Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation
Proceedings of the nineteenth annual symposium on Computational geometry
Anisotropic Centroidal Voronoi Tessellations and Their Applications
SIAM Journal on Scientific Computing
Building triangulations using ε-nets
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Locally uniform anisotropic meshing
Proceedings of the twenty-fourth annual symposium on Computational geometry
Discrete one-forms on meshes and applications to 3D mesh parameterization
Computer Aided Geometric Design
Orphan-Free Anisotropic Voronoi Diagrams
Discrete & Computational Geometry
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Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphan-free (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its dual have properties that parallel those of ordinary Voronoi diagrams: the primal's vertices, edges, and faces are connected, and the dual triangulation has a simple, closed boundary. Additionally, if the underlying metric has bounded anisotropy (ratio of eigenvalues), the dual is guaranteed to triangulate the convex hull of the sites. These results apply to the duals of anisotropic Voronoi diagrams of any set of sites, so long as their Voronoi diagram is orphan-free. By combining this general result with existing conditions for obtaining orphan-free anisotropic Voronoi diagrams, a simple and natural condition for a set of sites to form an embedded anisotropic Delaunay triangulation follows.