Marc Vigo

Computing directional constrained Delaunay triangulations

This work presents two generalizations of the algorithm for obtaining a constrained Delaunay triangulation of a general planar graph presented in [Vig97] and [Vig95]. While the first generalization works with elliptical distances, the second one can deal with a set of deforming ellipses associated to each point of the plane. The pseudo-code of the procedures involved in the algorithms is included, the suitability of the algorithms is analyzed, and several examples are presented.Abstract In this work, two generalizations of the algorithm for obtaining a constrained Delaunay triangulation of a general planar graph set forth in Vigo (Technical Report LSI-95-UR-R, Universitat Politecnica de Catalunya, 1995; Computer & Graphics 1997;21(2):215–23) are presented. While the first generalization works with elliptical distances, the second one can deal with a set of deforming ellipses associated to each point of the plane. The pseudo-code of the procedures involved in the algorithms is included, the suitability of the algorithms is analyzed, and several examples are given.

Regular triangulations of dynamic sets of points

The Delaunay triangulations of a set of points are a class of triangulations which play an important role in a variety of different disciplines of science. Regular triangulations are a generalization of Delaunay triangulations that maintain both their relationship with convex hulls and with Voronoi diagrams. In regular triangulations, a real value, its weight, is assigned to each point.In this paper a simple data structure is presented that allows regular triangulations of sets of points to be dynamically updated, that is, new points can be incrementally inserted in the set and old points can be deleted from it. The algorithms we propose for insertion and deletion are based on a geometric interpretation of the history data structure in one more dimension and use lifted flips as the unique topological operation. This results in rather simple and efficient algorithms. The algorithms have been implemented and experimental results are given.The Delaunay triangulations of a set of points are a class of triangulations which play an important role in a variety of different disciplines of science. Regular triangulations are a generalization of Delaunay triangulations that maintain both their relationship with convex hulls and with Voronoi diagrams. In regular triangulations, a real value, its weight, is assigned to each point. In this paper a simple data structure is presented that allows regular triangulations of sets of points to be dynamically updated, that is, new points can be incrementally inserted in the set and old points can be deleted from it. The algorithms we propose for insertion and deletion are based on a geometrical interpretation of the history data structure in one more dimension and use lifted flips as the unique topological operation. This results in rather simple and efficient algorithms. The algorithms have been implemented and experimental results are given.

Piecewise Linear Approximation of Trimmed Surfaces

Stereolithography applications require a surface model of the modeled object consisting of a mesh of triangular facets. This model can also be used for mechanical analysis through finite element methods. In this paper, a new algorithm for the piecewise linear approximation of trimmed surfaces is presented. The algorithm generates a triangulation that approximates the initial surface within a predefined tolerance. The approximation is conformai, without cracks in edges: a closed polyhedron is obtained in the case of a closed initial surface. The algorithm first builds a quadtree-structured bound on the patch curvatures for every surface patch, and then works by first discretizing trimming curves and afterwards relaxing the location of a sufficient number of vertices inside the trimmed region in every patch. The resulting triangulation satisfies the max-min criterion in parametric space.

Skeleton computation of orthogonal polyhedra

Skeletons are powerful geometric abstractions that provide useful representations for a number of geometric operations. The straight skeleton has a lower combinatorial complexity compared with the medial axis. Moreover, while the medial axis of a polyhedron is composed of quadric surfaces the straight skeleton just consist of planar faces. Although there exist several methods to compute the straight skeleton of a polygon, the straight skeleton of polyhedra has been paid much less attention. We require to compute the skeleton of very large datasets storing orthogonal polyhedra. Furthermore, we need to treat geometric degeneracies that usually arise when dealing with orthogonal polyhedra. We present a new approach so as to robustly compute the straight skeleton of orthogonal polyhedra. We follow a geometric technique that works directly with the boundary of an orthogonal polyhedron. Our approach is output sensitive with respect to the number of vertices of the skeleton and solves geometric degeneracies. Unlike the existing straight skeleton algorithms that shrink the object boundary to obtain the skeleton, our algorithm relies on the plane sweep paradigm. The resulting skeleton is only composed of axis‐aligned and 45° rotated planar faces and edges.

Efficient algorithms for boundary extraction of 2D and 3D orthogonal pseudomanifolds

In this paper we present algorithms to extract the boundary representation of orthogonal polygons and polyhedra, either manifold or pseudomanifold. The algorithms we develop reconstruct not only the polygons of the boundaries but also the hole-face inclusion relationship. Our algorithms have a simple input so they can be used to convert many different kinds of models to B-Rep. In the 2D case, the input is the set of vertices, and in the 3D case, some small additional information must be supplied for every vertex. All proposed algorithms run in O(nlogn) time and use O(n) space, where n is the number of vertices of the input. Moreover, we explain how to use our proposal to extract the boundary from the well-known voxel and octree models as well as from three vertex-based models found in the related literature: the neighbourhood, the EVM, and the weighted vertex list models.

Skeleton computation of an image using a geometric approach

In this work we develop two algorithms to compute the skeleton of a binary 2D image. Both algorithms follow a geometric approach and work directly with the boundary of the image which is an orthogonal polygon (OP). One of these algorithms processes the edges of the polygon while the other one uses its vertices. Compared with a thinning method, the presented algorithms show a good performance. This is a work in progress as our final goal is to extend the vertex-based algorithm method to 3D in order to compute the surface skeleton of a binary volume.

A practical and robust method to compute the boundary of three-dimensional axis-aligned boxes

The union of axis-aligned boxes results in a constrained structure that is advantageous for solving certain geometrical problems. A widely used scheme for solid modelling systems is the boundary representation (B-rep). We present a method to obtain the B-rep of a union of axis-aligned boxes. Our method computes all boundary vertices, and additional information for each vertex that allows us to apply already existing methods to extract the B-rep. It is based on dividing the three-dimensional problem into two-dimensional boundary computations and combining their results. The method can deal with all geometrical degeneracies that may arise. Experimental results prove that our approach outperforms existing general methods, both in efficiency and robustness.

Splat representation of parametric surfaces

Point and splat-based representations have become a suitable technique both for modeling and rendering complex 3D shapes. Converting other kinds of models as parametric surfaces to splat-based representations will allow to mix surface and splat-based models and to take advantage of the existing point-based rendering methods.In this work, we present an approach to convert a parametric surface into a splat-based representation. It works in parametric space, performs an adaptive sampling based on the surface curvature and a given error tolerance and uses power Voronoi diagrams. The goal is to approximate the surface with an optimized set of elliptical splats.

Two triangulation methods based on edge refinement

In this paper two curvature adaptive methods of surface triangulation are presented. Both methods are based on edge refinement to obtain a triangulation compatible with the curvature requirements. The first method applies an incremental and constrained Delaunay triangulation and uses curvature bounds to determine if an edge of the triangulation is admissible. The second method uses this function also in the edge refinement process, i.e. in the computation of the location of a refining point, and in the re-triangulation needed after the insertion of this refining point. Results are presented, comparing both approaches.