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VRONI: an engineering approach to the reliable and efficient computation of Voronoi diagrams of points and line segments
 Comput. Geom. Theory and Appl
"... We discuss the design and implementation of a topologyoriented algorithm for the computation of Voronoi diagrams of points and line segments in the twodimensional Euclidean space. The main focus of our work was on designing and engineering an algorithm that is completely reliable and fast in pract ..."
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Cited by 54 (4 self)
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We discuss the design and implementation of a topologyoriented algorithm for the computation of Voronoi diagrams of points and line segments in the twodimensional Euclidean space. The main focus of our work was on designing and engineering an algorithm that is completely reliable and fast in practice. The algorithm was implemented in ANSI C, using standard floatingpoint arithmetic. In addition to Sugihara and Iri’s topologyoriented approach, it is based on a very careful implementation of the numerical computations required, an automatic relaxation of epsilon thresholds, and a multilevel recovery process combined with “desperate mode”. The resulting code, named vroni, was tested extensively on realworld data and turned out to be reliable. CPUtime statistics document that it is always faster than other popular Voronoi codes. In our computing environment, vroni needs about 0.01n log2 n milliseconds to compute the Voronoi diagram of n line segments, and this formula holds for a wide variety of synthetic and realworld data. In particular, its CPUtime consumption is hardly affected by the actual distribution of the input data. Vroni also features a function for computing offset curves, and it has been
A generic library of floatingpoint numbers and its application to exact computing
 In 14th International Conference on Theorem Proving in Higher Order Logics
, 2001
"... Abstract. In this paper we present a general library to reason about floatingpoint numbers within the Coq system. Most of the results of the library are proved for an arbitrary floatingpoint format and an arbitrary base. A special emphasis has been put on proving properties for exact computing, i. ..."
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Cited by 52 (6 self)
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Abstract. In this paper we present a general library to reason about floatingpoint numbers within the Coq system. Most of the results of the library are proved for an arbitrary floatingpoint format and an arbitrary base. A special emphasis has been put on proving properties for exact computing, i.e. computing without rounding errors. 1
Classroom examples of robustness problems in geometric computations
 In Proc. 12th European Symposium on Algorithms, volume 3221 of Lecture Notes Comput. Sci
, 2004
"... ..."
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 44 (26 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
Highlevel filtering for arrangements of conic arcs
 In Proc. ESA 2002
, 2002
"... Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for impleme ..."
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Cited by 41 (9 self)
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Abstract. Many computational geometry algorithms involve the construction and maintenance of planar arrangements of conic arcs. Implementing a general, robust arrangement package for conic arcs handles most practical cases of planar arrangements covered in literature. A possible approach for implementing robust geometric algorithms is to use exact algebraic number types — yet this may lead to a very slow, inefficient program. In this paper we suggest a simple technique for filtering the computations involved in the arrangement construction: when constructing an arrangement vertex, we keep track of the steps that lead to its construction and the equations we need to solve to obtain its coordinates. This construction history can be used for answering predicates very efficiently, compared to a naïve implementation with an exact number type. Furthermore, using this representation most arrangement vertices may be computed approximately at first and can be refined later on in cases of ambiguity. Since such cases are relatively rare, the resulting implementation is both efficient and robust. 1
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 (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
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 37 (16 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 37 (14 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.
Esolid  a system for exact boundary evaluation
 ComputerAided Design
, 2002
"... We present a system, ESOLID, that performs exact boundary evaluation of lowdegree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algor ..."
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Cited by 28 (2 self)
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We present a system, ESOLID, that performs exact boundary evaluation of lowdegree curved solids in reasonable amounts of time. ESOLID performs accurate Boolean operations using exact representations and exact computations throughout. The demands of exact computation require a different set of algorithms and efficiency improvements than those found in a traditional inexact floating point based modeler. We describe the system architecture, representations, and issues in implementing the algorithms. We also describe a number of techniques that increase the efficiency of the system based on lazy evaluation, use of floating point filters, arbitrary floating point arithmetic with error bounds, and lower dimensional formulation of subproblems. ESOLID has been used for boundary evaluation of many complex solids. These include both synthetic datasets and parts of a Bradley Fighting Vehicle designed using the BRLCAD solid modeling system. It is shown that ESOLID can correctly evaluate the boundary of solids that are very hard to compute using a fixedprecision floating point modeler. In terms of performance, it is about an order of magnitude slower as compared to a floating point boundary evaluation system on most cases. 1