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15
On the design of CGAL a computational geometry algorithms library
 Softw. – Pract. Exp
, 1998
"... 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 90 (15 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. Copyright c ○ 1999 John Wiley & Sons, Ltd. KEY WORDS: computational geometry; software library; C++; generic programming;
A Separation Bound for Real Algebraic Expressions
 In Lecture Notes in Computer Science
, 2001
"... Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the si ..."
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Cited by 38 (4 self)
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Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real. 1
An adaptable and extensible geometry kernel
 In Proc. Workshop on Algorithm Engineering
, 2001
"... ii ..."
Applications of the Generic Programming Paradigm in the Design of CGAL
, 1998
"... We report on the use of the generic programming paradigm in the computational geometry algorithms library cgal. The parameterization of the geometric algorithms in cgal enhances exibility and adaptability and opens an easy way for abolishing precision and robustness problems by exact but neverthe ..."
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Cited by 15 (2 self)
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We report on the use of the generic programming paradigm in the computational geometry algorithms library cgal. The parameterization of the geometric algorithms in cgal enhances exibility and adaptability and opens an easy way for abolishing precision and robustness problems by exact but nevertheless ecient computation. Furthermore we discuss circulators, which are an extension of the iterator concept to circular structures. Such structures arise frequently in geometric computing. 1 Introduction cgal is a C++ library of geometric algorithms and data structures. It is developed by several sites in Europe and Israel. The goal is to enhance the technology transfer of the algorithmic knowledge developed in the eld of computational geometry to applications in industry and academia. Computational geometry is the subarea of algorithm design that deals with the design and analysis of algorithms for geometric problems involving objects like points, lines, polygons, and polyhedra. Ove...
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 14 (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.
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.
A package for exact kinetic data structures and sweepline algorithms
, 2007
"... In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to ex ..."
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Cited by 7 (0 self)
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In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability. 1
Online Zone Construction in Arrangements of Lines in the Plane
 In Proc. of the 3rd Workshop of Algorithm Engineering
, 1999
"... Given a finite set L of lines in the plane we wish to compute the zone of an additional curve fl in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by fl, where fl is not given in advance but rather provided online portion by por ..."
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Cited by 6 (4 self)
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Given a finite set L of lines in the plane we wish to compute the zone of an additional curve fl in the arrangement A(L), namely the set of faces of the planar subdivision induced by the lines in L that are crossed by fl, where fl is not given in advance but rather provided online portion by portion. This problem is motivated by the computation of the area bisectors of a polygonal set in the plane. We present four algorithms which solve this problem efficiently and exactly (giving precise results even on degenerate input). We implemented the four algorithms. We present implementation details, comparison of performance, and a discussion of the advantages and shortcomings of each of the proposed algorithms. 1 Introduction Given a finite collection L of lines in the plane, the arrangement A(L) is the subdivision of the plane into vertices, edges and faces induced by L. Arrangements of lines in the plane, as well as arrangements of other objects and in higher dimensional spaces, ...
Designing a Computational Geometry Algorithms Library
 Lecture Notes for Advanced School on Algorithmic Foundations of Geographic Information Systems, CISM
, 1996
"... Introduction Geometric problems arise in many areas. Computer graphics, robotics, manufacturing, and geographic information systems are some examples. Often the same geometric subproblems are to be solved. Hence a library providing solutions for core problems in geometric computing has a wide range ..."
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Cited by 6 (1 self)
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Introduction Geometric problems arise in many areas. Computer graphics, robotics, manufacturing, and geographic information systems are some examples. Often the same geometric subproblems are to be solved. Hence a library providing solutions for core problems in geometric computing has a wide range of applications and can be very useful. The success of LEDA [16], a library of efficient data types and algorithms, has shown that the existence of a library can make a tremendous difference for taking advanced techniques in data structures and algorithms from theory to practice. The field of computational geometry is now very close to a state where it can provide such a library of geometric algorithms. Over the past twenty years many algorithms for geometric problems have been developed by computational geometers. Many of these algorithms clearly have no direct impact for geometric computing in practice, because they are efficient compared to other solutions only for huge problem i