*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.