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  A geometric transformation that portrays a spherical world on a flat map (see map image and map). The alternative — a globe — is cumbersome, expensive to reproduce and useless to viewers eager to look from place to place at will, without lifting or turning a bulky three-dimensional model. Unless viewed directly, a globe also distorts the relative appearance of size, shape, distance and direction.

Map projection is readily understood as a two-stage process. In stage one, shrinking the world to a hypothetical globe establishes the map\'s stated scale (Maling, 1992, pp. 82-3). In stage two, placing this globe in contact with a plane, cone or cylinder allows the mathematical transfer of meridians, parallels, coasts and boundaries onto a flat or flattenable surface (see figure). In some projections a third stage readjusts locations and shapes, as when the sinusoidal projection corrects for the enlarged poleward areas on the equirectangular projection by bending meridians inward toward a central meridian.

Flattening a globe distorts distances. scale is no longer the same everywhere and in all directions away from all points. In general, a projection maintains its stated scale only at the point (for projection onto a plane) or line of contact (for a cone or cylinder). Moreover, distortion grows with increased distance from the tangent point or standard line. Allowing the projection surface to penetrate the globe affords a broader, more balanced pattern of distortion by providing a circular band of low distortion in the case of a plane or two bands of low distortion on a cone or cylinder. A cartographer can tailor a projection to a specific country or region by carefully selecting the projection surface and its orientation to the globe (Robinson and Snyder, 1991).

Because of many possible perspectives and orientations, map projection is a source of frustration as well as insight (Snyder and Voxland, 1989). Although all projections distort most distances, an equidistant projection might preserve true distance from the equator or from a pole or another point of interest. Similarly, an equivalent projection can preserve the true relative areas of countries and continents, whereas a conformal projection, which preserves small shapes as well as angles around points, is especially useful on large-scale maps of small areas. Unfortunately, equidistance, equivalence and conformality are mutually exclusive properties.

Projections can be controversial (Monmonier, 1995, pp. 1-44). The Mercator projection, a conformal mapping onto a cylinder, is valuable to navigators because straight lines are lines of constant geographical direction. But as a general view of the world, the Mercator map diminishes the size and presumably the importance of developing nations, nearer than most developed countries to the map\'s low-distortion zone along the equator. By contrast, the Gall-Peters map, an equivalent projection proposed as \'fair to all peoples\', accurately portrays relative area but grossly distorts the shapes of Africa and Latin America (Snyder, 1993, pp. 164-6). Although other projections offer a more balanced treatment of area and shape, authors eager to focus on people, not land area, can map socioeconomic data on a cartogram on which area represents population. (MM)

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projection principal developable surfaces (above) generate distinctive projection grids (below)

References Maling, D.H. 1992: Coordinate systems and map projections, 2nd edn. Oxford: Pergamon Press. Monmonier, M. 1995: Drawing the line: tales of maps and cartocontroversy. New York: Henry Holt. Robinson, A.H. and Snyder, J.P., eds, 1991: Matching the map projection to the need. Rockville, Md.: American Congress on Surveying and Mapping. Snyder, J.P. 1993: Flattening the earth: two thousand years of map projections. Chicago: University of Chicago Press. Snyder, J.P. and Voxland, P.M. 1989: An album of map projections, Professional Paper 1453.Washington: US Geological Survey.



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