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.
Polelines avoid the crowded appearance of projections like the sinusoidal and Mollweide at the cost of horizontal scale distortion, which is infinite at the poles.
Cartographers such as K.H. Wagner and M. Eckert developed entire “families” of flat-polar projections. Other projections with polelines include Nell's pseudocylindrical (the first nontrivial one), Winkel I and Winkel II.
In 1906 the German professor Max Eckert (later Eckert-Greifendorff) published six pseudocylindrical projections sharing some features in the normal aspect:
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.
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.
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 the projection of Ortelius's oval map, which lacks constant scale along parallels and thus is not pseudocylindrical.
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.
Several other pseudocylindrical projections, most notably by Siemon, McBryde and Thomas, and a series by Wagner, are also based on polelines and sinusoidal meridians.
The shape of Karl D.P. Rosén's pseudocylindrical projection (1926) is derived from the sinusoidal, with latitudes compressed: the poles are mapped to lines corresponding to the parallels approximately 53°7′48″N and S, thus changing the aspect ratio from 2:1 to about 3.388:1. This is similar to the compression of coordinates in projections by Aitoff, Hammer, Eckert-Greifendorff, but without a compensating expansion. The projection is scaled up and parallels are spaced to make the whole map equal-area.
Rosén's approach was later generalized by Urmayev.
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.
Instead of applying a geometric perspective process such as that used for classic azimuthal projections, or achieving some previously defined goal, like Mercator's, or using 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 (the interpolation method is not rigidly specified). Like in all pseudocylindrical projections, in the normal aspect meridians are equally spaced along all parallels, which are horizontal and straight (and, between 38°N and 38°S, regularly spaced). 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.
Another compromise design which preserves neither shape nor areas, Kavrayskiy's seventh projection of 1939 is a flat-polar pseudocylindrical with elliptical meridians and correct scale at the central meridian. In contrast with Robinson's, its mathematical derivation and description are much simpler.
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), the flat-polar quartic projection, has fourth-order curves as meridians and is equal-area.