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ALGEBRAIC GEOMETRY
"... Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is ..."
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Cited by 426 (6 self)
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Algebraic geometry is the mathematical study of geometric objects by means of algebra. Its origins go back to the coordinate geometry introduced by Descartes. A classic example is the circle of radius 1 in the plane, which is
Residual Resultant over the Projective Plane and the Implicitization Problem
, 2001
"... In this article, we first generalize the recent notion of residual resultant of a complete intersection [4] to the case of a local complete intersection of codimension 2 in the projective plane, which is the necessary and su#cient condition for a system of three polynomials to have a solution " ..."
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Cited by 20 (8 self)
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In this article, we first generalize the recent notion of residual resultant of a complete intersection [4] to the case of a local complete intersection of codimension 2 in the projective plane, which is the necessary and su#cient condition for a system of three polynomials to have a solution "outside" a variety, defined here by a local complete intersection of codimension 2. We give its degree in the coe#cients of each polynomial and compute it as the gcd of three polynomials or as a product of two determinants divided by another one. In a second part we use this new type of resultant to give a new method to compute the implicit equation of a rational surface with base points in the case where these base points are a local complete intersection of codimension 2.
Using projection operators in Computer Aided Geometric Design
 In Topics in Algebraic Geometry and Geometric Modeling
, 2003
"... We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention di#erent formulations of resultants, including the projective resultant, the toric resultant, and the residual resultants. In the second part we illustrate these tools, and ..."
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Cited by 15 (7 self)
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We give an overview of resultant theory and some of its applications in computer aided geometric design. First, we mention di#erent formulations of resultants, including the projective resultant, the toric resultant, and the residual resultants. In the second part we illustrate these tools, and others projection operators, on typical problems as surface implicitization, inversion, intersection, and detection of singularities of a parameterized surface.
Curve/surface intersection problem by means of matrix representation
 In SNC’09: Proceedings of the International Conference on Symbolic Numeric Computation. Kyoto, Japan.ACM
, 2009
"... In this paper, we introduce matrix representations of algebraic curves and surfaces for Computer Aided Geometric Design (CAGD). The idea of using matrix representations in CAGD is quite old. The novelty of our contribution is to enable non square matrices, extension which is motivated by recent rese ..."
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Cited by 9 (4 self)
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In this paper, we introduce matrix representations of algebraic curves and surfaces for Computer Aided Geometric Design (CAGD). The idea of using matrix representations in CAGD is quite old. The novelty of our contribution is to enable non square matrices, extension which is motivated by recent research in this topic. We show how to manipulate these representations by proposing a dedicated algorithm to address the curve/surface intersection problem by means of numerical linear algebra techniques. 1
Approximate implicitization via curve fitting
, 2003
"... We discuss methods for fitting implicitly defined (e.g. piecewise algebraic) curves to scattered data, which may contain problematic regions, such as edges, cusps or vertices. As the main idea, we construct a bivariate function, whose zero contour approximates a given set of points, and whose gradie ..."
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Cited by 9 (6 self)
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We discuss methods for fitting implicitly defined (e.g. piecewise algebraic) curves to scattered data, which may contain problematic regions, such as edges, cusps or vertices. As the main idea, we construct a bivariate function, whose zero contour approximates a given set of points, and whose gradient field simultaneously approximates an estimated normal field. The coefficients of the implicit representation are found by solving a system of linear equations. In order to allow for problematic input data, we introduce a criterion for detecting points close to possible singularities. Using this criterion we split the data into segments and develop methods for propagating the orientation of the normals globally. Furthermore we present a simple fallback strategy, that can be used when the process of orientation propagation fails. The method has been shown to work successfully
Comparative Benchmarking of Methods for Approximate Implicitization
 GEOMETRIC DESIGN AND COMPUTING, NASHBORO
, 2004
"... Recently a variety of algorithms for the approximation of parametric curves and surfaces by algebraic representations have been developed. We test three of these methods on several test cases, keeping track of time and memory consumption of the implementations and the quality of the approximation. A ..."
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Cited by 8 (5 self)
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Recently a variety of algorithms for the approximation of parametric curves and surfaces by algebraic representations have been developed. We test three of these methods on several test cases, keeping track of time and memory consumption of the implementations and the quality of the approximation. Additionally we discuss some qualitative aspects of the different methods.
IMPLICITIZATION OF RATIONAL SURFACES USING TORIC VARIETIES
, 2004
"... Abstract. We extend to the toric case two methods for computing the implicit equation of a rational parameterized surface. The first approach gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local comp ..."
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Cited by 8 (1 self)
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Abstract. We extend to the toric case two methods for computing the implicit equation of a rational parameterized surface. The first approach gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any nonzero maximal minor of this matrix. The second method is the toric extension of the method of moving surfaces, and involves finding linear and quadratic relations (syzygies) among the input polynomials. When there are no base points, we show that these can be put together into a square matrix whose determinant is the implicit equation. Its extension to the case where there are base points is also explored. 1.