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polymake: a Framework for Analyzing Convex Polytopes
, 1999
"... polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few ..."
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Cited by 100 (15 self)
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polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook. The tutorial starts with the very basics and ends up with a few polymake applications to research problems. Then we present the main features of the system including the interfaces to other software products. polymake is free software; it is available on the Internet at http://www.math.tuberlin.de/diskregeom/polymake/.
On the design of CGAL a computational geometry algorithms library
 SOFTW. – PRACT. EXP
, 1999
"... CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major ..."
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Cited by 97 (16 self)
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CGAL is a Computational Geometry Algorithms Library written in C++, which is being developed by research groups in Europe and Israel. The goal is to make the large body of geometric algorithms developed in the field of computational geometry available for industrial application. We discuss the major design goals for CGAL, which are correctness, flexibility, easeofuse, efficiency, and robustness, and present our approach to reach these goals. Generic programming using templates in C++ plays a central role in the architecture of CGAL. We give a short introduction to generic programming in C++, compare it to the objectoriented programming paradigm, and present examples where both paradigms are used effectively in CGAL. Moreover, we give an overview of the current structure of the CGALlibrary and consider software engineering aspects in the CGALproject.
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 25 (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.
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
One Sided Error Predicates in Geometric Computing
 Proc. 15th IFIP World Computer Congress, Fundamentals  Foundations of Computer Science
, 1998
"... A conservative implementation of a predicate returns true only if the exact predicate is true. That is, we accept a one sided error for the implementation. For geometric predicates, such as orientation or incircletests, this allows efficient floating point implementations of the predicates with ra ..."
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Cited by 6 (1 self)
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A conservative implementation of a predicate returns true only if the exact predicate is true. That is, we accept a one sided error for the implementation. For geometric predicates, such as orientation or incircletests, this allows efficient floating point implementations of the predicates with rare occurrences of the one sided error. We discuss the use of such conservative implementations for convex hull and triangulation algorithms for point sets in the plane. The resulting programs show a minor slowdown compared to an implementation that completely ignores the finite precision issue. However, our programs always produce output that satisfies basic desirable properties. The output can be easily checked for correctness and  if necessary  it can be repaired with an exact implementation of the needed predicates. Although (or since?) conservative implementations of predicates cannot detect degeneracies, the programs work for degenerate input. In fact, in our experiments the advanta...
Planar Nef Polyhedra and . . .
, 2001
"... We present two generic software projects that are part of the software library CGAL. The first part describes the design of a geometry kernel for higherdimensional Euclidean geometry and the interaction with application programs. We describe the software structure, the interface concepts, and their ..."
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Cited by 1 (0 self)
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We present two generic software projects that are part of the software library CGAL. The first part describes the design of a geometry kernel for higherdimensional Euclidean geometry and the interaction with application programs. We describe the software structure, the interface concepts, and their models that are based on coordinate representation, number types, and memory layout. In the higherdimensional software kernel the interaction between linear algebra and the geometric objects and primitives is one important facet. In the actual design our users can replace number types, representation types, and the traits classes that inflate kernel functionality into our current application programs: higherdimensional convex hulls and Delaunay tedrahedralisations. In the second part we present the realization of planar Nef polyhedra. The concept of Nef polyhedra subsumes all kinds of rectilinear polyhedral subdivisions and is therefore of general applicability within a geometric software library. The software is based on the theory of extended points and segments that allows us to reuse classical algorithmic solutions like plane sweep to realize binary operations of Nef polyhedra.
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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 iss ..."
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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.
Abstract Discrete Optimization
, 2005
"... Complete and robust nofit polygon generation ..."
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