# Creation of Hyperbolic Ornaments

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Hyperbolic ornaments are pictures which are invariant under a discrete symmetry group of isometric transformations of the hyperbolic plane. They are the hyperbolic analogue of Euclidean ornaments, including but not limited to those Euclidean ornaments which belong to one of the 17 wallpaper groups. The creation of hyperbolic ornaments has a number of applications. They include artistic goals,communication of mathematical structures and techniques, and experimental research in the hyperbolic plane. Manual creation of hyperbolic ornaments is an arduous task. This work describes two ways in which computers may help with this process. On the one hand, a computer may provide a real-time drawing tool, where any stroke entered by the user will be replicated according to the rules of some previously selected symmetry group. Finding a suitable user interface for the intuitive selection of the symmetry group is a particular challenge in this context. On the other hand, existing Euclidean ornaments can be transported to the hyperbolic plane by changing the orders of their centers of rotation. This requires a deformation of the fundamental domains of the ornament, and one particularlywell suited approach uses conformal deformations for this step, approximated using discrete conformality concepts from discrete differential geometry. Both tools need a way to produce high quality renderings of the hyperbolic ornament, dealing with the fact that in general an infinite number of fundamental domains will be visible in the finite model of the hyperbolic plane. To deal with this problem, an approach similar to ray tracing can be used, variations of which are discussed as well.

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### BibTeX entry

@article{CreationofHyperbolicOrnaments, title = {Creation of Hyperbolic Ornaments}, abstract = {Hyperbolic ornaments are pictures which are invariant under a discrete symmetry group of isometric transformations of the hyperbolic plane. They are the hyperbolic analogue of Euclidean ornaments, including but not limited to those Euclidean ornaments which belong to one of the 17 wallpaper groups. The creation of hyperbolic ornaments has a number of applications. They include artistic goals,communication of mathematical structures and techniques, and experimental research in the hyperbolic plane. Manual creation of hyperbolic ornaments is an arduous task. This work describes two ways in which computers may help with this process. On the one hand, a computer may provide a real-time drawing tool, where any stroke entered by the user will be replicated according to the rules of some previously selected symmetry group. Finding a suitable user interface for the intuitive selection of the symmetry group is a particular challenge in this context. On the other hand, existing Euclidean ornaments can be transported to the hyperbolic plane by changing the orders of their centers of rotation. This requires a deformation of the fundamental domains of the ornament, and one particularlywell suited approach uses conformal deformations for this step, approximated using discrete conformality concepts from discrete differential geometry. Both tools need a way to produce high quality renderings of the hyperbolic ornament, dealing with the fact that in general an infinite number of fundamental domains will be visible in the finite model of the hyperbolic plane. To deal with this problem, an approach similar to ray tracing can be used, variations of which are discussed as well.}, url = {http://martin.von-gagern.net/publications/2014-phd/ http://mediatum.ub.tum.de/doc/1210572/document.pdf}, year = 2014, author = {Martin von Gagern}, comment = {}, urldate = {2019-08-08}, collections = {art,basically-computer-science,geometry} }