Map Projections: Interrupted Maps

Map Projections

Interrupted Maps

Introduction

Globe wrapped in interrupted sinusoidal map

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 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 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 neighbor points on the sphere become widely spaced in the map; therefore, too many lobes negate the benefits of interruption. As mentioned for oblique maps, such distortion actually happens at the edges of any ordinary projection.

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

Simple Interrupted Maps

Interrupted Sanson-Flamsteed (3+3 lobes)

Interrupted sinusoidal map, with three full lobes per hemisphere

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

Interrupted Sanson-Flamsteed map (9 + 9)

Interrupted sinusoidal map, each hemisphere split in nine lobes

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 a different lobe arrangement and much more primitive projection methods).

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 corresponding conterminous map.

Maps with lobes 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.

Interrupted Sanson-Flamsteed map emphasizing oceans

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

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 sinusoidal projection, all other properties still hold, including the mapped area. Asymmetrical lobes are featured in classic interrupted maps by Goode, Boggs and McBryde.