<P> More generally, the surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid; see below . Even more generally, projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds . However, "map projection" refers specifically to a cartographic projection . </P> <P> Maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the Earth's surface at once; and they are cheaper to produce and transport . These useful traits of maps motivate the development of map projections . </P> <P> However, Carl Friedrich Gauss's Theorema Egregium proved that a sphere's surface cannot be represented on a plane without distortion . The same applies to other reference surfaces used as models for the Earth . Since any map projection is a representation of one of those surfaces on a plane, all map projections distort . Every distinct map projection distorts in a distinct way . The study of map projections is the characterization of these distortions . </P> <P> Projection is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate . Rather, any mathematical function transforming coordinates from the curved surface to the plane is a projection . Few projections in actual use are perspective . </P>

What does it mean when a map projection distorts the earth