Abstract not found

We show how to compute and maintain the two-dimensional arrangement on a quadric that is induced by intersection curves with other quadrics. The key idea is to parameterize the quadric by two variables, which then allows to implicitly compute the arrangement in a modified parameter space. We give details of a possible parameterization and explain how to implement the needed geometric and topological predicates.

Nef polyhedra in d-dimensional space are the closure of half-spacesunder boolean set operation. In consequence, they can represent non-manifold situations, open and closed sets, mixed-dimensionalcomplexes and they are closed under all boolean and topological operations, such as complement and boundary. They were intro-duced by W. Nef in his seminal 1978 book on polyhedra. We presented in previous work a new data structure for theboundary representation of three-dimensional Nef polyhedra with efficient algorithms for boolean operations. These algorithms weredesigned for correctness and can handle all cases, in particular all degeneracies. To this extent we rely on exact arithmetic to avoidwell known problems with floating-point arithmetic. In this paper, we present important optimizations for the algo-rithms. We describe the chosen implementations for the pointlocation and the intersection-finding subroutines, a kd-tree and afast box-intersection algorithm, respectively. We evaluate this optimized implementation with extensive experiments that supplementthe runtime analysis from our previous paper and that illustrate the effectiveness of our optimizations. We compare our implementa-tion with the A

Many applications dealing with large data structures can benefit from keeping them in compressed form. Compression has many benefits: it can allow a representation to fit in main memory rather than swapping out to disk, and it improves cache performance since it allows more data to fit into the cache. However, a data structure is only useful if it allows the application to perform fast queries (and updates) to the data.

This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

We give an overview of the tutorial for theCGAL:: Polyhedron 3 and its use in subdivision algorithms. The full tutorial and the accompanying source code are available

Abstract not found

We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3. Our implementation is complete in the sense that it does not assume general position, namely, it can handle degenerate input, and produces exact results. Our software also includes applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3. The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with a naïve approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. Our method is significantly faster. The video demonstrates the techniques used on simple cases as well as on degenerate cases. The relevant programs, source code, data sets, and documentation are available at

In this paper we discuss a class of adaptive refinement algorithms for generating unstructured meshes in two and three dimensions. We focus on Skeleton Based Refinement (SBR) algorithms as proposed by Plaza and Carey [25] and provide an extension that involves the introduction of the graph of the skeleton for meshes consisting of simplex cells. By the use of data structures derived from the graph of the skeleton, we reformulate the Skeleton Based Refinement scheme and devise a more natural and consistent approach for this class of adaptive refinement algorithms. As an illustrative case, we discuss in detail the graphs for 2D refinement of triangulations and for 3D we propose a corresponding new face-based data structure for tetrahedra. Experiments using the two dimensional algorithm and exploring the properties of the associated graph are provided.