*27.08.2011*
*Map projections are used to map the surface of a mathematical Earth model
like a sphere or
ellipsoid onto a plane based on geometrical or mathematical rules, principles or constraints.*

*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.