Map Projections

Interrupted Maps

Maps wrapped on a globe show advantages and limitations of selected projections
Comparing equatorial maps at identical scales tangent to a globe is an interesting exercise. In particular, which map could be glued to the global surface more neatly? why is South America so similar in the second and fifth maps, but not in the fourth one (which uses the same projection as the fifth one)? why are Africa and Western Europe nearly identical in the second, fourth and fifth maps? which is the "best" version of Greenland (check the globe)?
Lambert's cylindrical: equal-area, with severe horizontal stretching and vertical compression near the poles (Greenland and Antarctica are barely visible but presented in entirety)	Equidistant cylindrical: has correct vertical scale (the total height equals half a circumference), but severe areal exaggeration near the poles, where horizontal scale becomes infinite	Mercator's projection (partially transparent here) compensates the horizontal scaling by stretching the meridians; in order to preserve local angles, the poles are moved to infinity	Ordinary sinusoidal: pseudocylindrical and equal-area; total height and scale are correct along the central meridian, but shapes are badly skewed far from it	A symmetrically interrupted sinusoidal remains pseudocylindrical and equal-area; it has correct scale along the central meridian of every lobe, but many discontinuities

Introduction

No map projection can preserve shape and size simultaneously, and the larger the mapped area, the more pronounced the total distortion. Rectangular world maps are prone to excessive area and distance stretching, while those using circular and elliptical projections usually present too much shape distortion at the periphery.

Interrupted sinusoidal map, with three full lobes per hemisphere

Interrupted maps attempt a compromise, cutting the terrestrial surface along some arbitrarily chosen lines and projecting each section, or lobe (or gore, in case interruptions repeat periodically along related lines like meridians), separately with lower stretching. Commonly lobe boundaries are designed to fall upon less important (for the map's purpose) areas, like oceans.

In a sense, interrupting a map creates another kind of distance distortion, since points adjacent on the sphere become widely spaced in the map; therefore, too many lobes would negate the benefits of interruption. As mentioned for oblique maps, any ordinary flat map is actually interrupted in at least one point or line which becomes the map's outer edge. That distortion is almost always overlooked, and here I refer only to nontrivial "internal" interruptions.

Interrupted projections were used by Martin Waldseemüller (1507, 12 gores), Henricus Glareanus (1527, 12 gores) and Leonardo da Vinci (ca. 1514, 8 gores), among others. Another early example was a variation of Werner's projection by Mercator (1538). Some modern designs, like the HEALPix projection, were explicitly created with interruption in mind.

Interrupted sinusoidal map, each hemisphere split in nine lobes

Simple Interrupted Maps

The sinusoidal (also known as Sanson-Flamsteed) projection has a simple construction and interesting features: pseudocylindrical, equal-area and constant vertical scale (i.e., parallels are uniformly spaced). On a whole-world equatorial sinusoidal map, the polar regions at extreme longitudes suffer from strong shape distortion (shearing). Interrupting the map along meridians preserves its better features with lesser shearing.

Interrupted sinusoidal map with asymmetrical lobe boundaries emphasizing oceans over land.

Interrupted sinusoidal map with asymmetrical lobe boundaries emphasizing lands over oceans, after Deetz and Adams.

Clearly there is a trade-off: increasing the number of lobes further reduces shape distortion as each lobe is centered around its own different meridian, until the discontinuities make the map more a curiosity than something useful in its planar form. However, a lobed map could, if printed on a sheet of flexible material, cut and joined at the borders, make up a fairly good globe; interestingly enough, ancient gore maps had exactly that purpose, albeit with more primitive geometry than the sinusoidal's. Most early maps survived only as crude copies or rough descriptions; for instance, only the shape and size of Glareanus's gores are known with certainty, but not the graticule details; Waldseemüller's is also obscure.

Another gore map. Since it is based on the polyconic projection, parallels are curved and it is not equal-area. Global areal distortion is not as pronounced as in the original conterminous map.

Maps with lobes arranged in a row along the Equator make clear why cylindrical projections necessarily distort polar regions: they must horizontally stretch and fasten them together in order to force a rectangular map.

Globe showing Penelope's round-the-world odyssey. Diameter 10cm, interrupted polyconic projection with 12 gores.

Finally, as usual, designing an interrupted map reflects the author's particular point of view. An asymmetrical arrangement of lobe boundaries can avoid cutting the three major oceans instead of land masses. In the case of a pseudocylindrical projection as the sinusoidal, all other properties still hold, including the mapped area. Asymmetrical lobes are featured in classic interrupted maps by John P.Goode, Samuel W.Boggs/Oscar S.Adams and Felix W.McBryde/Paul D.Thomas. The sinusoidal projection itself was used by Goode (ca. 1916) and Charles H.Deetz and O.S.Adams (1934, clipped beyond 80°N and 70°S, complete in a USGS publication of 1978).

A gore map using Apian's first projection. It is not equal-area as the sinusoidal, and distorts shape more than the polyconic version, but its geometry is simpler to build with a ruler and a pair of compasses.

Myriahedral Maps

Traditionally, interrupted maps attempt to minimize distortion in area, shape, or both, while keeping discontinuities at a minimum. By abandoning this last requisite, J.J.van Wijk's myriahedral maps (2008) are practically equal-area and conformal, but with a number of lobes approaching infinity.

Myriahedral maps are produced by a flexible algorithm which first approximates the Earth by a polyhedron, then creates a tree graph covering all its numerous faces. Edges are split and the polyhedron is flattened, each tree leaf becoming the end of a lobe. By assigning appropriate weights to where cuts are desirable, the result may resemble traditional designs (e.g., cuts only along meridians yield azimuthal or cylindrical projections, depending on alignment constraints; cuts only along parallels yield polyconic maps) — or a completely arbitrary criterion may be chosen, e.g., irregular lobes leaving either continents or oceans uninterrupted. With the power of digital computers and numerical optimization methods, a surprising range of maps is possible.

Due to their unusual nature and recent history, it is still difficult finding practical applications for myriahedral projections as ordinary maps. Among others, the author suggests their use as a qualitative measure of distortion. For instance, to preserve the conformal aspect ratio while avoiding areal stretching, lobes become increasingly thin; consequently, seen from a distance, areas of greater distortion appear more and more transparent.

www.progonos.com/furuti    November 21, 2012