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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...
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...
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 16 (5 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 16 (2 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 12 (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
, 2007
"... We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR³, 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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Cited by 7 (0 self)
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We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR³, 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.
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: I. The Generic Algorithm ⋆
"... We present an exact and efficient algorithm for computing a proper parametric representation of the intersection of two quadrics in threedimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection in projective space are polyno ..."
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We present an exact and efficient algorithm for computing a proper parametric representation of the intersection of two quadrics in threedimensional real space given by implicit equations with rational coefficients. The output functions parameterizing the intersection in projective space are polynomial, whenever it is possible, which is the case when the intersection is not a smooth quartic (for example, a singular quartic, a cubic and a line, and two conics). Furthermore, the parameterization is 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 wellidentified cases where there may be an extra square root. In addition, the algorithm is practical: a complete and efficient C++ implementation is described in Lazard et al. (2006). In Part I, we present an algorithm for computing a parameterization of the intersection of two arbitrary quadrics which we prove to be nearoptimal in the generic, smooth quartic, case. Parts II and III treat the singular cases. We present in Part II the first classification of pencils of quadrics according to the real type of the intersection and we show how this classification can be used to efficiently determine the type of the real part of the intersection of two arbitrary quadrics. This classification is at the core of the design of our algorithms for computing nearoptimal parameterizations of the real part of the intersection in all singular