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Quantitative Properties
A sphere, unlike a cone or cylinder, cannot be simply reformed into a plane. In order to view the surface of a round body on a two-dimensional flat plane, you must first define a developable surface (i.e., one that can be cut and flattened onto a plane without stretching) and devise a means of systematically representing all or part of the spherical surface onto the plane. This inevitably leads to distortions of one kind or another. Three characteristic properties of map projections are subject to distortion: area, shape, and scale. A given projection cannot retain more than one of these properties over a large area of the Earth.
An equal-area projection is one in which equal spatial units at different points on the map represent the same true area measurement. For example, consider a coin of any size as our spatial unit. The area of the Earth covered by the coin on one part of the map display is exactly equal to that covered by the coin on any other part. Shape and scale must be distorted, although there are some equal-area maps designed to minimize these distortions for at least parts of the map and perhaps preserve shape or scale at specific points.
Other terms for equal-area projections include equivalent, homolographic or homalographic (from the Greek homalos or homos, same, and graphos, write), authalic (from the Greek autos, same, and ailos, area), and equireal.
A conformal projection is one that preserves the relative local angles (shape) about every point on the map, except for one or more possible singular points. Since relative local angles are correct, meridians intersect parallels at right angles (90 degrees). No map can be both equal-area and conformal. Another term for conformal is orthomorphic (from the Greek orthos, straight, and morphe, shape).
An equidistant projection shows true scale between one or two points and every other point on the map, or along every meridian. No map projection shows scale correctly in all directions throughout the entire map.
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