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13
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 46 (9 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 40 (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 37 (16 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
 International Journal of Robotics Research
"... 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 24 (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. KEY WORDS—Computational geometry, robustness and precision, arrangements, algorithmic motion planning 1.
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 15 (10 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
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Advanced Programming Techniques Applied to Cgal’s Arrangement Package
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Code Flexibility and Program Eciency by Genericity: Improving Cgal's Arrangements?
"... 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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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
exibility of the code does not come at the expense of eciency 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