Map Projections: Pseudocylindrical Projections

Map Projections

Pseudocylindrical Projections

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.

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.

Introduction

In all cylindrical projections there is strong shape distortion, and usually area is also greatly exaggerated, at higher latitudes (in the normal aspect). 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:

straight horizontal parallels, not necessarily equidistant
arbitrary curves for meridians, equidistant along every parallel

Some properties, inherited from cylindrical maps, are noteworthy:

horizontal parallels visually preserve latitude relations, thus easing correlation of phenomena which mostly depend on distance from Equator, e.g., daylight periods, climate, winds and greenhouse warming
constant scale at any point of a parallel eases measurements in its direction

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.

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.

Common Pseudocylindrical Projections

Sinusoidal (Sanson-Flamsteed) Projection

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

Despite the common name, this projection was not first studied by either Sanson (ca. 1650) or Flamsteed (1729, posthumously), but probably by Mercator (at least it was included in later editions of Mercator's atlas). It is also called sinusoidal and Mercator equal-area, and can be easily deduced. It preserves area and also distances along the horizontals, i.e., all parallels in a normal map are standard lines. Although the equatorial band is reproduced with little distortion, polar caps can be hard to read. The partially constant scale and simple construction still recommend this projection for continents like Africa and South America, frequently after convenient recentering.

Both the Equator and central meridian are standard lines, thus the whole map is twice wide as tall.

Mollweide Projection

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.

Collignon Projection


Collignon map, symmetrical diamond form

Interrupted Collignon map	Common Collignon map

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 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.

Quartic Authalic Projection

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.

Flat-Polar Pseudocylindrical Projections

Maps with polelines, called flat-polar, represent the poles as straight lines instead of points. Of course all cylindrical projections are flat-polar in the equatorial aspect, but the term is commonly applied to a large group of pseudocylindrical designs. Cartographers such as K. H. Wagner and M. Eckert developed whole "families" of flat-polar projections.

Polelines avoid the crowded appearance of projections like the sinusoidal and Mollweide at the cost of scale distortion.

Flat Polar Quartic Projection

Flat polar quartic map

Several projections created by Felix W. McBryde and Paul Thomas have polelines one-third as long as the Equator. The fourth and best known (1949) has fourth-order curves as meridians and is equal-area.

Flat-polar Projections by Eckert

In odd-numbered Eckert projections (Eckert I here), the graticule is nearly square at the center

In 1906 the German professor Max Eckert (later Eckert-Greifendorff) published six pseudocylindrical projections sharing some features in the normal aspect:

Eckert's projection II is similar to his first design, but nonconstant parallel spacing makes it equal-area

the central meridian is straight, half as long as the Equator, and a standard line in odd-numbered projections
poles are flat, half as long as the Equator
even-numbered projections are equal-area
odd-numbered projections have equally spaced parallels

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.

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.

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.

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.

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.

Robinson Projection

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.

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.