Polygonal and Polyhedral Delaunay Meshing

We consider construction of a polyhedral Delaunay partition as a limit of the sequence of radical partitions (power diagrams), while the dual Voronoi diagram is obtained as a limit of sequence of weighted Delaunay partitions. Using a lifting analogy, this problem is reduced to the construction of a pair of dual convex polyhedra, inscribed and superscribed around circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron, while the sequence of dual polyhedra converges to the inscribed polyhedron. Different rules can be used to build sequences of dual polyhedra. We are mostly interested in the case when the vertices of primal polyhedra can move or glue together, meaning that no new faces are allowed for dual polyhedra. These rules essentially define the transformation of the set of initial spheres defining power diagram into the set of Delaunay spheres using sphere movement, radius variation, and sphere elimination as admissible operations. Even though rigorous foundations (existence theorems) for this problem are still unavailable, we suggest a functional which measures deviation of the convex polyhedron from the one inscribed into paraboloid. This functional is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from paraboloid. The absolute minimizer of this functional is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by means of a simple translation. This formulation of the Dirichlet functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. Numerical examples illustrate polygonal Delaunay meshing in planar domains. In this work we concentrate on the experimental confirmation of the viability of suggested approach putting aside mesh quality problems. The gradient of the above functional defines a manifold describing evolution of Delaunay spheres hence one can optimize Delaunay-Voronoi meshes using this manifold as a constraint.