Map projections

Systematic methods of transforming the spherical representation of parallels, meridians, and geographic features of the Earth's surface to a nonspherical surface, usually a plane. Map projections have been of concern to cartographers, mathematicians, and geographers for centuries because globes and curved-surface reproductions of the Earth are cumbersome, expensive, and difficult to use for making measurements. Although the term “projection” implies that transformation is accomplished by projecting surface features of a sphere to a flat piece of paper using a light source, most projections are devised mathematically and are drawn with computer assistance. The task can be complex because the sphere and plane are not applicable surfaces. As a result, each of the infinite number of possible projections deforms the geometric relationships among the points on a sphere in some way, with directions, distances, areas, and angular relationships on the Earth never being completely recreated on a flat map.

Although the most commonly used map projections are accessible through mapping software, their fundamental attributes are best appreciated by considering the problem of the transformation of a spherical surface to a plane.

 

Deformation patterns

 

It is impossible to transfer spherical coordinates to a flat surface without distortion caused by compression, tearing, or shearing of the surface. Conceptually, the transformation may be accomplished in two ways: (1) by geometric transfer to some other surface, such as a tangent or intersecting cylinder, cone, or plane, which can then be developed, that is, cut apart and laid out flat; or (2) by direct mathematical transfer to a plane of the directions and distances among points on the sphere. Patterns of deformation can be evaluated by looking at different projection families. Whether a projection is geometrically or mathematically derived, if its pattern of scale variation is like that which results from geometric transfer, it is classed as cylindrical, conic, or in the case of a plane, azimuthal or zenithal.  See also: Cartography; Terrestrial coordinate system

 

Cylindrical projections

 

These result from symmetrical transfer of the spherical surface to a tangent or intersecting cylinder. True or correct scale can be obtained along the great circle of tangency or the two homothetic small circles of intersection. If the axis of the cylinder is made parallel to the axis of the Earth, the parallels and meridians appear as perpendicular lines. Points on the Earth equally distant from the tangent great circle (Equator) or small circles of intersection (parallels equally spaced on either side of the Equator) have equal scale departure. The pattern of deformation therefore parallel the parallels, as change in scale occurs in a direction perpendicular to the parallels. A cylinder turned 90° with respect to the Earth's axis creates a transverse projection with a pattern of deformation that is symmetric with respect to a great circle through the Poles. Transverse projections based on the Universal Transverse Mercator grid system are commonly used to represent satellite images, topographic maps, and other digital databases requiring high levels of precision. If the turn of the cylinder is less than 90°, an oblique projection results (Fig. 1). All cylindrical projections, whether geometrically or mathematically derived, have similar patterns of deformation.  See also: Great circle, terrestrial

 

 

Fig. 1  Oblique Mercator projection on a cylinder. (After American Oxford Atlas, Oxford, 1951)

 

 

 

 

 

 

 

Conic projections

 

Transfer to a tangent or intersecting cone is the basis of conic projections. For these projections, true scale can be found along one or two small circles in the same hemisphere. Conic projections are usually arranged with the axis of the cone parallel to the Earth's axis. Consequently, meridians appear as radiating straight lines and parallels as concentric angles. Conical patterns of deformation parallel the parallels; that is, scale departure is uniform along any parallel (Fig. 2). Several important conical projections are not true conics in that their derivation either is based upon more than one cone (polyconic) or is based upon one cone with a subsequent rearrangement of scale variation. Because conic projections can be designed to have low levels of distortion in the midlatitudes, they are often preferred for representing countries such as the United States.

 

 

Fig. 2  Conical projection, based on origin 45°N, 10°E. (After American Oxford Atlas, Oxford, 1951)

 

 

 

 

 

 

 

Azimuthal projections

 

These result from the transfer to a tangent or intersecting plane established perpendicular to a right line passing through the center of the Earth. All geometrically developed azimuthal projections are transferred from some point on this line. Points on the Earth equidistant from the point of tangency or the center of the circle of intersection have equal scale departure. Hence the pattern of deformation is circular and concentric to the Earth's center. All azimuthal projections, whether geometrically or mathematically derived, have two aspects in common: (1) all great circles that pass through the center of the projection appear as straight lines; and (2) all azimuths from the center are truly displayed (Fig. 3).

 

 

Fig. 3  Azimuthal projections with tangent planes have been raised above the sphere surface. (a) Polar azimuthal. (b) Oblique azimuthal. (After American Oxford Atlas, Oxford, 1951)

 

 

 

 

 

 

Nongeometric projections show no consistency in their patterns of deformation; nevertheless, many are useful. One such group, called pseudocylindrical projections, is derived by mathematically arranging the spherical surface within a shape bordered by a series of specified smooth curves, such as ellipses, parabolas, or sine or quartic curves (Fig. 4).

 

 

Fig. 4  Four pseudocylindrical projections. (University of Wisconsin-Madison Cartographic Laboratory)

 

 

 

 

 

 

 

 

Properties

 

The retention anywhere on the projection of some specific geometric quality of the sphere, such as uniformity of compass rose, or the creation of some useful attribute, such as straight rhumb lines, is called a property. Projections offer possibilities for maintaining single properties or combinations of properties. Ultimately, a map's intended use should determine the properties to be retained in the transformation process. All properties are the result of arranging the magnitudes and directions of scale variation.

Conformality

 

Conformality (orthomorphism) is the retention of angular relationships at each point, resulting in the preservation of the shapes of small features. It is obtained by arranging the scale so that at each point the scale is uniform in all directions. Since the spheroid and plane are not applicable, scale must change from point to point. Thus, all conformal projections deform the relative sizes of Earth areas (Fig. 5). Conformal projections are most widely used for navigational, engineering, and topographic maps, since observable angles may be measured on the map with a protractor.

 

 

Fig. 5  The property of equivalence is not maintained on the Mercator projection, making Greenland and South America appear similar in size. The inset map shows that South America is actually about 15 times larger than Greenland.

 

 

 

 

 

 

 

Equivalence

 

Equivalence (equal-area) is the retention of the relative sizes of areas. It is obtained by arranging the scale at each point so that the product of the sides of an infinitesimal Earth rectangle projected anywhere on the map is the same. Since the two surfaces are nonapplicable, the scale at a point must be different in different directions. Therefore, the properties of conformality and equivalence are mutually exclusive. Equivalent projections are most widely used for mapping geographic, economic, and similar types of data whenever the areal extent of the phenomena is important. The extent of Earth area within administrative subdivisions can best be derived by measurement from equivalent projections.

 

Azimuthality

 

Azimuthality is the retention of azimuths (correct directional relations along a great circle bearing) from one point to another. Within limited areas, this, like all properties, can be approached on many projections but can be precisely retained from only one or two points. The centers of all projections theoretically or actually constructed on a tangent plane have this property. Azimuthality is useful in graphic aids to communication and for observations dealing with electronic impulses that travel along great circles.

 

Significant lines

 

Significant reference lines on the Earth include great circles, rhumb lines, and small circles, which are utilized in navigation, communication, and area and space analysis. By arranging the scale, these may be made to appear as straight or consistently curved lines on a map. Of special significance are great circles and rhumb lines in air and sea navigation. Because they represent the shortest distance between two locations on the Earth's surface, aircraft follow great circle routes. The practical requirement of guiding movement by compass angle requires that headings be along rhumb lines (loxodromes). Scale on projections can be arranged so that entire families of these kinds of critical lines may be made to appear as straight lines. Similarly, directional concepts, such as a specific cardinal direction, can be made parallel anywhere on the map. Any circle on the Earth can be made to appear as a circle on a map. Many other similar attributes are possible with map projections.

 

 

Examples

 

Although an infinite number of different projections is possible, only a relatively few are widely used. There is no best projection for any specific mapping problem, and in most cases several different projections can be applied to any given project.

 

Mercator

 

The Mercator projection is a mathematically derived, cylindrical-type, conformal projection that is conventionally based on the Equator as the tangent great circle. The projection was developed in 1569 by the Flemish cartographer Gerhardus Mercator to represent his world map. Linear scale on the Mercator increases in a direction perpendicular to the Equator as the secant of the latitude on a sphere. All rhumb lines are straight lines on the conventional form, making the Mercator the most widely used projection for nautical navigation. The transverse Mercator (and similar variants for the spheroid, for example, Gauss-Krüger) employs a meridian pair as the tangent great circle. Rhumb lines are not straight lines on the transverse form.

 

Sinusoidal

 

The sinusoidal, also called the Sanson-Flamsteed or Sanson-Mercator projection, is an equal-area projection with a straight central meridian and straight, equally spaced parallels. An area of least distortion is found along the Equator and central meridian, making it a popular projection for representing Africa and South America. Since the projection comes to a point at the Poles, shapes are distorted in the high latitudes.

 

Mollweide

 

An equal-area projection with straight parallels, the Mollweide resembles the sinusoidal but forms an ellipse without pointed poles. In contrast to the sinusoidal, low distortion is found along the 40th parallels north and south latitude rather than at the center. When centered on the prime meridian, the Mollweide is a popular projection for representing continental Europe.

 

Gnomonic

 

The gnomonic (azimuthal) projection is obtained by geometric transfer from the center of a sphere to a tangent plane. Scale is greatly exaggerated away from its center, making the projection impractical for representing more than 45° of latitude in a single hemisphere. Because all arcs of great circles appear as straight lines, this projection is popular for navigation.

 

Azimuthal equidistant

 

The azimuthal equidistant projection is formed by straight meridians radiating from a single point of plane tangency with the surface. Parallels form concentric circles and, when measured from the point of tangency, all points are true in distance and global direction. Although commonly used to represent Arctic and Antarctic regions, the projection can be centered on any location, making it ideal for plotting seismic or radio waves.

 

Stereographic

 

The stereographic (azimuthal and conformal) projection is derived by geometric transfer to a tangent plane from a point opposite the point of tangency. All circles on the Earth appear as circles on the map. It is widely used as a framework for topographic maps of small areas and for navigation and grid systems in polar areas.

 

Lambert conformal conic

 

The Lambert conformal conic with two true-scale (standard) parallels is widely used for aeronautical and meteorological charts, as the framework for topographic series, and for rectangular grid systems of smaller areas. Scale increases away from the standard parallels, and great circles are nearly straight lines.

 

Albers equal-area conic

 

The Albers equal-area conic with two standard parallels is similar in appearance to the Lambert. The projection is commonly used to show middle latitudes for administrative and area research maps, especially for the United States, Europe, and Russia.

 

Polyconic

 

Although the polyconic (the ordinary, or American, polyconic) is neither equal-area or conformal; each parallel is true scale. It has been widely used for large-scale topographic maps, each sheet being individually developed and essentially error-free. Sheets fit together north-south but not east-west and cannot be mosaicked in all directions without gaps. The U.S. Geological Survey ceased using the polyconic for topographic maps in favor of either the transverse Mercator or the Lambert conformal conic. The modified polyconic employs straight meridians and distributes scale error over the map sheet.

 

Goode's homolosine

 

Interrupted projections, such as the Goode's homolosine, make it possible to maintain a degree of equivalence and conformality on a single world map. The projection is created by inserting breaks over land or ocean areas and is homolographic from 40° north to the North Pole and from 40° south to the South Pole. Elsewhere, from 40° north to 40° south, the projection is sinusoidal. Goode's homolosine is a popular projection for use in atlases and textbooks.

Most other forms of projection are used for small-scale, general or thematic maps. Most widely used for these are equivalent or near-equivalent varieties. The display of imagery from a satellite orbiting about the rotating Earth has required complex mathematical developments in map projections.

 

 

Computers

 

Computers have become extremely valuable tools for the assisting in selecting the most appropriate projection to fit the unique constraints of each mapping problem. Mapping software allows the cartographer to experiment with alternative projections or to make parameter adjustments after a projection has been selected.

Arthur H. Robinson

Thomas A. Wikle

 

Bibliography

 

 

  • B. D. Dent, Cartography: Thematic Map Design, 4th ed., 1996
  • D. H. Maling, Coordinate Systems and Map Projections, 2d ed., 1993
  • F. Pearson II, Map Projection Methods, 1984
  • A. H. Robinson, Elements of Cartography, 6th ed., 1995
  • J. P. Snyder and P. M. Voxland, Album of Map Projections, 1989
  •  alifazeli=egeology.blogfa.com