Discrete & Computational Geometry

, Volume 46, Issue 2, pp 223–251

Rigidity of Quasicrystallic and Zγ-Circle Patterns

Article

DOI: 10.1007/s00454-011-9349-5

Cite this article as:
Bücking, U. Discrete Comput Geom (2011) 46: 223. doi:10.1007/s00454-011-9349-5

Abstract

The uniqueness of the orthogonal Zγ-circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is, they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic Zγ-circle patterns.

Keywords

Circle patternRigidityQuasicrystallic

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany