Quantitative Properties of Map Projections
A sphere, unlike a polyhedron, cone, or cylinder, cannot be reformed into a plane. To portray the surface of a sphere on a plane, you must first define a developable surface (a surface that you can cut and flatten onto a plane without stretching or creasing it), and then create rules for representing part or all of the surface on the plane.
All projections result in distortion. Depending on the projection, you can preserve these map properties: shape, distance, direction, scale, and area. No projection can retain more than one of these properties over a large portion of a sphere.
For a description of the properties that specific map projections preserve, see Summary and Guide to Projections.
Map projections that preserve shape locally (within "small" areas) are conformal. A projection is conformal when, at any point, the scale of the map is the same in all directions. When a projection preserves shape, the meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. An older term for conformal is orthomorphic.
Map projections that preserve distance are equidistant. A projection is equidistant when it preserves distances from the center of the projection to all other points on the map. Maps are also described as equidistant when the separation between the parallels is uniform (as in, the projection maintains distances along meridians). No map projection maintains distance in all directions from any arbitrary point.
A map projection preserves direction when the projection portrays azimuths (angles from the central point or from a point on a line to another point) correctly in all directions. Many azimuthal projections preserve direction.
Scale is the ratio between a distance portrayed on a map and the same extent on the sphere. No projection maintains constant scale over large areas, but some are able to limit scale variation to one or two percent.
Map projections that preserve area are equal-area or equivalent. Equal-area projections portray areas in proportional relationships to the areas on the sphere that they represent. Older terms for equal-area are homolographic, homalographic, authalic, and equireal. No map can be both equal-area and conformal.