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25
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 32 (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
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 31 (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.
Towards an open curved kernel
 In Proc. Annual ACM Symp. on Computational Geometry
, 2004
"... 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 ar ..."
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Cited by 31 (14 self)
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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, and isolating points. Thirdly, our experiments on circular arcs show that our ∗ Work partially supported by the IST Programme of the EU as a
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
Efficient Exact Geometric Predicates for Delaunay Triangulations
, 2002
"... A time efficient implementation of the exact computation paradigm relies on arithmetic filters which are used to speed up the exact computation of easy instances of the geometric predicates. Depending of what is called easy instances, we usually classify filters as static or dynamic and also some in ..."
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Cited by 16 (5 self)
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A time efficient implementation of the exact computation paradigm relies on arithmetic filters which are used to speed up the exact computation of easy instances of the geometric predicates. Depending of what is called easy instances, we usually classify filters as static or dynamic and also some in between categories often called semistatic. In this
CGAL  The Computational Geometry Algorithm Library
 Sandia National Laboratory
, 2001
"... The Cgal project (www.cgal.org) is a collaborative e#ort of several research institutes in Europe. The mission of the project is to make the most important of the solutions and methods developed in computational geometry available to users in industry and academia. ..."
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Cited by 15 (0 self)
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The Cgal project (www.cgal.org) is a collaborative e#ort of several research institutes in Europe. The mission of the project is to make the most important of the solutions and methods developed in computational geometry available to users in industry and academia.
Exact and Efficient Construction of Planar Minkowski Sums using the Convolution Method
"... The Minkowski sum of two sets A, B ∈ IR d, denoted A⊕B, is defined as {a + b  a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR 2 using the convolution of the polygon boundaries. This method allows for faster computation of th ..."
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Cited by 14 (0 self)
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The Minkowski sum of two sets A, B ∈ IR d, denoted A⊕B, is defined as {a + b  a ∈ A, b ∈ B}. We describe an efficient and robust implementation for the construction of Minkowski sums of polygons in IR 2 using the convolution of the polygon boundaries. This method allows for faster computation of the sum of nonconvex polygons in comparison to the widelyused methods for Minkowskisum computation that decompose the input polygons into convex subpolygons and compute the union of the pairwise sums of these convex subpolygon. Our source code, as well as the data sets we used in our experiments, can be downloaded from:
Exact implementation of arrangements of geodesic arcs on the sphere with applications
 In Abstracts of 24th Eur. Workshop Comput. Geom
, 2008
"... Recently, the Arrangement 2 package of Cgal, the Computational Geometry Algorithms Library, has been greatly extended to support arrangements of curves embedded on twodimensional parametric surfaces. The general framework for sweeping a set of curves embedded on a twodimensional parametric surface ..."
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Cited by 5 (4 self)
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Recently, the Arrangement 2 package of Cgal, the Computational Geometry Algorithms Library, has been greatly extended to support arrangements of curves embedded on twodimensional parametric surfaces. The general framework for sweeping a set of curves embedded on a twodimensional parametric surface was introduced in [3]. In this paper we concentrate on the specific algorithms and implementation details involved in the exact construction and maintenance of arrangements induced by arcs of great circles embedded on the sphere, also known as geodesic arcs, and on the exact computation of Voronoi diagrams on the sphere, the bisectors of which are geodesic arcs. This class of Voronoi diagrams includes the subclass of Voronoi diagrams of points and its generalization, power diagrams, also known as Laguerre Voronoi diagrams. The resulting diagrams are represented as arrangements, and can be passed as input to consecutive operations supported by the Arrangement 2 package and its derivatives. The implementation is complete in the sense that it handles degenerate input, and it produces exact results. An example that uses real world data is included. Additional material is available at
Arrangements on parametric surfaces II: Concretizations and applications
 IN COMPUTER SCIENCE
, 2010
"... We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamental ..."
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Cited by 4 (4 self)
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We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables exact construction, maintenance, and manipulation of arrangements embedded on certain twodimensional orientable parametric surfaces in threedimensional space. The fundamentals of the framework are described in a companion paper. Our work covers arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets, and documented the practical efficiency of our approach.