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18
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...
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 88 (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...
A Core Library For Robust Numeric and Geometric Computation
 In 15th ACM Symp. on Computational Geometry
, 1999
"... Nonrobustness is a wellknown problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (Core) for robust numeric and geometric computation ba ..."
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Cited by 60 (9 self)
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Nonrobustness is a wellknown problem in many areas of computational science. Until now, robustness techniques and the construction of robust algorithms have been the province of experts in this field of research. We describe a new C/C++ library (Core) for robust numeric and geometric computation based on the principles of Exact Geometric Computation (EGC). Through our library, for the first time, any programmer can write robust and efficient algorithms. The Core Library is based on a novel numerical core that is powerful enough to support EGC for algebraic problems. This is coupled with a simple delivery mechanism which transparently extends conventional C/C++ programs into robust codes. We are currently addressing efficiency issues in our library: (a) at the compiler and language level, (b) at the level of incorporating EGC techniques, as well as the (c) the system integration of both (a) and (b). Pilot experimental results are described. The basic library is available at http://cs.nyu.edu...
MAPC: A library for Efficient and Exact Manipulation of Algebraic Points and Curves
"... We present MAPC, a library for exact representation of geometric objects  specifically points and algebraic curves in the plane. Our library makes use of several new algorithms, which we present here, including methods for nding the sign of a determinant, finding intersections between two curves, ..."
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Cited by 28 (8 self)
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We present MAPC, a library for exact representation of geometric objects  specifically points and algebraic curves in the plane. Our library makes use of several new algorithms, which we present here, including methods for nding the sign of a determinant, finding intersections between two curves, and breaking a curve into monotonic segments. These algorithms are used to speed up the underlying computations. The library provides C++ classes that can be used to easily instantiate, manipulate, and perform queries on points and curves in the plane. The point classes can be used to represent points known in a variety of ways (e.g. as exact rational coordinates or algebraic numbers) in a unified manner. The curve class can be used to represent a portion of an algebraic curve. We have used MAPC for applications dealing with algebraic points and curves, including sorting points along a curve, computing arrangement of curves, medial axis computations, and boundary evaluation on curved primitives. As compared to earlier algorithms and implementations utilizing exact arithmetic, our library is able to achieve more than an order of magnitude improvement in performance.
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 22 (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
Efficient and accurate brep generation of low degree sculptured solids using exact arithmetic
 In ACM/SIGGRAPH Symposium on Solid Modeling
, 1997
"... We present efficient representations and algorithms for exact boundary computation on low degree sculptured CSG solids using exact arithmetic. Most of the previous work using exact arithmetic has been restricted to polyhedral models. In this paper, we generalize it to higher order objects, whose bou ..."
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Cited by 20 (8 self)
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We present efficient representations and algorithms for exact boundary computation on low degree sculptured CSG solids using exact arithmetic. Most of the previous work using exact arithmetic has been restricted to polyhedral models. In this paper, we generalize it to higher order objects, whose boundaries are composed of rational parametric surfaces. The use of exact arithmetic and representation guarantees that a geometric algorithm is numerically accurate and is likely to be required for perturbation techniques which handle degeneracies. We present efficient algorithms for computing the intersection curves of trimmed parametric surfaces, decomposing them into multiple components for e cient point location queries inside the trimmed regions, and computing the boundary of the resulting solid using topological information and component classification
Recent Progress in Exact Geometric Computation
 English Today
, 2001
"... Computational geometry has produced an impressive wealth of efficient algorithms. ..."
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Cited by 20 (6 self)
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Computational geometry has produced an impressive wealth of efficient algorithms.
Robustness Issues in Geometric Algorithms
 Applied Computational Geometry
, 1996
"... ic predicates. If an instance of a predicate is nearly degenerate, then the value of the corresponding expression can be 1 This note was adapted from a chapter of the report "Computational geometry: perspectives and challenges", edited by Bernard Chazelle. very small, less than the rounding error ..."
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Cited by 16 (0 self)
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ic predicates. If an instance of a predicate is nearly degenerate, then the value of the corresponding expression can be 1 This note was adapted from a chapter of the report "Computational geometry: perspectives and challenges", edited by Bernard Chazelle. very small, less than the rounding error in the floatingpoint evaluation of the expression. Hence the sign of the computed expression may well be erroneous. Usually it is possible to argue that the computed expression sign is the true sign for slightly perturbed coordinate data. Since coordinate data may well be imprecise originally, the erroneous sign may appear to be innocuous. The difficulty arises with multiple predicate evaluations; there is no guarantee that any single global perturbation produces all the computed predicate values. Indeed, the computed predicate values may be geometrically inconsistent. Catastrophic implementation failure can easily result. There are two broad categories of methods to d
Efficient and Exact Manipulation of Algebraic Points and Curves
, 2000
"... An important part of solid modeling systems based on curved primitives is the representation and manipulation of algebraic plane curves with rational coefficients and points with algebraic coordinates. These objects are often approximated by floatingpoint numbers and spline curves, which are easy t ..."
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Cited by 14 (2 self)
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An important part of solid modeling systems based on curved primitives is the representation and manipulation of algebraic plane curves with rational coefficients and points with algebraic coordinates. These objects are often approximated by floatingpoint numbers and spline curves, which are easy to manipulate, but are subject to accuracy and robustness problems. Exact computation can eliminate these numerical robustness problems, but efficient exact methods have not been available. Moreover, it is widely believe that exact arithmetic is impractical for manipulating curved primitives.