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 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.Postprint (published version

Aproximació facetada de superfícies paramètriques retallades

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Two triangulations 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 approachesPostprint (published version

Skeletal representations of orthogonal shapes

In this paper we present two skeletal representations applied to orthogonal shapes of R^n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in the plane and compare the resulting skeleton with other skeletal representations.Postprint (published version

Funciones en el modelado de solidos y paradigmas de diseño

En este artículo se describen y analizan las diferentes funciones de modelado de sólidos más comunes que ofrecen los actuales sistemas de CAD. Se explican también las limitaciones e inconsistencias a qué pueden dar lugar el uso de dichas funciones. Por otro lado, también se analizan y comparan los diferentes paradigmas de diseño con los que trabajan los actuales sistemas. Se describe, evalúa y compara el paradigma clásico basado en operaciones booleanas, con los paradigmas basados en el diseño paramétrico y el diseño basado en características. También se describe cual es la interrelación a través de la interfaz de usuario entre los paradigmas de diseño y las funciones de modelado.Postprint (published version

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 (Brep). 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.)Peer ReviewedPostprint (author’s final draft

From Degenerate Patches to Triangular and Trimmed Patches

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Splat representation of parametric surfaces

Point-based geometry representations and their splat-based generalizations have become a suitable technique both for modeling and rendering complex 3D shapes. So, it seems interesting to convert other kind of models to a point or splat-based representations. In this work, we present an approach to convert a parametric surface to an elliptical splat-based representation. Although this conversion supposes a loss of information going from an analytical to an approximate model, it will allow to locally modify zones with complex features, to mix surface and splat-based models and to take advantage of the existing point-based rendering methods. The presented approach works in the parametric space and performs an adaptive sampling based on the surface curvature and a given error tolerance. The goal is to obtain an optimized set of elliptical splats that completely covers the surface. Two strategies are presented, one based on a quadrangular subdivision of the parametric space and the other on power Voronoi diagrams. Finally, some open problems are enumerated.Postprint (published version

Skeleton computation of an image using a geometric approach

In this work we develop two algorithms to compute the skeleton of a binary 2D images. Both algorithms follow a geometric approach and work directly with the boundary of the image wich is an orthogonal polygon (OP). One of these algorithms processes the edges of the polygon while the other uses its vertices. Compared to a thinning method, the presented algorithms show a good performance.Postprint (published version

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.Postprint (published version