On the Validity of Implicitization by Moving Quadrics for Rational Surfaces with No Base Points

Cox, David A.; Goldman, Ron; Zhang, Ming

doi:10.1006/jsco.1999.0325

Cited by 78 publications

(71 citation statements)

References 7 publications

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“…When V(a, b, c) = ∅, this is proved in [2]. The analogous result for φ : P 2 −→ P 3 , namely Theorem 1.6, is proved in [1], but with no restriction at all on the degree of the syzygies.…”

Section: Curvilinear Base Points and Koszul Syzygiesmentioning

confidence: 75%

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Hoffman

Wang

2006

Trans. Amer. Math. Soc.

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Abstract. Let I = f 1 , f 2 , f 3 be a bigraded ideal in the bigraded polynomial ring k [s, u; t, v]. Assume that I has codimension 2. Then Z = V(I) ⊂ P 1 × P 1 is a finite set of points. We prove that if Z is a local complete intersection, then any syzygy of the f i vanishing at Z, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).

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Section: Curvilinear Base Points and Koszul Syzygiesmentioning

confidence: 75%

Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces

Hoffman

Wang

2006

Trans. Amer. Math. Soc.

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“…Practical problems associated with the exact implicitization of curves and surfaces are addressed in [22] and [5]. Gröbner bases can also be used [7].…”

Section: Approximate Implicitizationmentioning

confidence: 99%

Approximate Implicitization of Space Curves

Aigner

Jüttler

Poteaux

2011

Texts &Amp; Monographs in Symbolic Computation

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The process of implicitization generates an implicit representation of a curve or surface from a given parametric one. This process is potentially interesting for applications in Computer Aided Design, where the robustness and efficiency of intersection algorithm can be improved by simultaneously considering implicit and parametric representations. This paper gives an brief survey of the existing techniques for approximate implicitization of hyper surfaces. In addition it describes a framework for the approximate implicitization of space curves.

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“…However here, instead of computing symbolic determinants, we rely on constant coefficients of some Laurent series. The obtained algorithm is then applied to solve important questions in CAGD (Computer Aided Geometric Design), namely the Implicitization Problem, the determination of the Singular locus and the Intersection Problem for rational hypersurfaces (see [29], [7], [11], [13], [14], [16], [10]). …”

Section: §1 Introductionmentioning

confidence: 99%