Cartograms and Circular-Arc Simplification of Polygonal Subdivisions

Cartograms are proportional drawings of maps where the areas of all countries correspond to prescribed weights. Besides the property of being a proportional representation it is essential for the quality of a cartogram that the countries are not too badly deformed. A good clarity can only be achieved if all countries can easily be identified by the reader due to the fact that their principal topological properties have been carried over from the original map to the cartogram. In this thesis we investigate an approach to generate cartograms based on a deformation of borders between different countries. In particular, we allow to replace straight line segments with circular arcs in order to transfer area from one country to another. The countries in the resulting cartograms look thus a bit like clouds or flakes, depending on whether the edges were bent outwards or inwards. We introduce several problem variants and show that they are all NP -hard. Then we propose some relaxations aiming to simplify the problem so that it can be resolved efficiently. First a relaxation of the area constraint is considered where a deviation of the attained area from the prescribed area can be tolerated. Then we suggest relaxations of the vertex-related constraints. A heuristic based on the area relaxation and using a max-flow algorithm is then presented. This heuristic has been implemented and an evaluation has been conducted by showcasing some visual output from the implementation.