However, Carl Friedrich Gauss 's Theorema Egregium proved that a sphere 's surface can not 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.
what does it mean when a map projection distorts the earth
Theorema Egregium