Michal Etzion

Hexahedral Mesh Generation using the Embedded Voronoi Graph

Abstract. This work presents a new approach for automatic hexahedral meshing, based on the embedded Voronoi graph. The embedded Voronoi graph contains the full symbolic information of the Voronoi diagram and the medial axis of the object, and a geometric approximation to the real geometry. The embedded Voronoi graph is used for decomposing the object, with the guiding principle that resulting sub-volumes are sweepable. Sub-volumes are meshed independently, and the resulting meshes are easily combined and smoothed to yield the final mesh. The approach presented here is general and automatic. It handles any volume, even if its medial axis is degenerate. The embedded Voronoi graph provides complete information regarding proximity and adjacency relationships between the entities of the volume. Hence, decomposition faces are determined unambiguously, without any further geometric computations. The sub-volumes computed by the algorithm are guaranteed to be well-defined and disjoint. The size of the decomposition is relatively small, since every sub-volume contains a different Voronoi face. Mesh quality seems high since the decomposition avoids generation of sharp angles, and sweep and other basic methods are used to mesh the sub-volumes.

Computing Voronoi skeletons of a 3-D polyhedron by space subdivision

We tackle the problem of computing the Voronoi diagram of a 3-D polyhedron whose faces are planar. The main difficulty with the computation is that the diagram’s edges and vertices are of relatively high algebraic degrees. As a result, previous approaches to the problem have been non-robust, difficult to implement, or not provenly correct. We introduce three new proximity skeletons related to the Voronoi diagram: (1) the Voronoi graph(VG), which contains the complete symbolic information of the Voronoi diagram without containing any geometry; (2) the approximate Voronoi graph(AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the proximity structure diagram (PSD), which enhances the VG with a geometric approximation of Voronoi elements to any desired accuracy. The new skeletons are important for both theoretical and practical reasons. Many applications that extract the proximity information of the object from its Voronoi diagram can use the Voronoi graphs or the proximity structure diagram instead. In addition, the skeletons can be used as initial structures for a robust and efficient global or local computation of the Voronoi diagram. We present a space subdivision algorithm to construct the new skeletons, having three main advantages. First, it solves at most uni-variate quartic polynomials. This stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-variate system of equations. Second, the algorithm enables purely local computation of the skeletons in any limited region of interest. Third, the algorithm is simple to implement.  2002 Elsevier Science B.V. All rights reserved.

Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts

The paper presents an algorithm to construct the Voronoi diagram of a 3-D Iinear polyhedron. The robustness and simplicity of the algorithm are due to the separation between the computation of the symbolic and geometric parts of the diagram. The symbolic part of the diagram, the Voronoi graph, is computed by a space subdivision algorithm. The computation of the Voronoi graph utilizes only relatively simple 2-D geometric computations. Given the Voronoi graph, and a geometric approximation given by the space subdivision, the construction of the geometric part is simple and reliable. An important advantage of the algorithm is that it enables local and partial computation of the Voronoi diagram. In a previous paper we have given a detailed proof of correctness of the computation of the Voronoi graph. This paper complements the previous one by detailing the algorithm and its implementation. In addition, this paper describes the computation of the geometric part of the diagram. CR

One-dimensional selections for feature-based data exchange

In the parametric feature based design paradigm, most features possess arguments that are subsets of the boundary of the current model, subsets defined interactively by user selection of boundary entities. Any system for feature-based data exchange (FBDE) must support the exchange of such selections. In this paper we describe in detail an algorithm for supporting one-dimensional selections (sets of edges and curves) for FBDE. The algorithm is applicable to a wide class of FBDE architectures, including the Universal Product Representation (UPR) and the STEP parametrics specification.

On compatible star decompositions of simple polygons

The authors introduce the notion of compatible star decompositions of simple polygons. In general, given two polygons with a correspondence between their vertices, two polygonal decompositions of the two polygons are said to be compatible if there exists a one-to-one mapping between them such that the corresponding pieces are defined by corresponding vertices. For compatible star decompositions, they also require correspondence between star points of the star pieces. Compatible star decompositions have applications in computer animation and shape representation and analysis. They present two algorithms for constructing compatible star decompositions of two simple polygons. The first algorithm is optimal in the number of pieces in the decomposition, providing that such a decomposition exists without adding Steiner vertices. The second algorithm constructs compatible star decompositions with Steiner vertices, which are not minimal in the number of pieces but are asymptotically worst-case optimal in this number and in the number of added Steiner vertices. They prove that some pairs of polygons require /spl Omega/(n/sup 2/) pieces, and that the decompositions computed by the second algorithm possess no more than O(n/sup 2/) pieces. In addition to the contributions regarding compatible star decompositions, the paper also corrects an error in the only previously published polynomial algorithm for constructing a minimal star decomposition of a simple polygon, an error which might lead to a nonminimal decomposition.

Two-dimensional selections for feature-based data exchange

Proper treatment of selections is essential in parametric feature-based design. Data exchange is one of the most important operators in any design paradigm. In this paper we address two-dimensional selections (faces and surfaces) in feature-based data exchange (FBDE). We define the problem formally and present algorithms to address it, in general and in various cases in which feature rewrites are necessary. The general algorithm operates at a geometric level and does not require solving the persistent naming problem, which is required for selection support inside a single CAD system. All algorithms are applicable to the Universal Product Representation (UPR) FBDE architecture, and the general algorithm is also applicable to the STEP parametrics specification.