Quantitative Properties of Map Projections
A sphere, unlike a polyhedron, cone, or cylinder, cannot be reformed into a plane. In order to portray 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 or creasing) and devise rules for systematically representing all or part of the spherical surface on the plane. Any such process inevitably leads to distortions of one kind or another. Five essential characteristic properties of map projections are subject to distortion: shape, distance, direction, scale, and area. No projection can retain more than one of these properties over a large portion of the Earth. This is not because a sufficiently clever projection has yet to be devised; the task is physically impossible. The technical meanings of these terms are described below.
Shape (also called conformality)
Shape is preserved locally (within "small" areas) when the scale of a map at any point on the map is the same in any direction. Projections with this property are called conformal. In them, meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. An older term for conformal is orthomorphic (from the Greek orthos, straight, and morphe, shape).
Distance (also called equidistance)
A map projection can preserve distances from the center of the projection to all other places on the map (but from the center only). Such a map projection is called equidistant. Maps are also described as equidistant when the separation between parallels is uniform (e.g., distances along meridians are maintained). No map projection maintains distance proportionality in all directions from any arbitrary point.
A map projection preserves direction when azimuths (angles from the central point or from a point on a line to another point) are portrayed correctly in all directions. Many azimuthal projections have this property.
Scale is the ratio between a distance portrayed on a map and the same extent on the Earth. No projection faithfully maintains constant scale over large areas, but some are able to limit scale variation to one or two percent.
Area (also called equivalence)
A map can portray areas across it in proportional relationship to the areas on the Earth that they represent. Such a map projection is called equal-area or equivalent. Two older terms for equal-area are homolographic or homalographic (from the Greek homalos or homos, same, and graphos, write), and authalic (from the Greek autos, same, and ailos, area), and equireal. Note that no map can be both equal-area and conformal.
For a complete description of the properties that specific map projections maintain, see Summary and Guide to Projections.