Map Projections

Map Projections

For the representation of the entire surface of the earth without any kind of distortion, a map must have a spherical surface; a map of this kind is known as a globe. A flat map cannot accurately represent the rounded surface of the earth except for very small areas where the curvature is negligible. To show large portions of the earth's surface or to show areas of medium size with accuracy, the map must be drawn in such a way as to compromise among distortions of areas, distances, and direction. In some cases the cartographer may wish to achieve accuracy in one of these qualities at the expense of distortion in the others. The various methods of preparing a flat map of the earth's surface are known as projections and are classified as geometric or analytic, depending on the technique of development. Geometric projections are classified according to the type of surface on which the map is assumed to be developed, such as cylinders, cones, or planes; plane projections are also known as azimuthal or zenithal projections. Analytical projections are developed by mathematical computation.

Cylindrical Projections

In making a cylindrical projection, the cartographer regards the surface of the map as a cylinder that encircles the globe, touching it at the equator. The parallels of latitude are extended outward from the globe, parallel to the equator, as parallel planes intersecting the cylinder. Because of the curvature of the globe, the parallels of latitude nearest the poles when projected onto the cylinder are spaced progressively closer together, and the projected meridians of longitude are represented as parallel straight lines, perpendicular to the equator and continuing to the North and South poles. After the projection is completed, the cylinder is assumed to be slit vertically and rolled out flat. The resulting map represents the world's surface as a rectangle with equally spaced parallel lines of longitude and unequally spaced parallel lines of latitude. Although the shapes of areas on the cylindrical projection are increasingly distorted toward the poles, the size relationship of areas on the map is equivalent to the size relationship of areas on the globe.

The familiar Mercator projection, developed mathematically by the Flemish geographer Gerardus Mercator, is related to the cylindrical projection, with certain modifications. A Mercator map is accurate in the equatorial regions but greatly distorts areas in the high latitudes. Directions, however, are represented faithfully, and this is especially valuable in navigation. Any line cutting two or more meridians at the same angle is represented on a Mercator map as a straight line. Such a line, called a rhumb line, represents the path of a ship or an airplane following a steady compass course. Using a Mercator map, a navigator can plot a course simply by drawing a line between two points and reading the compass direction from the map.

Azimuthal Projection

This group of map projections is derived by projecting the globe onto a plane that may be tangent to it at any point. The group includes the gnomonic, orthographic, and stereographic plane projections. Two other types of plane projections are known as the azimuthal equal area and the azimuthal equidistant; they cannot be projected but are developed on a tangent plane. The gnomonic projection is assumed to be formed by rays projected from the center of the earth. In the orthographic projection the source of projecting rays is at infinity, and the resulting map resembles the earth as it would appear if photographed from outer space. The source of projecting rays for the stereographic projection is a point diametrically opposite the tangent point of the plane on which the projection is made.

The nature of the projection varies with the source of the projecting rays. Thus the gnomonic projection covers areas of less than a hemisphere, the orthographic covers hemispheres, the azimuthal equal area and the stereographic projections map larger areas, and the azimuthal equidistant includes the entire globe. In all these types of projection, however (except in the case of the azimuthal equidistant), the portion of the earth that appears on the map depends on the point at which the imaginary plane touches the earth. A plane-projection map with the plane tangent to the surface of the earth at the equator would represent the equatorial region, but would not show the entire region in one map; with the plane tangent at either of the poles, the map would represent the polar regions.

Because the source of the gnomonic projection is at the center of the earth, all great circles, that is, the equator, all meridians, and any other circles that divide the globe into two equal parts, are represented as straight lines. A great circle that connects any two points on the earth is always the shortest distance between the two points. The gnomonic map is therefore a great aid to navigation when used in conjunction with the Mercator.

Conic Projections

In preparing a conic projection a cone is assumed to be placed over the top of the globe. After projection, the cone is assumed to be slit and rolled out to a flat surface. The cone touches the globe at all points on a single parallel of latitude, and the resulting map is extremely accurate for all areas near that parallel, but becomes increasingly distorted for all other areas in direct proportion to the distance of the areas from the standard parallel.

To provide greater accuracy, the Lambert conformal conic projection assumes a cone that passes through a part of the surface of the globe, intersecting two parallels. Because the resulting map is accurate in the immediate vicinity of both parallels, the area represented between the two standard parallels is less distorted than the same area reproduced by a single conic projection.

The polyconic projection is a considerably more complicated projection in which a series of cones is assumed, each cone touching the globe at a different parallel, and only the area in the immediate vicinity of each parallel is used. By compiling the results of the series of limited conic projections, a large area may be mapped with considerable accuracy. Because a cone cannot be made to touch the globe in the extreme polar and equatorial regions, the various conic projections are used to map comparatively small areas in the temperate zones. Polyconic maps offer a good compromise in the representation of area, distance, and direction over small areas.

Mathematical Computation

For accurate delineation of large areas on a small scale, a number of so-called projections have been developed mathematically. Maps based on mathematical computation represent the entire earth in circles, ovals, or other shapes. For special purposes the earth often is drawn not within the original form of the projection but within irregular, joined parts. Maps of this type, called interrupted projections, include Goode's interrupted homolosine and Eckert's equal-area projection.