Implicitizing rational surfaces with base points using the method of moving surfaces

Zheng, Jinhai; Sederberg, Thomas W.; Chionh, Eng-Wee; Cox, David A.

doi:10.1090/conm/334/05980

Cited by 21 publications

(10 citation statements)

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“…Example: [Base Points] Given a surface defined by rational parametric equations Consider the following example from [24]:…”

Section: Motivationmentioning

confidence: 99%

“…Using the Dixon resultant matrix construction [11,18,22], the perturbed system can be computed in about 1 min, where as the rank submatrix construction takes less than a second. The equation computed using the moving surfaces in [24] is…”

Section: Motivationmentioning

confidence: 99%

“…A case in point is implicitization problems with base points. The resultant of such a polynomial system is 0, since for every specialization of its parametric representation, there is a common solution [24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Conditions for determinantal formula for resultant of a polynomial system

Chtcherba¹,

Kapur

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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Matrices constructed from a parameterized multivariate polynomial system are analyzed to ensure that such a matrix contains a condition for the polynomial system to have common solutions irrespective of whether its parameters are specialized or not. Such matrices include resultant matrices constructed using well-known methods for computing resultants over projective, toric and affine varieties. Conditions on these matrices are identified under which the determinant of a maximal minor of such a matrix is a nontrivial multiple of the resultant over a given variety. This condition on matrices allows a generalization of a linear algebra construction, called rank submatrix, for extracting resultants from singular resultant matrices, as proposed by Kapur, Saxena and Yang in ISSAC'94. This construction has been found crucial for computing resultants of non-generic, specialized multivariate polynomial systems that arise in practical applications. The new condition makes the rank submatrix construction based on maximal minor more widely applicable by not requiring that the singular resultant matrix have a column independent of the remaining columns. Unlike perturbation methods, which require introducing a new variable, rank submatrix construction is faster and effective. Properties and conditions on symbolic matrices constructed from a polynomial system are discussed so that the resultant can be computed as a factor of the determinant of a maximal non-singular submatrix.

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“…Example: [Base Points] Given a surface defined by rational parametric equations Consider the following example from [24]:…”

Section: Motivationmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

Conditions for determinantal formula for resultant of a polynomial system

Chtcherba¹,

Kapur

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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“…Various symbolic-computation-based methods for implicitization have been introduced, based on Gröbner bases [AGR95,Buc88], resultants [Bus01,MM02], moving curves and surfaces [ZSCC03,SSQK94], residue calculus [EM04] or on other methods [GV97,CD04]. These techniques produce an exact implicit representation.…”

Section: Introductionmentioning

confidence: 99%

Implicitization and Distance Bounds

Aigner

Szilágyi

Jüttler

et al. 2006

Algebraic Geometry and Geometric Modeling

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We address the following problem: given a curve in parametric form, compute the implicit representation of another one that approximates the parametric curve on a certain domain of interest. We study this problem from the numerical point of view: what happens with the output curve if the input curve is slightly changed? It is shown that for any approximate parameterization of the given curve, the curve obtained by an approximate implicitization with a given precision is contained within a certain perturbation region.

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“…The reverse process is called implicitization. The implicitization problem is always solvable, and there are several different approaches to deal with this problem, as described by Busé (2001), Buchberger (1988), Corless et al (2001), and Zheng et al (2003).…”

Section: Introductionmentioning

confidence: 99%

Local parametrization of cubic surfaces

Szilágyi

Jüttler

Schicho

2006

Journal of Symbolic Computation

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Algebraic surfaces-which are frequently used in geometric modeling-are represented either in implicit or parametric form. Several techniques for parameterizing a rational algebraic surface as a whole exist. However, in many applications, it suffices to parameterize a small portion of the surface. This motivates the analysis of local parametrizations, i.e., parametrizations of a small neighborhood of a given point P of the surface S. In this paper we introduce several techniques for generating such parameterizations for nonsingular cubic surfaces. For this class of surfaces, it is shown that the local parametrization problem can be solved for all points, and any such surface can be covered completely.

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Implicitizing rational surfaces with base points using the method of moving surfaces

Cited by 21 publications

References 0 publications

Conditions for determinantal formula for resultant of a polynomial system

Conditions for determinantal formula for resultant of a polynomial system

Implicitization and Distance Bounds

Local parametrization of cubic surfaces

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