Human beings have known that the shape of the Earth resembles a sphere and not a flat surface since classical times, and possibly much earlier than that. If the world were indeed flat, cartography would be much simpler because map projections would be unnecessary.
To represent a curved surface such as the Earth in two dimensions, you must geometrically transform (literally, and in the mathematical sense, "map") that surface to a plane. Such a transformation is called a map projection. The term projection derives from the geometric methods that were traditionally used to construct maps, in the fashion of optical projections made with a device called camera obscura that Renaissance artists relied on to render three-dimensional perspective views on paper and canvas.
While many map projections no longer rely on physical projections, it is useful to think of map projections in geometric terms. This is because map projection consists of constructing points on geometric objects such as cylinders, cones, and circles that correspond to homologous points on the surface of the planet being mapped according to certain rules and formulas.
The following sections describe the basic properties of map projections, the surfaces onto which projections are developed, the types of parameters associated with different classes of projections, how projected data can be mapped back to the sphere or spheroid it represents, and details about one very widely used projection system, called Universal Transverse Mercator.