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LEDA: A Platform for Combinatorial and Geometric Computing
, 1999
"... We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical ..."
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Cited by 635 (43 self)
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We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our webpages for more information.
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (20 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 94 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
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.
Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 72 (11 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Static Analysis Yields Efficient Exact Integer Arithmetic for Computational Geometry
 ACM Trans. Graph
, 1996
"... Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floatingpoint arithmetic for real arithmetic can fail, since geometric primitives depend upon signevaluation and may not be re ..."
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Cited by 58 (4 self)
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Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floatingpoint arithmetic for real arithmetic can fail, since geometric primitives depend upon signevaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extendedprecision arithmetic is required. We describe staticanalysis techniques that reduce the performance cost of exact integer arithmetic used to implement geometric algorithms. We have used the techniques for a number of examples, including linesegment intersection in two dimensions, Delaunay triangulations, and a threedimensional boundarybased polyhedral modeller. In general, the techniques are appropriate for algorithms that use primitives of relatively low algebraic total degree, e.g., those involving flat objects (...
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 37 (3 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
Exact Geometric Computation in LEDA
 In Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example MaxPlanckInstitut fur Informatik, ..."
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Cited by 35 (5 self)
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real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example MaxPlanckInstitut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. Supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). y Fachbereich 14, Informatik, Universitat des Saarlandes, 66041 Saarbrucken, Germany. z MartinLutherUniversitat Halle, Fachbereich Mathematik und Informatik, 06099 Halle, Germany. 0 arises in the computation of Voronoi diagrams of line segments [2]. For i, 1 i 3, let l i : a i x + b i
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.
Algebraic Approach of Residues and Applications
 The Mathematics of Numerical Analysis, volume 32 of Lectures in Applied Math
, 1996
"... In this paper, we give an overview of the theory of residues and its applications. We adopt the algebraic point of view and show how this theory can lead to effective computations. For this purpose, we analyze the properties of Bezoutians, objects which underly many problems related to polynomial sy ..."
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Cited by 28 (11 self)
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In this paper, we give an overview of the theory of residues and its applications. We adopt the algebraic point of view and show how this theory can lead to effective computations. For this purpose, we analyze the properties of Bezoutians, objects which underly many problems related to polynomial systems. We recall the main properties of a Gorenstein Algebra, where we can naturally speak of residues. We show how we can construct a residue, in the case of complete intersection, and we give the algebraic counterparts of some theorems of the analytical theory of residues. We also give some direct applications of this approach to the computations of Chow Forms and show how the information concerning multiple roots can be recovered from the local residues in a very simple way. Finally, we propose a new method for solving zerodimensional systems of n equations P1 = \Delta \Delta \Delta = Pn = 0 in n variables x1 , ..., xn . This method involves elementary manipulations on the matrices...