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.
It is impossible to transfer spherical coordinates to a flat surface without distortion caused by compression, tearing, or shearing of the surface (see illustration). 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.
Cylindrical projections 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. All cylindrical projections, whether geometrically or mathematically derived, have similar patterns of deformation.
Transfer to a tangent or intersecting cone is the basis of conicprojections. For these projections, true scale can be found along one or twosmall circles in the same hemisphere. Conic projections are usually arrangedwith the axis of the cone parallel to the Earth's axis. Consequently,meridians appear as radiating straight lines and parallels as concentricangles. Conical patterns of deformation parallel the parallels; that is, scaledeparture is uniform along any parallel. Several important conical projectionsare not true conics in that their derivation either is based upon more than onecone (polyconic) or is based upon one cone with a subsequent rearrangement ofscale variation. Because conic projections can be designed to have low levelsof distortion in the midlatitudes, they are often preferred for representingcountries such as the United States.
Azimuthal projections result from the transfer to a tangent or intersectingplane established perpendicular to a right line passing through the center ofthe Earth. All geometrically developed azimuthal projections are transferredfrom some point on this line. Points on the Earth equidistant from the point oftangency or the center of the circle of intersection have equal scaledeparture. Hence the pattern of deformation is circular and concentric to theEarth's center. All azimuthal projections, whether geometrically ormathematically derived, have two aspects in common: (1) all great circles thatpass through the center of the projection appear as straight lines; and (2) allazimuths from the center are truly displayed


Types of Projections

Cylindrical Projection

In a typical cylindrical projection, one imagines the paper to be wrapped as a cylinder around the globe, tangent to it along the equator. Light comes from a point source at the center of the globe or, in some cases, from a filament running from pole to pole along the globe's axis. In the former case the poles clearly cannot be shown on the map, as they would be projected along the axis of the cylinder out to infinity. In the latter case the poles become lines forming the top and bottom edges of the map. The Mercator projection, long popular but now less so, is a cylindrical projection of the latter type that can be constructed only mathematically. In all cylindrical projections the meridians of longitude, which on the globe converge at the poles, are parallel to one another; in the Mercator projection the parallels of latitude, which on the globe are equal distances apart, are drawn with increasing separation as their distance from the equator increases in order to preserve shapes. However, the price paid for preserving shapes is that areas are exaggerated with increasing distance from the equator. The effect is most pronounced near the poles; e.g., Greenland is shown with enormously exaggerated size, although its shape in small sections is preserved. The poles themselves cannot be shown on the Mercator projection. Students using the Mercator projection obtain an incorrect impression of the relative sizes of the countries of the world.

Conic Projection

In a conic projection a paper cone is placed on a globe like a hat, tangent to it at some parallel, and a point source of light at the center of the globe projects the surface features onto the cone. The cone is then cut along a convenient meridian and unfolded into a flat surface in the shape of a circle with a sector missing. All parallels are arcs of circles with a pole (the apex of the original cone) as their common center, and meridians appear as straight lines converging toward this same point. Some conic projections are conformal (shape preserving); some are equal-area (size preserving). A polyconic projection uses various cones tangent to the globe at different parallels. Parallels on the map are arcs of circles but are not concentric.

Azimuthal Projection

In an azimuthal projection a flat sheet of paper is tangent to the globe at one point. The point light source may be located at the globe's center (gnomonic projection), on the globe's surface directly opposite the tangent point (stereographic projection), or at some other point along the line defined by the tangent point and the center of the globe, e.g., at a point infinitely distant (orthographic projection). In all azimuthal projections, the tangent point is the central point of a circular map; all great circles passing through the central point are straight lines, and all directions from the central point are accurate. If the central point is a pole, then the meridians (great circles) radiate from that point and parallels are shown as concentric circles. The gnomonic projection has the useful property that all great circles (not just those that pass through the central point) appear as straight lines; conversely, all straight lines drawn on it are great circles. A navigator taking the shortest route between two points (always part of a great circle) can plot his course on a gnomonic projection by simply drawing a straight line between the two points.

Pseudocylindrical

A sinusoidal projection shows relative sizes accurately, but grossly distorts shapes. Distortion can be reduced by "interrupting" the map.
Pseudocylindrical projections represent the central meridian and each parallel as a straight line segment, but not the other meridians, except for the Collignon projection, which in its most common forms represents all meridians as straight lines from the poles to the equators as straight line segments. Each pseudocylindrical projection represents a point on the Earth along the straight line representing its parallel, at a distance which is a function of its difference in longitude from the central meridian.
• Sinusoidal: the north-south scale is the same everywhere at the central meridian, and the east-west scale is throughout the map the same as that; correspondingly, on the map, as in reality, the length of each parallel is proportional to the cosine of the latitude. Thus the shape of the map for the whole earth is the area between two symmetric rotated cosine curves
The true distance between two points on the same meridian corresponds to the distance on the map between the two parallels, which is smaller than the distance between the two points on the map; the meridians drawn on the map help the user realizing the distortion and mentally compensating for it

Hybrid

The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with the Collignon projection in polar areas.
Conical
• Equidistant conic
• Lambert conformal conic
• Albers conic

Pseudoconical

• Bonne
• Werner cordiform designates a pole and a meridian; distances from the pole are preserved, as are distances from the meridian (which is straight) along the parallels



Other Projections
Among the other commonly used map projections are the Mollweide homolographic and the sinusoidal, both of which are equal-area projections with horizontal parallels; they are especially useful for world maps. Goode's homolosine projection is a composite using the sinusoidal projection between latitudes 40°N and 40°S and the homolographic projection for the remaining parts. Interruptions, or splits, are often made in the ocean areas in order to show land areas with truer shapes. The A. H. Robinson projection, now used by the United States Geographic Service, has gained acceptance because it accurately represents relative size.