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80
A Core Library For Robust Numeric and Geometric Computation
 In 15th ACM Symp. on Computational Geometry
, 1999
"... Nonrobustness is a wellknown problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (Core) for robust numeric and geometric computation ba ..."
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Cited by 74 (11 self)
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Nonrobustness is a wellknown problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (Core) for robust numeric and geometric computation based on the principles of Exact Geometric Computation (EGC). Through our library, for the first time, any programmer can write robust and efficient algorithms. The Core Library is based on a novel numerical core that is powerful enough to support EGC for algebraic problems. This is coupled with a simple delivery mechanism which transparently extends conventional C/C++ programs into robust codes. We are currently addressing efficiency issues in our library: (a) at the compiler and language level, (b) at the level of incorporating EGC techniques, as well as the (c) the system integration of both (a) and (b). Pilot experimental results are described. The basic library is available at http://cs.nyu.edu...
Classroom examples of robustness problems in geometric computations
 In Proceedings of the 12th annual European Symposium (ESA
, 2004
"... Abstract The algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can g ..."
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Cited by 45 (13 self)
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Abstract The algorithms of computational geometry are designed for a machine model with exact real arithmetic. Substituting floating point arithmetic for the assumed real arithmetic may cause implementations to fail. Although this is well known, there is no comprehensive documentation of what can go wrong and why. In this paper, we study simple algorithms for planar convex hulls and 3d Delaunay triangulations and give examples which make the algorithms fail in many different ways. For the incremental planar convex hull algorithm our examples cover the negation space of the correctness properties of the algorithms. We also show how to construct failureexamples semisystematically and discuss the geometry of the floating point implementation of the orientation predicate. We hope that the paper will be useful for teaching computational geometry. The paper comes with a companion webpage (http://www.mpisb.mpg.de/ kettner/proj/NonRobust/).
A Perturbation Scheme for Spherical Arrangements with Application to Molecular Modeling
, 1997
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Advanced Programming Techniques Applied to Cgal’s Arrangement Package
, 2007
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Recent progress in exact geometric computation.
 Journal of Logic and Algebraic Programming,
, 2005
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Exact and Efficient 2DArrangements of Arbitrary Algebraic Curves
"... We show how to compute the planar arrangement induced by segments of arbitrary algebraic curves with the BentleyOttmann sweepline algorithm. The necessary geometric primitives reduce to cylindrical algebraic decompositions of the plane for one or two curves. We compute them by a new and efficient ..."
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Cited by 26 (10 self)
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We show how to compute the planar arrangement induced by segments of arbitrary algebraic curves with the BentleyOttmann sweepline algorithm. The necessary geometric primitives reduce to cylindrical algebraic decompositions of the plane for one or two curves. We compute them by a new and efficient method that combines adaptiveprecision root finding (the Bitstream Descartes method of Eigenwillig et al., 2005) with a small number of symbolic computations, and that delivers the exact result in all cases. Thus we obtain an algorithm which produces the mathematically true arrangement, undistorted by rounding error, for any set of input segments. Our algorithm is implemented in the EXACUS library AlciX. We report on experiments; they indicate the efficiency of our approach. 1
Efficient Exact Geometric Computation Made Easy
, 1999
"... We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easytowrite, correct and efficient geometric programs. ..."
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Cited by 25 (7 self)
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We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easytowrite, correct and efficient geometric programs.
Robust Geometric Computing in Motion
 INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH
, 2002
"... Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely is ..."
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Cited by 25 (2 self)
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Transforming a geometric algorithm into an effective computer program is a difficult task. This transformation is particularly made hard by the basic assumptions of most theoretical geometric algorithms concerning complexity measures and (more crucially) the handling of robustness issues, namely issues related to arithmetic precision and degenerate input. The paper starts with a discussion of the gap between the theory and practice of geometric algorithms, together with a brief review of existing solutions to some of the problems that this dichotomy brings about. We then turn to an overview of the CGAL project and library. The CGAL project is a joint effort by a number of research groups in Europe and Israel to produce a robust software library of geometric algorithms and data structures. The library is now available for use with significant functionality. We describe the main goals and results of the project. The central part of the paper is devoted to arrangements (i.e., space subdivisions induced by geometric objects) and motion planning. We concentrate on the maps and arrangements part of the CGAL library. Then we describe two packages developed on top of CGAL for constructing robust geometric primitives for motion algorithms.
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 24 (6 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.
Euclidean Voronoi diagram of 3D balls and its computation via tracing edges
, 2005
"... ... known as an additively weighted Voronoi diagram, in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute the Euclidean Voronoi diagram for 3D spheres with different radii. The presented algorithm follows Voronoi edges one by one until the constr ..."
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Cited by 21 (10 self)
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... known as an additively weighted Voronoi diagram, in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute the Euclidean Voronoi diagram for 3D spheres with different radii. The presented algorithm follows Voronoi edges one by one until the construction is completed in O(mn) time in the worstcase, where m is the number of edges in the Voronoi diagram and n is the number of spherical balls. As building blocks, we show that Voronoi edges are conics that can be precisely represented as rational quadratic Bézier curves. We also discuss how to conveniently represent and process Voronoi faces which are hyperboloids of two sheets.