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10
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 95 (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...
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 62 (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...
NearOptimal Parameterization of the Intersection of Quadrics: II. A Classification of Pencils
, 2005
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Computing Quadric Surface Intersections Based on an Analysis of Plane Cubic Curves
, 2002
"... Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of ..."
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Cited by 17 (6 self)
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Computing the intersection curve of two quadrics is a fundamental problem in computer graphics and solid modeling. We present an algebraic method for classifying and parameterizing the intersection curve of two quadric surfaces. The method is based on the observation that the intersection curve of two quadrics is birationally related to a plane cubic curve. In the method this plane cubic curve is computed first and the intersection curve of the two quadrics is then found by transforming the cubic curve by a rational quadratic mapping. Topological classification and parameterization of the intersection curve are achieved by invoking results from algebraic geometry on plane cubic curves.
Enhancing Levin's Method for Computing QuadricSurface Intersections
, 2002
"... Levin's method produces a parameterization of the intersection curve of two quadrics in the form p(u); a(u) :h d()V/, where a(u) and d(u) are polynomial vectorvalued functions, and s(u) is a quartic polynomial. ..."
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Cited by 17 (3 self)
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Levin's method produces a parameterization of the intersection curve of two quadrics in the form p(u); a(u) :h d()V/, where a(u) and d(u) are polynomial vectorvalued functions, and s(u) is a quartic polynomial.
Using multivariate resultants to find the intersection of three quadric surfaces
 ACM Trans. on Graphics
, 1991
"... Macaulay’s concise but explicit expression for nmltivariate resultants has many potential applications in computeraided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By B6zout’s theorem, three quadric surfaces have eit ..."
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Cited by 13 (1 self)
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Macaulay’s concise but explicit expression for nmltivariate resultants has many potential applications in computeraided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By B6zout’s theorem, three quadric surfaces have either at most eight or intlnitely many intersections. Our method finds the intersections, when there are finitely many, by generating a polynomial of degree at most eight whose roots are the intersection coordinates along an appropriate axis. Only addition, subtraction, and multiplication are required to find the polynomial. But when there are pmsibilities of extraneous roots, division and greatest common divisor computations are necessary to identify and remove them.
Signature Sequence of Intersection Curve of Two Quadrics for Exact Morphological Classification
, 2005
"... this paper  we enumerate all 35 di#erent morphologies of QSIC, and characterize each of these morphologies using a signature sequence that can exactly be computed using rational arithmetic. The third problem, not handled here, leads to a lengthy case by case study which depends a lot on the applic ..."
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Cited by 9 (1 self)
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this paper  we enumerate all 35 di#erent morphologies of QSIC, and characterize each of these morphologies using a signature sequence that can exactly be computed using rational arithmetic. The third problem, not handled here, leads to a lengthy case by case study which depends a lot on the application behind. Consider the intersection curve of two quadrics given by BX = 0, where X = (x, y, z, w) and A, B are 4 4 real symmetric matrices. The characteristic polynomial of (1) and f(#) = 0 is called the characteristic equation of B
Using signature sequences to classify intersection curves of two quadrics
 COMPUTER AIDED GEOMETRIC DESIGN
, 2009
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NearOptimal Parameterization of the Intersection of Quadrics: III. Parameterizing Singular Intersections
"... We conclude, in this third part, the presentation of an algorithm for computing an exact and proper parameterization of the intersection of two quadrics. The coordinate functions of the parameterizations in projective space are polynomial, whenever it is possible. They are also nearoptimal in the s ..."
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We conclude, in this third part, the presentation of an algorithm for computing an exact and proper parameterization of the intersection of two quadrics. The coordinate functions of the parameterizations in projective space are polynomial, whenever it is possible. They are also nearoptimal in the sense that the number of distinct square roots appearing in the coefficients of these functions is minimal except in a small number of cases (characterized by the real type of the intersection) where there may be an extra square root. Our algorithm builds on the classification of pencils of quadrics of P 3 (R) over the reals presented in Part II and the typedetection algorithm that we deduced from this classification. Moreover, since the algorithm presented in Part I is nearoptimal when the intersection is a nonsingular quartic, we focus here on the case where the intersection is singular and present, for all possible real types of intersection, algorithms for computing nearoptimal rational parameterizations. We also give examples covering all the possible situations, in terms of both the real type of intersection and the number and depth of square roots appearing in the coefficients of the parameterizations. Key words: Intersection of surfaces, pencils of quadrics, curve parameterization, singular intersections.
Abstract
"... We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR 3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility ..."
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We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR 3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with nondegenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic so as to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing Brep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives.