances along the horizontals, i.e. all parallels in an equatorial map are standard lines — but only the central meridian. Although the equatorial band is reproduced with little distortion, the polar caps suffer from poor legibility due to shearing. The partially constant scale and simple construction still recommend this projection for continents like Africa and South America, frequently after convenient recentering and, for world maps, interruption.

Both the Equator and central meridian are standard lines, thus the whole map is twice wide as tall, the same proportions of the Mollweide projection with which it has been frequently combined.

Mollweide's Projection

Normal Mollweide map
Normal Mollweide map
Normal Mollweide map Rescaled Mollweide projections by Bromley (above, with standard Equator) and Tobler (on the right, circular) are equal-area too. Normal Mollweide map

Created by the German Karl B. Mollweide, the eponymous pseudocylindrical projection is bounded by an ellipse; poles are points and its Equator is twice as long as the straight central meridian, but neither is a standard line. All other meridians are elliptical arcs, and parallels are unequally spaced in order to preserve areas. Only the intersections of the central meridian with the standard parallels 40°44'12"N and S are free of distortion. Even though its geometry is easily deduced, calculation is more complex than for the other classic pseudocylindrical still important today, the sinusoidal — this and the loss of uniform scale along the central meridian are the price paid for lesser crowding in polar areas.

Despite the equal area property and its pleasant shape, Mollweide's projection received little recognition since publication in 1805, becoming better known only after the French Jacques Babinet presented it as the homalographic in 1857. Historically, its other common aliases include elliptical, Babinet, homolographic (from the Greek homo for "same", thus equal-area).

Profoundly influential, this projection was combined with the sinusoidal in fused (John P. Goode's homolosine, Allen K. Philbrick's Sinu-Mollweide, György Érdi-Krausz) and averaged (Samuel W. Boggs and Oscar S. Adams's eumorphic; some authors also relate it to the Winkel II) designs. Interrupted variants, alone or combined, have also been popular.

Other variations include oblique aspects like John Bartholomew's Atlantis and simple rescaling by orthogonal reciprocal factors, which preserves areal equivalence while changing both aspect ratio and the angular distortion pattern. For instance, Waldo Tobler (1962) suggested making the whole map circular with standard parallels 73°7'43.85" N and S; Robert H. Bromley's projection (1965) elongates the ellipse merging the standard parallels at the Equator.

Mollweide's 2:1 ellipse is occasionally mistaken for Aitoff's and Hammer's projections, neither of which are pseudocylindricals, although the latter is also equal-area. Much more easily confused is an elliptical full-world extension of Apian's second globular projection, mathematically much simpler and not equal-area.

Foucault stereographic equivalent map
Foucault's stereographic equivalent projection

Foucault's Stereographic Equivalent

One of the pseudocylindrical projections presented by De Prépetit Foucault in 1862 is the stereographic equivalent: its straight parallels have spacing identical to the one in the equatorial aspect of the azimuthal stereographic and Braun's stereographic cylindrical, while meridians, calculated in order to preserve area, are fifth-order curves.

Despite being one of the first "modern" pseudocylindrical projections, after the sinusoidal and Mollweide's, the stereographic equivalent is little more than a novelty: since the parallel spacing increases faster at higher latitudes, the horizontal scale shrinks accordingly, creating sharp poles with excessive angular distortion. Another equal-area pseudocylindrical projection with parallel spacing derived from azimuthal principles, the quartic authalic, was only marginally more successful, but in practice is considerably more interesting and useful. The same concept, applied to the azimuthal orthographic, evidently leads to Lambert's cylindrical equal-area projection; from the azimuthal equidistant to the sinusoidal; and is impractical for the gnomonic.

Maps with Straight Graticules

Collignon map in diamond form
Collignon map, symmetrical diamond form
Collignon map in diamond form Normal Mollweide map
Interrupted Collignon map Collignon map in its most common form

The graticule in the equatorial aspect of a few pseudocylindrical projections comprises only straight lines, possibly broken at the Equator. Those include the trapezoidal, one of the oldest projections, Eckert's I and II, composites like the HEALPix grid, and Snyder's cartographic pun with a serious message.

Collignon's Projection

Édouard Collignon's projection, introduced in 1865, preserves areas but strongly distorts shape. In the equatorial aspect, both northern and southern hemispheres can be either a isosceles triangle with base on the Equator and height half the Equator's length, or a isosceles trapezium (British: trapezoid) with the small base at the Equator. All graticule lines are straight but the meridians are optionally broken at the Equator.

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; more recently, it has found practical application as part of the HEALPix format.

Craster parabolic map
Craster's parabolic projection

The Parabolic Equal-area Projection

In 1929, Lt.Col. John E.E.Craster presented the features of three pseudocylindrical equal-area projections with meridians based on conic sections. He rejected the elliptical and hyperbolic versions, and included an abbreviated table of ordinates for the parabolic design. Mathematical details were later presented by Charles H.Deetz and Oscar S.Adams (1934), including an interrupted version, and again by Adams (1945).

Although with the same 2:1 proportions and superficially resembling the sinusoidal, Craster's parabolic projection has more convex meridians, somewhat reducing shape distortion in high latitudes near the boundary meridians. Its mathematical development is an interesting application of solving cubic functions.

As part of a series (1934) of pseudocylindrical projections based on trigonometric and conic curves, Reinholds Putniņš duly noted his P4 projection as identical with Craster's parabolic.

Projections by Siemon and Others

Quartic authalic map
Quartic authalic map
Loximuthal maps with central meridian 0°. The reference latitude is 0° (top) and 51.5°N, near Greenwich (above). Loximuthal maps must be created for each point of interest, like these examples.

In prewar Germany, Karl Siemon developed four pseudocylindrical projections, the first one undoubtably the best known. All but one were later independently rediscovered by other authors.

The Loximuthal Projection

Dubbed Wegtreue Ortskurskarte by Siemon, his first projection (1935) is today almost always referred to as the loximuthal, after W.Tobler who published it in 1966. The portmanteau nicely summarizes its unique feature: given a reference latitude, all straight lines touching its intersection with the central meridian are loxodromes. In addition, they are all standard lines with correct azimuth, analogous to straight lines through the center of an azimuthal equidistant map.

In contrast, on a Mercator map all loxodromes are straight lines, but scale quickly changes not only between different loxodromes but also along the same lines; azimuths are also not uniform.

The loximuthal projection is not equal-area, and is conformal only where the central meridian intersects the reference (also standard) parallel. Unlike most pseudocylindrical projections, it is not symmetrical with respect to the Equator, unless the Equator is selected to be the reference latitude.

The Quartic Authalic Projection

Quartic authalic map
Quartic authalic map

Like Foucault's stereographic equivalent, the quartic authalic is another equal-area (quartic because its meridians are fourth-order curves, authalic from Greek autos ailos, "same area") projection derived from azimuthal principles; it is Siemon's third published projection (1937) and was independently developed by O.S.Adams (presented in 1944, then again in 1945 with a detailed analysis, without a name).

Since its parallels are spaced exacly like the equatorial aspect of Lambert's azimuthal equal-area projection, it is the limiting case of a transformation which led to the much more famous Hammer projection, among others. Adams, who preferred an interrupted version in two hemispheres, emphasized how its poles are less pointed than the sinusoidal's, while the equatorial band avoids the vertical exaggeration of Mollweide's. Nevertheless, the quartic authalic was generally ignored, except as an inspiration for the flat-polar quartic projection by McBryde and Thomas.

Finally, Siemon's fourth projection is simply the quartic authalic with reciprocal horizontal and vertical rescalings to preserve area while fitting a 2 : 1 aspect ratio.

Kavrayskiy's Fifth Projection

Kavrayskiy V map
Kavrayskiy's fifth projection

Besides conic and cylindrical projections, the Russian Vladimir V.Kavrayskiy also proposed three pseudocylindrical designs, of which the V and VI are based on trigonometric functions and are equal-area. The fifth (1933) has pointed poles, standard lines at the 35°N and S parallels and meridians calculated from sinusoids, but not sinusoidal themselves. The sixth (1936) is identical to Wagner's first projection of 1932, with sinusoidal meridians.

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