Cartographic projections are obtained through geometric transformations aimed at representing the terrestrial globe on a flat surface. The globe is an ellipsoid, but for the sake of geometric simplicity it is treated, by convention, as a sphere divided into meridians and parallels. The flat surface on which the map is drawn may be tangent to the globe’s surface or may be obtained from the projection of another solid surrounding the Earth’s sphere, typically a cylinder or a cone.
In the first case, the projection is called azimuthal; it can be polar if the point of tangency is one of the poles, equatorial if the point lies on the Equator, or oblique if the point lies anywhere else on the globe. If the surrounding solid is a cylinder, the projection is called a pseudocylinder projection and may consist in an actual projection from the center of the Earth, as in the Mercator (1512-1594) projection, or a non-projective geometric transformation, as in the maps by Ptolemy (ca. 100-ca. 178) and in sea charts.
Depending on the position of the projection point, the cartographic projection is called gnomonic if the point lies at the center of the Earth, stereographic if the point lies on the surface of the globe opposite the projection plane, perspective if the point lies outside the Earth, or orthographic if the projection point is located at infinity. The perspective projection, in turn, is called polar if the point lies at one of the poles, or equatorial if it lies on the Equator.
Ptolemy adopted perspective projection to represent the heavens. For the map of the inhabited world, he used a pseudoconic projection, an oblique pseudo-azimuthal projection, and a perspective projection.