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34
Towards Exact Geometric Computation
, 1994
"... Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To rec ..."
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Cited by 90 (6 self)
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Exact computation is assumed in most algorithms in computational geometry. In practice, implementors perform computation in some fixedprecision model, usually the machine floatingpoint arithmetic. Such implementations have many wellknown problems, here informally called "robustness issues". To reconcile theory and practice, authors have suggested that theoretical algorithms ought to be redesigned to become robust under fixedprecision arithmetic. We suggest that in many cases, implementors should make robustness a nonissue by computing exactly. The advantages of exact computation are too many to ignore. Many of the presumed difficulties of exact computation are partly surmountable and partly inherent with the robustness goal. This paper formulates the theoretical framework for exact computation based on algebraic numbers. We then examine the practical support needed to make the exact approach a viable alternative. It turns out that the exact computation paradigm encomp...
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 (...
Evaluating Signs of Determinants Using SinglePrecision Arithmetic
, 1994
"... We propose a method to evaluate signs of 2 x 2 and 3 x 3 determinants with bbit integer entries using only b and (b + 1)bit arithmetic respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been ..."
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Cited by 40 (5 self)
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We propose a method to evaluate signs of 2 x 2 and 3 x 3 determinants with bbit integer entries using only b and (b + 1)bit arithmetic respectively. This algorithm has numerous applications in geometric computation and provides a general and practical approach to robustness. The algorithm has been implemented and experimental results show that it slows down the computing time by only a small factor with respect to floatingpoint calculation.
Snap Rounding Line Segments Efficiently in Two and Three Dimensions
, 1997
"... We study the problem of robustly rounding a set S of n line segments in R 2 using the snap rounding paradigm. In this paradigm each pixel containing an endpoint or intersection point is called "hot," and all segments intersecting a hot pixel are rerouted to pass through its center. We show that a ..."
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Cited by 32 (4 self)
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We study the problem of robustly rounding a set S of n line segments in R 2 using the snap rounding paradigm. In this paradigm each pixel containing an endpoint or intersection point is called "hot," and all segments intersecting a hot pixel are rerouted to pass through its center. We show that a snaprounded approximation to the arrangement defined by S can be built in an outputsensitive fashion, and that this can be done without first determining all the intersecting pairs of segments in S. Specifically, we give a deterministic planesweep algorithm running in time O(n log n + P h2H jhj log n), where H is the set of hot pixels and jhj is the number of segments intersecting a hot pixel h 2 H. We also give a simple randomized incremental construction whose expected running time matches that of our deterministic algorithm. The complexity of these algorithms is optimal up to polylogarithmic factors. This research is supported by NSF grant CCR9625289 and by U.S. ARO grant DAAH04...
Efficient exact evaluation of signs of determinants
 IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... This paper presents a theoretical and experimental study on two different methods to evaluate the sign of a determinant with integer entries. The first one is a method based on the GramSchmidt orthogonalisation process which has been proposed by Clarkson [5]. We review the analysis of Clarkson an ..."
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Cited by 28 (2 self)
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This paper presents a theoretical and experimental study on two different methods to evaluate the sign of a determinant with integer entries. The first one is a method based on the GramSchmidt orthogonalisation process which has been proposed by Clarkson [5]. We review the analysis of Clarkson and propose a variant of his method. The second method is an extension to n \Theta n determinants of the ABDPY method [1] which works only for 2 \Theta 2 and 3 \Theta 3 determinants. Both methods compute the signs of a n \Theta n determinants whose entries are integers on b bits, by using an exact arithmetic on only b + O(n) bits. Furthermore, both methods are adaptive, dealing quickly with easy cases and resorting to the fulllength computation only for null determinants.
Polyhedral Modeling With Multiprecision Integer Arithmetic
, 1996
"... this paper appeared in the Third Symposium on Solid Modeling and Applications [7]. ..."
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Cited by 25 (3 self)
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this paper appeared in the Third Symposium on Solid Modeling and Applications [7].
Sign Determination in Residue Number Systems
, 1997
"... Sign determination is a fundamental problem in algebraic as well as geometric computing. It is the critical operation when using real algebraic numbers and exact geometric predicates. We propose an exact and efficient method that determines the sign of a multivariate polynomial expression with ra ..."
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Cited by 23 (9 self)
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Sign determination is a fundamental problem in algebraic as well as geometric computing. It is the critical operation when using real algebraic numbers and exact geometric predicates. We propose an exact and efficient method that determines the sign of a multivariate polynomial expression with rational coefficients. Exactness is achieved by using modular computation. Although this usually requires some multiprecision computation, our novel techniques of recursive relaxation of the moduli and their variants enable us to carry out sign determination and comparisons by using only single precision. Moreover, to exploit modern day hardware, we exclusively rely on floating point arithmetic, which leads us to a hybrid symbolicnumeric approach to exact arithmetic. We show how our method can be used to generate robust and efficient implementations of real algebraic and geometric algorithms including Sturm sequences, algebraic representation of points and curves, convex hull and Voronoi...
Delaunay Triangulations of Imprecise Points in Linear Time after Preprocessing
, 2008
"... An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one ..."
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Cited by 20 (5 self)
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An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.