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“Restricted Voronoi Diagrams for (Re)-Meshing Surfaces and Volumes”

Bruno Lévy

Curves and Surfaces, 2014

Abstract: We propose an efficient and robust algorithm to compute the intersection between a Voronoi diagram and a simplicial complex, possibly embedded in high dimensional space (typically 6d to 10d). Such a high-dimensional setting is interesting for some anisotropic mesh generation algorithms. Efficiency is obtained from a simple characterization of the bisectors that contribute to each Voronoi cell. This observation results in an algorithm that is practical even in high-dimensional space. Robustness is obtained through a combination of arithmetic filters, expansion arithmetics and symbolic perturbation. To avoid degeneracies, similarly to what is done for the in-sphere predicate, we consider the Voronoi diagram as a power diagram with all its weights set to zero, and apply symbolic perturbation to the weights. We give the derivation of the three predicates (resp. four) used by the surfacic (resp. volumetric) versions of the algorithm. All the derivations are expressed in function of the dot product (kernel) in the embedding space. The algorithm is implemented in the PCK (Predicate Construction Kit) tool, part of the GEOGRAM library.

## BibTex reference

@INPROCEEDINGS{L:PCK:2014,
TITLE = "Restricted Voronoi Diagrams for (Re)-Meshing Surfaces and Volumes",
AUTHOR = "Bruno Lévy",
BOOKTITLE = "Curves and Surfaces",
YEAR = "2014",
}

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