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35
Implicitization of surfaces in P 3 in the presence of base points
 J. Algebra Appl
"... Abstract. We show that the method of moving quadrics for implicitizing surfaces in P 3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local co ..."
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Cited by 21 (9 self)
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Abstract. We show that the method of moving quadrics for implicitizing surfaces in P 3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit
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 "outsi ..."
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Cited by 19 (7 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 13 (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.
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 10 (7 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
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 8 (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
Implicitizing rational surfaces with base points using the method of moving surfaces
 Contemporary Mathematics Series, AMS
, 2003
"... Abstract. The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have ..."
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Cited by 7 (0 self)
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Abstract. The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M. The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)—thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and “cornercut ” surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I ′ , where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra. To Bruno Buchberger in honor of his achievements in computational algebra 1.
Equations of parametric surfaces with base points via syzygies
, 2003
"... Let S be a parametrized surface in P 3 given as the image of φ: P 1 × P 1 → P 3. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This ..."
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Cited by 7 (0 self)
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Let S be a parametrized surface in P 3 given as the image of φ: P 1 × P 1 → P 3. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of S when certain base points are present. This work extends the algorithm provided by Cox [5] for when φ has no base points, and it is analogous to some of the results of Busé, Cox, and D’Andrea [2] for the case when φ: P 2 → P 3 has base points.
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 7 (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.
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 6 (6 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.
On the Homology of TwoDimensional Elimination
"... We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two, after ..."
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Cited by 6 (2 self)
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We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two, after observing that the structure of its irreducible ideals (always complete intersections by a classical theorem of Serre) leads to a natural approach to the calculation of Sylvester determinants, we introduce a computerassisted method (with a minimal intervention by the computer) which succeeds, in degree ≤ 5, in producing the full sets of equations of the ideals. In the process, it answers affirmatively some questions raised by D. Cox ((9)).