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The Design and Implementation of Planar Maps in CGAL
 Special Issue, selected papers of the Workshop on Algorithm Engineering (WAE
, 1999
"... this paper has been supported in part by ESPRIT IV LTR Projects No. 21957 (CGAL) and No. 28155 (GALIA), by the USAIsrael Binational Science Foundation, by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), by ..."
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Cited by 39 (17 self)
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this paper has been supported in part by ESPRIT IV LTR Projects No. 21957 (CGAL) and No. 28155 (GALIA), by the USAIsrael Binational Science Foundation, by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Center for Geometric Computing and its Applications), by a FrancoIsraeli research grant "factory of the future" (monitored by AFIRST/France and The Israeli Ministry of Science), and by the Hermann Minkowski  Minerva Center for Geometry at Tel Aviv University
Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 33 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
Exacus: Efficient and exact algorithms for curves and surfaces
 IN ESA, VOLUME 1669 OF LNCS
, 2005
"... We present the first release of the EXACUS C++ libraries. We aim for systematic support of nonlinear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of ..."
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Cited by 30 (12 self)
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We present the first release of the EXACUS C++ libraries. We aim for systematic support of nonlinear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of the libraries, which currently compute arrangements of curves and curve segments of low algebraic degree, and boolean operations on polygons bounded by such segments.
Robust Geometric Computing in Motion
, 2000
"... In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular. ..."
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Cited by 24 (2 self)
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In this paper we discuss the gap between the theory and practice of geometric algorithms. We then describe effors to settle this gap and facilitate the successful implementation of geometric algorithms in general and of algorithms for geometric arrangements and motion planning in particular.
An adaptable and extensible geometry kernel
 In Proc. Workshop on Algorithm Engineering
, 2001
"... ii ..."
TwoDimensional Arrangements in CGAL and Adaptive Point Location for Parametric Curves
 In Proc. of the 4th Workshop of Algorithm Engineering
, 2000
"... . Given a collection C of curves in the plane, the arrangement of C is the subdivision of the plane into vertices, edges and faces induced by the curves in C. Constructing arrangements of curves in the plane is a basic problem in computational geometry. Applications relying on arrangements arise ..."
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Cited by 14 (9 self)
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. Given a collection C of curves in the plane, the arrangement of C is the subdivision of the plane into vertices, edges and faces induced by the curves in C. Constructing arrangements of curves in the plane is a basic problem in computational geometry. Applications relying on arrangements arise in fields such as robotics, computer vision and computer graphics. Many algorithms for constructing and maintaining arrangements under various conditions have been published in papers. However, there are not many implementations of (general) arrangements packages available. We present an implementation of a generic and robust package for arrangements of curves that is part of the CGAL 1 library. We also present an application based on this package for adaptive point location in arrangements of parametric curves. 1
Parametric search made practical
 SoCG: 18th Symposium on Computational Geometry
, 2002
"... In this paper we show that in sortingbased applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole’s optimization of certain parametricsearch algorithms may be unnecessary under realistic assumptions about the inpu ..."
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Cited by 10 (1 self)
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In this paper we show that in sortingbased applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole’s optimization of certain parametricsearch algorithms may be unnecessary under realistic assumptions about the input. Furthermore, we present a generic, flexible, and easytouse framework that greatly simplifies the implementation of algorithms based on parametric search. We use our framework to implement an algorithm that solves the Fréchetdistance problem. The implementation based on parametric search is faster than the binarysearch approach that is often suggested as a practical replacement for the parametricsearch technique.
Complete, Exact and Efficient Implementation for Computing the Adjacency Graph of an Arrangement of Quadrics
, 2007
"... We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the in ..."
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Cited by 7 (2 self)
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We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation.
Precise global collision detection in multiaxis NCmachining
 ComputerAided Design
, 2004
"... We introduce a new approach to the problem of collision detection in multiaxis NCmachining. Due to the directional nature (tool axis) of multiaxis NCmachining, space subdivision techniques are adopted from raytracing algorithms and extended to suit the peculiarities of the problem in hand. We e ..."
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Cited by 2 (1 self)
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We introduce a new approach to the problem of collision detection in multiaxis NCmachining. Due to the directional nature (tool axis) of multiaxis NCmachining, space subdivision techniques are adopted from raytracing algorithms and extended to suit the peculiarities of the problem in hand. We exploit the axialsymmetry inherent in the tool's rotational motion to derive a highly precise polygontool intersection algorithm which, combined with the proper data structure, also yields efficient computation times. Other advantages of the proposed method is the separation of the entire computation into a preprocessing stage that is executed only once, allowing more than one toolpath to be efficiently verified thereafter, and the introduced ability to test for collisions against arbitrary shaped tools such as flatend or ballend, or even test for interference with the tool holder or other parts of the NCmachine. Keywords: NCmachining, 5axis machining, collision detection and verification, space subdivision, ray tracing, lower envelopes. 1.
Complete, Exact and Efficient Implementation for Computing the Adjacency Graph of an Arrangement of Quadrics
 in "15th Annual European Symposium on Algorithms  ESA 2007
"... We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the int ..."
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Cited by 2 (0 self)
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We present a complete, exact and efficient implementation to compute the adjacency graph of an arrangement of quadrics, i.e. surfaces of algebraic degree 2. This is a major step towards the computation of the full 3D arrangement. We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the adjacency graph of the arrangement. Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points. It is exact in that it always computes the mathematically correct result. It is efficient measured in running times, i.e. it compares favorably to the only previous implementation. Key words: Arrangement, intersection of surfaces, quadrics, pencils of quadrics, curve parameterization.