The map as a work of art with mathematical fundamentals

**Cartography is the science of mapmaking.** Therefor the spherical shape of the Earth needs to be put onto a plain piece of paper. As a model for the Earth you take the sphere (or the rotation ellipsoid).

Now the **challenge of the (mathematical) cartography** consists in finding suitable **mapping functions** (also called projections or map projections) which map the surface of the sphere onto the plane in consideration of certain **side conditions** as

*preserving lenght*of special curves,*area-fidelity*of regions,*conformality*, etc. .

The projections which specifically map the system of longitudes and latitudes are called here *GridMaps* .

While viewing a map you should always be aware of the following **important fact**:

As a result of the **THEOREMA EGREGIUM by C.F. Gauss** it is **impossible to construct maps which are an exact image of the Earth's surface**.

Single curves however can be projected lenght-preserving.

The task consists in keeping the possible **distortions within tolerable limits**.

To get an initial impression of the **manifoldness of beautiful map projections** I would like to present you one of the many possibilities to systematize map projections which based on the **appearance of the map grid**. Here it is a matter of division into

See for instance [2], [3] and [4] under Literature.