Equal-Area and Conformal Projection Types

A foundational result of mathematical cartography, formalized in Carl Friedrich Gauss's Theorema Egregium (1827), establishes that a curved surface such as a sphere cannot be flattened onto a plane without distorting at least one of the fundamental geometric properties of shape, area, distance, or direction, so no single projection can ever preserve all of them simultaneously. Cartographers therefore classify projections by the property each one holds true: conformal projections preserve local shape and angles at every point at the cost of area; equal-area, or equivalent, projections preserve the relative size of regions at the cost of shape; equidistant projections preserve true distance from one or two chosen points or along particular lines; and azimuthal projections preserve true direction, or bearing, from a single central point to everywhere else on the map.