Geodesic
Hyperbolization of Euclidean ornaments
The electronic journal of combinatorics, The Björner Festschrift volume, Tome 16 (2009) no. 2
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In this article we outline a method that automatically transforms an Euclidean ornament into a hyperbolic one. The necessary steps are pattern recognition, symmetry detection, extraction of a Euclidean fundamental region, conformal deformation to a hyperbolic fundamental region and tessellation of the hyperbolic plane with this patch. Each of these steps has its own mathematical subtleties that are discussed in this article. In particular, it is discussed which hyperbolic symmetry groups are suitable generalizations of Euclidean wallpaper groups. Furthermore it is shown how one can take advantage of methods from discrete differential geometry in order to perform the conformal deformation of the fundamental region. Finally it is demonstrated how a reverse pixel lookup strategy can be used to obtain hyperbolic images with optimal resolution.
DOI : 10.37236/78
Classification : 51M09, 52C26, 51F15, 53A35, 68T10
Mots-clés : wallpaper groups, symmtry patterns, hyperbolic ornament 
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     author = {Martin von Gagern and J\"urgen Richter-Gebert},
     title = {Hyperbolization of {Euclidean} ornaments},
     journal = {The electronic journal of combinatorics},
     year = {2009},
     volume = {16},
     number = {2},
     doi = {10.37236/78},
     zbl = {1168.51310},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/78/}
}
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Martin von Gagern; Jürgen Richter-Gebert. Hyperbolization of Euclidean ornaments. The electronic journal of combinatorics, The Björner Festschrift volume, Tome 16 (2009) no. 2. doi: 10.37236/78

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