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Parametrization of Generalized Primal-Dual Triangulations

Memari, Pooran, Patrick Mullen, and Mathieu Desbrun

20th International Meshing Roundtable, Springer-Verlag, pp.237-253, October 23-26 2011

IMR
PROCEEDINGS

20th International Meshing Roundtable
Paris, France
October 23-26, 2011

California Institute of Technology, CNRS - LTCI, Telecom Paris Tech
Email:pmemari@caltech.edu

Summary
Motivated by practical numerical issues in a number of modeling and simulation problems, we introduce the notion of a compatible dual complex to a primal triangulation, such that a simplicial mesh and its compatible dual complex (made out of convex cells) form what we call a primal-dual triangulation. Using algebraic and computational geometry results, we show that compatible dual complexes exist only for a particular type of triangulation known as weakly regular. We also demonstrate that the entire space of primal-dual triangulations, which extends the well known (weighted) Delaunay/Voronoi duality, has a convenient, geometric parametrization. We finally discuss how this parametrization may play an important role in discrete optimization problems such as optimal mesh generation, as it allows us to easily explore the space of primal-dual structures along with some important subspaces..

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