We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions. The algorithm protects edges with weighted points to avoid the difficulty posed by small angles between adjacent input elements. These weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement algorithm using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. Guaranteed bounds on the aspect ratios, normal variation and dihedral angles are also provided. To this end, we present new concepts and results including a new definition of local feature size and a proof for a generalized topological ball property.

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This paper presents an algorithm for sampling and triangulating a generic C 2-smooth surface Σ ⊂ R 3 that is input with an implicit equation. The output triangulation is guaranteed to be homeomorphic to Σ. We also prove that the triangulation has well-shaped triangles, large dihedral angles, and a small size. The only assumption we make is that the input surface representation is amenable to certain types of computations, namely computations of the intersection points of a line and Σ, computations of the critical points in a given direction, and computations of certain silhouette points.

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher

IntroductionWe describe the following data structures.For halfspace range reporting, in S-space using expected preprocessing time O(n log n), worst-case storage O(n log log n) and worst-case reporting time O(log n + k), where n is the number of data points and k the number of points reported; in d-space, with d even, using worst-case preprocessing time O(nlogn), storage O(n) and reporting time O(n1-1/Ld/21 log'n + k), where c is a constant. For ray shooting in a convex polytope in d-space determined by n facets, using deterministic preprocessing time 0( (n/ log n)ldi2J log' n) and storage 0( (n/ log n) ld/2J2c'os' ") and with query time O(logn).For ray shooting in arbitrary direction amon n hyperplanes using preprocessing O(nd/logld/2 n) and query time O(logn). BWe also describe a randomized algorithm for constructing the k-level of n planes in S-space. In the case of planes dual to points in convex position, in which the size of the k-level is O(nk), the algorithm uses nearly optimal expected time O(n logn + nk2c'0g* "). By a standard geometric transformation the same time bound applies for the construction of the k-order Voronoi diagram of n sites in the plane.