Computer-aided drawing in the symmetry groups of the euclidean plane

Abstract

This diploma thesis deals with theoretical and practical foundations of a computer program for drawing ornaments in the symmetry groups of the euclidean plane. Concepts for orbit calculation and for efficient display of patterns using overlapping buffers are explained. The use of voronoi polygons as user-modifiable fundamental domains is presented and a method for their calculation described. Different approaches for exporting to various formats are discussed. Furthermore, steps for the recognition of symmetry groups of patterns from external images are demonstrated, based on autocorrelation and the calculation of areas of dominance.

Full text

The full text is currently only available in German. It is available as a print quality PDF file of about 13MB. A smaller version for downloads as well was a version in HTML are planned, at least in German. There are no plans for an english translation in the near future, but if you are particularly interested in some aspects and don't understand them well enough in German, feel free to ask me for an explanation or translation in English.

Acknowledgements

I completely missed the opportunity to include acknowledgements into this work, so I'll remedy that now. First and foremost I thank my advisor, Prof. Jürgen Richter-Gebert, for his great support, his time and his many valuable suggestions. I also thank my family, who lightened the load for me. Likewise did my girlfriend Melanie; who furthermore read the work and, being a mathematical layman, helped a lot with the general comprehensibility. Special thanks for that as well.

Program

The program described in this work is available for free and can be downloaded from its project page.