MAP PROJECTIONS AND GRID SYSTEMS Most maps are printed on paper or other flat surface. Since the Earth is nearly spherical, it is impossible to represent its features on a plane without introducing distortions. The method by which the information is transferred determines the nature of the map projection, which is geometrically defined on the map by the graticule, representing parallels of latitude and meridians of longitude. The Greek astronomer, Hipparchus, is credited with superimposing on the Earth the system of parallels and meridians to provide the means by which the locations of features can be stated uniquely. Since only one meridian and one parallel can be drawn through a given point, the intersection of a meridian and a parallel defines a single location. When meridians and parallels are represented on a flat surface by means of a projection, points on the Earth can be represented uniquely on that surface to form a map. PROPERTIES OF PROJECTIONS All maps contain distortions inherent in their particular projections. These distortions affect the portrayal of area, shape, direction, and distance. Projections can be selected to eliminate or minimize one or more of the distortions at the expense of others, or to partially control several and thereby minimize the general distortion. Some projections maintain a constant areal scale. In simple terms this means that an object, such as a coin, placed on a map covers the same amount of geographical area wherever it is placed. Such projections are termed "equal-area" and are used when measurements or comparisons of areas are of primary importance. To achieve equal-area properties (equivalency) a projection must distort shapes and angles. In contrast, a projection that retains shapes and angles is called conformal and cannot be equal-area. Conformal projections show small areas, such as lakes and ponds, with the same shapes as they have on the globe. To do so, the parallels and meridians must meet at right angles, and the local scale around any point must not vary. Most modern maps, particularly at larger scales, are constructed on conformal projections because of the importance of true shape and direction. All projections have lines along which distances are shown correctly. They are called standard lines and are usually selected meridians or parallels. Cer tain projections show true distances from a selected point and are termed equidistant. Most directions between points on the Earth are correctly shown on all conformal projections, but the longest distances are less accurately represented. When exact directions from a specific point are needed, azimuthal (zenithal) projections are used. DEVELOPABLE SURFACES The term projection can be explained by the concept of a point light source shining through a model of the Earth causing the graticule to cast its shadow on an object. The object can be any number of geometric surfaces, but the most commonly used are the cone and its limiting shapes, the plane (cone with an altitude of 0 (zero) and an apex of 180°) and the cylinder (cone with an altitude of infinity and an apex angle of 0°). Both the cone and the cylinder can be flattened to a plane without further distortions and thus are known as developable surfaces. Conic projections are transferred to an imaginary cone placed over the Earth, sometimes obliquely but usually so their axes coincide. The side of the cone can be tangent to the Earth along a selected parallel of latitude or can intersect the Earth along two parallels. Parallels of tangency and secancy are called standard parallels and maintain a constant scale. Distortions increase away from the standard parallels. Azimuthal projections are transferred to a plane that is intersecting the Earth, tangent to it, or neither. The perspective center (light source) of the projection can be at the center of the Earth, on the surface, somewhere between, or at a point in space. Cylindrical projections are transferred to an imaginary cylinder placed tangent to or intersecting the Earth. The lines of tangency can be standard parallels, standard meridians, or any selected great circle (a circle whose center lies at the center of the Earth and whose diameter equals that of the Earth). Lines of intersection (secancy) are standard lines and need not coincide with lines of the graticule. Variations in projections also occur with changes in the orientation of the developable surface with respect to the Earth's axis. The orientation of a plane can be polar (perpendicular to the axis and centered at a Pole), equatorial (parallel to the axis and centered at a point on the Equator), or oblique (intersecting the axis at an acute angle). Likewise, the axis of a cylinder can be coincident with, perpendicular to, or oblique from the Earth's axis. Generally, a cone's axis coincides with that of the Earth. Another fundamental of projections is the method of transfer (projection) from the Earth's surface. Transfer by extending rays from the perspective center is a simple geometric projection in which rays connect the Earth's surface features to the projection surface. However, most map projections are mathematically transformed and are not defined geometrically. The mathematical transformation is defined to provide the specific property sought, such as conformality or equivalency. Thus, a given projection involves three basic considerations, (1) selection of a developable surface, (2) orientation of the surface to the Earth's axis, and (3) a transformation (usually mathematical) by which the Earth's surface features are transferred to the projection surface. The projection surface is then developed into a plane, resulting in a map with defined geometric characteristics. COMMONLY USED PROJECTIONS Map projections are too numerous to give descriptions of all of them here. Therefore, only a few of the projections more commonly used by engineers are discussed. Many other projections are available for particular uses, especially in small-scale mapping. A number are described by Dietz and Adams (1944). Lambert Conformal Conic Projection The Lambert conformal conic projection was devised in 1772 by Johann Heinrich Lambert. It assumes a cone intersecting (secant to) the Earth along two standard parallels (fig. 5) passing through the mapped area. The axis of the cone coincides with the Earth's axis. Scale is correct along both standard parallels, too small between them, and too large beyond them. The distortions grow as one moves away from the standard parallels. Because scale is correct along two parallels, the Lambert projection is often preferred to the simple conic projection with only one standard parallel. Because of the north-south distortions, this projection is most suitable for mapping areas that are elongated east-west. It is used for the 1:1,000,000-scale world aeronautical charts, the 1:500,000scale sectional aeronautical charts, the 1:500,000 A E F F FIGURE 5.-Lambert conformal conic projection; a secant cone having two standard parallels, lines AEB and CFD. Line EF is the central meridian. scale State base maps, and the 1:24,000-scale 7.5-min topographic quadrangles that lie in zones where the Lambert projection is the base for the State plane coordinate system. Polyconic Projection The polyconic projection (fig. 6) was devised by Ferdinand Hassler, the first superintendent of the U.S. Coast Survey (later U.S. Coast and Geodetic Survey, now NOS). It projects the Earth's surface on a series of cones, each tangent to the Earth along a different selected parallel of latitude. Scale is correct along each standard parallel and along the central meridian, but not elsewhere. The principal virtue of the polyconic projection is that it is easy to construct and plot by hand on map grids can be superimposed on them without significant error. Although it has minimal distortion over small areas, the polyconic projection is neither conformal nor equal-area. The polyconic projection was widely used as the standard for medium- and large-scale maps of the United States. USGS used the projection for quadrangle maps until the 1950's when rectangular coordinate plotters were adopted for plotting base sheets. Mercator Projection The Mercator projection takes its name from the Latin surname of Gerhard Kramer, who devised it. The projection first appeared in 1569 when Mercator published a map of the world. The Earth's surface is projected on a cylinder tangent at the Equator or secant along two parallels with its axis coincident with the Earth's axis (fig. 7), then the cylinder is cut and unrolled to a flat surface. The Mercator projection is conformal. All meridians are straight parallel lines uniformly spaced. Lines of latitude are also straight and parallel and are perpendicular to the meridians, but are not uniformly spaced. Scale is correct at the Equator, but increases rapidly with latitude. The original Mercator map became the prototype for nautical charts. The projection is particularly A B FIGURE 6-Polyconic projection uses a series of cones on identical axes. Line AB is the standard parallel for the largest cone shown; dashed lines parallel to AB are standard parallels for other cones. base sheets. Also, it minimizes the effects of all distortions over a limited area. For this reason it is suitable for large-scale sectional maps such as quadrangle maps. Because of the small distortion of polyconic projections, the State plane coordinate FIGURE 7.-Mercator projection using a cylinder tangent at the Equator, line AB. A B FIGURE 8.-Transverse Mercator projection using a cylinder tangent at a standard meridian, line AB. suitable for navigation because lines of fixed azimuth (rhumb lines) form straight lines. Most of the nautical charts issued by NOS are on the Mercator projection. Transverse Mercator Projection The transverse Mercator projection (fig. 8), originally devised by Lambert, is essentially the standard Mercator rotated through 90°. However, its appearance, characteristics, and use are quite different. The curved surface of the Earth is projected to a cylinder that is tangent along a central meridian or usually secant along small circles parallel to and equidistant from the central meridian. The cylinder is then cut and unrolled to a plane. Scale is correct along the central meridian when a tangent cylinder is used, but increases rapidly east or west. Except for the central meridian, all meridians and parallels are curved lines. Because the greatest distortions increase to the east and west, the projection is suitable for mapping areas that are elongated north-south. The transverse Mercator projection is used for large-scale mapping throughout the world (in Europe it is sometimes called the Gauss-Kruger projection). Many USGS 7.5-min quadrangle maps are cast on it. A special use of this projection is as the basis for the Universal Transverse Mercator (UTM) grid, discussed later. Combination Projections When the size or shape of an area is not ideally suited to the characteristics of a particular projec tion that might otherwise be desirable, the area can be divided into sections or zones, each to be mapped on its own projection, If an area is too large to be mapped on a single sheet at the desired scale, the projection can be designed for the entire area or zone, and each sheet mapped as part of the comprehensive projection. GRID SYSTEMS It is difficult to measure the relations between points referenced to the graticule because on most projections parallels or meridians are curved. Running plane land surveys with geographic coordinates would require complex computations. Using plane rectangular grids alleviates the probem. Plane rectangular grids are constructed as two sets of straight parallel lines that intersect each other at right angles to form squares. A grid is superimposed on the graticule of a projection in such a way that there is a precise mathematical relationship between the grid lines and the graticule. Then every grid intersection has a unique relationship to every graticule intersection, and every point on the map (or on the Earth) has a unique latitude and longitude plus a unique location expressed in values of x and y on the selected grid. Thus, coordinates and other relations in one system can be easily converted mathematically to the other. Grid systems greatly simplify the use of maps and reduce computations of distances, directions, coordinates, and areas to the realm of plane trigonometry. Nevertheless, the grids are subject to the same distortions as projections. State Plane Coordinate Systems Most modern large-scale maps show the State plane coordinate grid in addition to the graticule. There is a plane coordinate system for each of the 50 States and for the oceanic islands. All the systems are presently expressed in feet, except the metric grid for Guam. However, USGS maps of Puerto Rico show plane coordinates in meters. NOS publishes tables for converting positions between geographic and plane coordinates for the 50 States plus Puerto Rico and the Virgin Islands, but not Guam. Zones in which the longer dimension is northsouth have a grid based on the transverse Mercator projection. Those in which the longer dimension is east-west have a grid based on the Lambert conformal conic projection. Table 2 lists the zones and |