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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 91 (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.
The CGAL Kernel: A Basis for Geometric Computation
, 1996
"... A large part of the cgalproject is devoted to the development of a Computational Geometry Algorithms Library, written in C++. We discuss design issues concerning the cgalkernel which is the basis for the library and hence for all geometric computation in cgal. ..."
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Cited by 47 (10 self)
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A large part of the cgalproject is devoted to the development of a Computational Geometry Algorithms Library, written in C++. We discuss design issues concerning the cgalkernel which is the basis for the library and hence for all geometric computation in cgal.
A Probabilistic Analysis of the Power of Arithmetic Filters
 PUBLISHED IN DISCRETE AND COMPUTATIONAL GEOMETRY 20:523–547
, 1998
"... The assumption of realnumber arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In ..."
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Cited by 31 (5 self)
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The assumption of realnumber arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) ε, which represents the largest error that may arise in the evaluation of the expression. The threshold ε depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are errorfree. A pair (arithmetic engine,threshold) is an arithmetic filter. In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates ”whichside ” and ’insphere”. We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value V, with emphasis on small V, is Θ(V) for the whichside predicate, while for the insphere predicate it is Θ(V 2 3) in dimension 1, O(V 1 2) in dimension 2, and O(V 1 2 ln 1) in higher dimensions. Constants are small, and are given in the paper.
Efficient Exact Geometric Computation Made Easy
, 1999
"... We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easytowrite, correct and efficient geometric programs. ..."
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Cited by 20 (5 self)
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We show that the combination of the Cgal framework for geometric computation and the number type leda_real yields easytowrite, correct and efficient geometric programs.
The LEDA class real number
 MaxPlanck Institut Inform
, 1996
"... We describe the implementation of the LEDA [MN95, Nah95] data type real. Every integer is a real and reals are closed under the operations addition, subtraction, multiplication, division and squareroot. The main features of the data type real are ffl The userinterface is similar to that of the bu ..."
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Cited by 16 (5 self)
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We describe the implementation of the LEDA [MN95, Nah95] data type real. Every integer is a real and reals are closed under the operations addition, subtraction, multiplication, division and squareroot. The main features of the data type real are ffl The userinterface is similar to that of the builtin data type double.
Exact Geometric Predicates using Cascaded Computation
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various floatingpoint filters together with arbitrary precision packages, we develop an easytouse expression compiler called EXPCOMP. EXPCOMP supports all common op ..."
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Cited by 15 (2 self)
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In this paper we talk about a new efficient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various floatingpoint filters together with arbitrary precision packages, we develop an easytouse expression compiler called EXPCOMP. EXPCOMP supports all common operations +; \Gamma; \Delta; =; p . Applying a new semistatic filter, EXPCOMP combines the speed of static filters with the power of dynamic filters. The filter stages deal with all kinds of floatingpoint exceptions, including underflow. The resulting programs show a very good runtime behaviour. 1 Introduction When computer scientists design geometric algorithms, they usually assume the availability of exact arithmetic on real numbers. Since no computer directly provides exact arithmetic on real numbers, programmers implementing these algorithms must find some substitution. Quite commonly, they resort to floatingpoint arithmetic due to its support by hardand software as well as its co...
Interval Arithmetic: An Efficient Implementation And An Application To Computational Geometry
 In Workshop on Applications of Interval Analysis to systems and Control
, 1999
"... We discuss interval techniques for speeding up the exact evaluation of geometric predicates and describe a C++ implementation of interval arithmetic that is strongly influenced by the rounding modes of the widely used IEEE 754 standard. Using this approach we engineer an efficient floating point fil ..."
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Cited by 12 (1 self)
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We discuss interval techniques for speeding up the exact evaluation of geometric predicates and describe a C++ implementation of interval arithmetic that is strongly influenced by the rounding modes of the widely used IEEE 754 standard. Using this approach we engineer an efficient floating point filter for the computation of geometric predicates. We validate our approach experimentally, comparing it with other static, dynamic and semistatic filters.
Exact Geometric Computation Using Cascading
, 2000
"... In this paper we talk about a new ecient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various oatingpoint lters together with arbitrary precision packages, we develop an easytouse expression compiler called EXPCOMP. EXPCOMP supports all common operat ..."
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Cited by 10 (0 self)
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In this paper we talk about a new ecient numerical approach to deal with inaccuracy when implementing geometric algorithms. Using various oatingpoint lters together with arbitrary precision packages, we develop an easytouse expression compiler called EXPCOMP. EXPCOMP supports all common operations +; ; ; =; p . Applying a new semistatic lter, EXPCOMP combines the speed of static lters with the power of dynamic lters. The lter stages deal with all kinds of oatingpoint exceptions, including underow. The resulting programs show a very good runtime behaviour. Keywords: oatingpoint lter, exact arithmetic, expression compiler 1. Introduction When computer scientists design geometric algorithms, they usually assume the availability of exact arithmetic on real numbers. Since no computer directly provides exact arithmetic on real numbers, programmers implementing these algorithms must nd some substitution. Quite commonly, they resort to oatingpoint arithmetic due t...
Implementation of a Sweep Line Algorithm for the Straight Line Segment Intersection Problem
 TR MPII94105, MAXPLANCKINSTITUT FUR INFORMATIK, SAARBRUCKEN
, 1994
"... We describe a robust and efficient implementation of the BentleyOttmann sweep line algorithm [7] based on the LEDA library of efficient data types and algorithms [7]. The program computes the planar graph G induced by a set S of straight line segments in the plane. The nodes of G are all endpoints ..."
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Cited by 9 (2 self)
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We describe a robust and efficient implementation of the BentleyOttmann sweep line algorithm [7] based on the LEDA library of efficient data types and algorithms [7]. The program computes the planar graph G induced by a set S of straight line segments in the plane. The nodes of G are all endpoints and all proper intersection points of segments in S. The edges of G are the maximal relatively open subsegments of segments in S that contain no node of G. All edges are directed from left to right or upwards. The algorithm runs in time O((n + s) log n) where n is the number of segments and s is the number of vertices of the graph G. The implementation uses exact arithmetic for the reliable realization of the geometric primitives and it uses floating point filters to reduce the overhead of exact arithmetic.
Infimaximal Frames  A Technique For Making Lines Look Like Segments
 INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY AND APPLICATION
, 2000
"... Many geometric algorithms that are usually formulated for points and segments generalize easily to inputs also containing rays and lines. The sweep algorithm for segment intersection is a prototypical example. Implementations of such algorithms do, in general, not extend easily. For example, ..."
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Cited by 7 (3 self)
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Many geometric algorithms that are usually formulated for points and segments generalize easily to inputs also containing rays and lines. The sweep algorithm for segment intersection is a prototypical example. Implementations of such algorithms do, in general, not extend easily. For example,