Computers with multiple processor cores using shared memory are now ubiquitous. In this paper, we present several parallel geometric algorithms that specifically target this environment, with the goal of exploiting the additional computing power. The d-dimensional algorithms we describe are (a) spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) kd-tree construction, (c) axis-aligned box intersection computation, and finally (d) bulk insertion of points in Delaunay triangulations for mesh generation algorithms or simply computing Delaunay triangulations. We show experimental results for these algorithms in 3D, using our implementations based on the Computational Geometry Algorithms Library (CGAL, http://www.cgal.org/). This work is a step towards what we hope will become a parallel mode for CGAL, where algorithms automatically use the available parallel resources without requiring significant user intervention.

We present fundamental geometric data structures and algorithms offered by CGAL, the Computational Geometry Algorithms Library. As geometry is ubiquitous this library is used by application developers in medical imaging, VLSI, CAD/CAM, geophysics, computer graphics and last but not least GIS. In this demo we focus on those parts of CGAL which are relevant for geographic information systems software development.

The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. From a topological standpoint, the connection has recently been made between these cell complexes and the Morse theory of distance functions. In particular, in the generic setting, algorithms have been proposed to compute the flow complex--the stable and unstable manifolds associated to the critical points of the distance function to a point set. As algorithms ignoring degenerate cases and numerical issues are bound to fail on general inputs, this paper develops the first complete and robust algorithm to compute the flow complex.

Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the CGAL library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic arcs in the plane. Our first contribution is the design, implementation and testing of a kernel for computing arrangements of circular arcs.A preliminary C++ implementation exists also for arbitrary conic curves. We discuss the representation and predicates of the geometric objects. Our implementation is targeted for inclusion in the CGAL library. Our second contribution concerns exact and efficient algebraic algorithms for the case of conics. They treat all inputs, including degeneracies, and they are implemented as part of the library SYNAPS 2.1.Our tools include Sturm sequences, resultants, Descartes' rule, andisolating points. Thirdly, our experiments on circular arcs show that our methods compare favorably to existing alternatives using CORE 1.6x and LEDA 4.5.

Constructive root bounds is the fundamental technique needed to achieve guaranteed accuracy, the critical capability in Exact Geometric Computation. Known bounds are overly pessimistic in the presense of general rational input numbers. In this paper, we introduce a method which greatly improves the known bounds for k-ary rational input numbers. Since majority of input numbers in scientific and engineering applications are such numbers, this could lead to a significant speedup for a large class of applications. We apply our method to the BFMSS Bound. Implementation and experimental results based on the Core Library are reported.

Given a triangulation in the plane or a tetrahedralization in 3-space, we investigate the efficiency of locating a point by walking in the structure with different strategies.

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International audience

International audience