Flattening the Earth includes all the processes of map projections:

Map projections: Flattening the Earth by the way of a mapping surface
[Voser 1998,Voser 2003]

Map projections have advantages for calculating geometric properties of spatial entities compared to the calculations of these properties on a curved Earth model. In the plane of the map projection, the calculation of distances, angles, directions and areas may be done based on the rules of the classical geometry (Euclidean geometry).

In opposition, the disadvantages of map projections are their geometric distortions which depend on the position together with the projection method, its instatiation and implementation. This results by the fact that it is not possible to map from a curved surface like a sphere or spheroid onto a plane without distortions.

The analysis of the deformations is done by applying principles of differential geometry: the laws of surface theory. There, its first fundamental treats the geometric intrinsics (metrics on surfaces). Thereby, the rules to describe lengths, angles, areas are derived on the Gaussian fundamentals.

The analysis of these geometric properties says, that there is no way to map from the surface of a sphere or ellipsoid onto a plane without distortion. Generically, angles, areas and length are distorted. But there exist ways to controll the mentioned deformations in an infinitesian matter.

Because of these distortions, map projections cover a wide field in mathematical cartography, or moreover, in geomatics. More than 200 types of map projections are known, and already the Ancient Greeks dealt this topic.

There exist various ways to classify map projections:

 

In the application, there exist much more individual instances of coordinate reference systems of type map projection. They vary not only in distortion properties, but also in their parameters as well as their method implementations. Important to know when working with map projections is the underlying Earth model and its geodetic datum.