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A map projection is any of many methods used in cartography (mapmaking) to represent the two-dimensional curved surface of the earth or other body on a plane. Methods for constructing a projection may be mathematical, graphical, or geometric. Regardless of the method, in the end any projection can be expressed mathematically.
Flat maps could not exist without map projections. Flat maps can be more useful than models globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer diplays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the earth's surface at once; and they are cheaper to produce and transport. These useful traits of flat maps motivate the development of map projections.
Many properties can be measured on the earth's surface independently of its geography. Some of these properties are
Map projections can be constructed to preserve one or some of these properties, though not all of them simultaneously. Each projection preserves or compromises or approximates basic metric properties in different ways. The purpose of the map, then, determines which projection should form the base for the map. Since many purposes exist for maps, so do many projections exist upon which to construct them.
Another major concern that drives the choice of a projection is the compatibility of data sets. Data sets are geographic information. As such, their collection depends on the chosen model of the earth. Different models assign slightly different coordinates to the same location, so it is important that the model be known and that chosen projection is compatible with that model. On small areas (large scale) data compatibility issues are more important since metric distortions are minimal at this level. In very large areas (small scale), on the other hand, distortion is a more important factor to consider.
The creation of a map projection involves three steps.
Because the real earth's shape is irregular, information is lost in the first step, in which an approximating, regular model is chosen. Reducing the scale may be considered to be part of transforming geographic coordinates to plane coordinates.
Most map projections, both practically and theoretically, are not "projections" in any physical sense. Rather, they depend on mathematical formulae that have no direct physical interpretation. However, in understanding the concept of a map projection it is helpful to think of a globe with a light source placed at some definite point with respect to it, projecting features of the globe onto a surface. The following discussion of developable surfaces is based on that concept.
If a surface can be transformed onto another surface without stretching, tearing, or shrinking, then the surface is said to be an applicable surface. The sphere and ellipsoid are not applicable with a plane surface, so any projection that attempts to project them on a flat sheet will have to distort the image (similar to the impossibility of making a flat sheet from an orange peel). A surface that can be unfolded or unrolled into a flat plane or sheet without stretching, tearing or shrinking is called a 'developable surface'. The cylinder, cone and of course the plane are all developable surfaces.
The most complex part of a projection involves transforming the global model onto one of these developable surfaces. That process inevitably distorts some properties of the globe. Once the globe has been transformed onto a developable surface, the developable surface may be unfolded without further distortion.
Once a choice is made between projecting onto a cylinder, cone, or plane, the orientation of the shape must be chosen. The orientation is how the shape is placed with respect to the globe. The orientation of the projection surface can be normal (inline with the earth's axis), transverse (at right angles to the earth's axis) or oblique (any angle in between). These surfaces may also be either tangent or secant to the spherical or ellipsoidal globe. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Insofar as preserving metric properties go, it is never advantageous to move the developable surface away from contact with the globe, so that practice is not discussed here.
The globe is the only way to represent the earth without distorting one or more of the metric properties discussed above. Scale in particular suffers when adopting a map. Only a globe can have a constant scale throughout the entire map surface. The scale for flat maps will vary from point to point and may also vary in different directions from a single point (as in azimuthal maps). The scale for a flat map can only be true at specific points or along specific paths, and never across areas of any extent. The 'scale factor is therefore used to measure the difference between the idealized scale and the actual scale at a particular point on the map and in a particular direction at that point.
Projection construction is also affected by how the shape of the earth is approximated. In the following discussion on projection categories, a sphere is assumed. However, the Earth is not exactly spherical but is closer in shape to an oblate ellipsoid, a shape which bulges around the equator. Selecting a model for a shape of the earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to jusify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large and medium scale maps that need to accurately depict the land surface.
A third model of the shape of the earth is called a geoid, which is a complex and more or less accurate representation of the global mean sea level surface that is obtained through a combination of terrestrial and satellite gravity measurements. This model is not used for mapping due to its complexity but is instead used for control purposes in the construction of geographic datums. A geoid is used to construct a datum by adding irregularities to the ellipsoid in order to better match the Earth's actual shape (it takes into account the large scale features in the Earth's gravity field associated with mantle convection patterns, as well as the gravity signatures of very large geomorphic features such as mountain ranges, plateaus and plains). Historically datums have been based on ellipsoids that best represent the geoid within the region the datum is intended to map. Each ellipsoid has a distinct major and minor axis. Different controls (modifications) are added to the ellipsoid in order to construct the datum, which is specialized for a specific geographic regions (such as the North American Datum). A few modern datums, such as the one used in the Global Positioning System GPS, are optimized to represent the entire earth as well as possible with a single ellipsoid, at the expense of some accuracy in smaller regions.
A fundamental projection classification is based on type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are cylindrical (e.g., Mercator), conic (e.g., Albers), and azimuthal or plane (e.g., stereographic). Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic.
Another way to classify projections is through the properties they retain. Some of the more common categories are
NOTE: It is impossible to construct a map projection that is both equal-area and conformal.
Cylindrical projections are constructed by wrapping a cylinder around the Earth and then projecting onto the cylinder.
Pseudocylindrical projections are created mathematically, representing the central meridian and each parallel as a straight line. Each pesudocylindrical 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.
Azimuthal projections touch the earth to a plane at one tangent point; angles from that tangent point are preserved, and distances from that point are computed by a function independent of the angle.
Some azimuthal projections are true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a points of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane.
Conformal map projections preserve angles locally.
These projections preserve area.
These preserve distance from some standard point or line.
Compromise projections give up the idea of perfectly preserving metric properties, seeking instead to strike a balance between distortions, or to simply make things "look right".
See also: Cartographer, GIS
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