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Polygon Decomposition for Efficient Construction of Minkowski Sums
, 2000
"... Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various ..."
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Cited by 40 (7 self)
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Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various wellknown decompositions as well as with several new decomposition schemes. We report on our experiments with various decompositions and different input polygons. Among our findings are that in general: (i) triangulations are too costly (ii) what constitutes a good decomposition for one of the input polygons depends on the other input polygon  consequently, we develop a procedure for simultaneously decomposing the two polygons such that a "mixed" objective function is minimized, (iii) there are optimal decomposition algorithms that significantly expedite the Minkowskisum computation, but the decomposition itself is expensive to compute  in such cases simple heuristics that approximate the optimal decomposition perform very well.
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
Advanced programming techniques applied to Cgal’s arrangement package
 Computational Geometry: Theory and Applications
, 2005
"... Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, whi ..."
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Cited by 31 (15 self)
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Arrangements of planar curves are fundamental structures in computational geometry. Recently, the arrangement package of Cgal, the Computational Geometry Algorithms Library, has been redesigned and reimplemented exploiting several advanced programming techniques. The resulting software package, which constructs and maintains planar arrangements, is easier to use, to extend, and to adapt to a variety of applications. It is more efficient space and timewise, and more robust. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. In this paper we describe how various programming techniques were used to accomplish specific tasks within the context of computational geometry in general and Arrangements in particular. These tasks are exemplified by several applications, whose robust implementation is based on the arrangement package. Together with a set of benchmarks they assured the successful application of the adverted programming techniques. 1
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.
Code flexibility and program efficiency by genericity: Improving cgal’s arrangements
 In Proc. 12th Annu. Euro. Sympos. Alg
, 2004
"... Abstract. Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This impr ..."
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Cited by 19 (13 self)
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Abstract. Arrangements of planar curves are fundamental structures in computational geometry. We describe the recent developments in the arrangement package of Cgal, the Computational Geometry Algorithms Library, making it easier to use, to extend and to adapt to a variety of applications. This improved flexibility of the code does not come at the expense of efficiency as we mainly use genericprogramming techniques, which make dexterous use of the compilation process. To the contrary, we expedited key operations as we demonstrate by experiments. 1
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
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.