|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,
|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|
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 seek a compromise, cutting the terrestrial surface along some arbitrarily chosen lines, then projecting each section, or lobe (or gore, in case interruptions repeat periodically along related lines like meridians), separately with lower overall deformation. Often lobe boundaries are designed to fall on less important (regarding the map's purpose) areas, like oceans.
Although the Earth's surface is a continuous, unbound region, any flat map has a definite border which invariably introduces a kind of scale distortion: two points which are neighbors on Earth can have their counterparts widely separated, at opposite ends of the map. Therefore there is a practical limit to any additional interruptions, to a point where discontinuities negate the benefits of additional lobes (but see myriahedral maps below). Sometimes a single point on Earth is represented in the map's border by lines — e.g., in full-world azimuthal, most cylindrical, and all flat-polar pseudocylindrical projections — with an infinite scale factor. And obviously at least two points on the border actually represent the same point on Earth. This kind of 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|
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 a Swedish atlas and Deetz and Adams (including polar regions).|
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. Maximum areal distortion is much smaller than 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 like the sinusoidal, and distorts shape more than the polyconic projection, but its geometry is simpler to build with a ruler and a pair of compasses.|
|O.S.Adams, one of the authors of the quartic authalic projection, favored an interrupted version in two hemispheres|
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.
|Trivial interruptions: where do the edges of the map come from?|
|O.M.Miller proposed a classification of projections by the origin of discontinuities, trivial or otherwise, in seven categories. They are illustrated here with maps conventionally centered on the intersection of the Equator and the prime meridian.|
|1: a single point, like the azimuthal equal-area (pictured on the right) and the azimuthal equidistant|
|2: two points, like most cylindrical (Lambert's pictured) and conic projections; same as category 4 for flat maps since the projection surface must be unrolled|
|3: two small circles in parallel planes, like the Mercator (pictured) and central cylindrical projections, which cannot show the whole world|
|4: half a great circle, like all uninterrupted pseudocylindrical projections (flat-polar quartic pictured)|
|5: a great circle, like the azimuthal orthographic (pictured), which is limited to a hemisphere|
|6: segments of great circles, like star projections and most interrupted pseudocylindrical projections (Goode's homolosine pictured)|
|7: other curves, like the azimuthal stereographic and gnomonic, the conformal conic (pictured), Raisz's orthoapsidals and many others|
|In normal aspects, the small circles involved are usually parallels, and the great circles are pairs of opposite meridians.|