Map projection

A map projection is a way of showing the surface of a three-dimensional sphere on a flat surface. Such projections are necessary to create maps. The problem with projections is that they change the surface in some way; this is called distortion. Depending on the purpose of the map, some distortions may be less bad. This is why cartographers use different mathematical projections for different maps.
Carl Friedrich Gauss showed that is not possible to project the surface of a sphere to a plane, without distorting it. This theorem is known as Theorema egregium today.
To understand the projections, it is easier to imagine a light source, a globe, and another geometric object. The light source shines "a shadow" of each point of the globe onto the geometric object. At the end the surface of the geometric object is unrolled, this yields the map. It is possible to tell the projections apart by the type of helper object used.