Using multivariate resultants to find the implicit equation of a rational surface

Chionh, Eng-Wee; Goldman, Ron

doi:10.1007/bf01902137

Cited by 28 publications

(24 citation statements)

References 8 publications

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“…Proper parametrizations play an important role in many practical applications in computer aided geometric design, such as in visualization (see Hoffmann, 1993;Hoffmann et al, 1997;Hoschek and Lasser, 1993) or rational parametrization of offsets (see Arrondo et al, 1997). Also, they provide an implicitization approach based on resultants (see Theorem 7); similar results on implicitization can be found in Chionh and Goldman (1992) and Cox et al (1998). For other related questions of proper parametrizations we refer to Gao and Chou (1992).…”

Section: Introductionmentioning

confidence: 91%

Tracing index of rational curve parametrizations

Sendra

Winkler

2001

Computer Aided Geometric Design

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A rational parametrization of an algebraic curve establishes a rational correspondence of this curve with the affine or projective line. This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational parametrization determines a linear tracing of the curve, when the parameter takes values in the algebraic closure of the ground field. Such a parametrization might trace the curve once or several times. We formally introduce the concept of the tracing index of a parametrization, we show how to compute it, and we relate it to the degree of rational reparametrizations as well as to the degree of the curve. In addition, we show how to apply these results to the case of real curves, where we introduce the notion of real tracing index.

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Section: Introductionmentioning

confidence: 91%

Tracing index of rational curve parametrizations

Sendra

Winkler

2001

Computer Aided Geometric Design

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“…Recent developments include the possibility of handling intersection singularities [10,49]. Intersections of offsets (see Section 2.4) of parametric surfaces are analyzed in [85].…”

Section: Intersection Of Curves and Surfacesmentioning

confidence: 99%

Computational Methods for Geometric Processing. Applications to Industry

Iglesias

Gálvez

Puig-Pey

2001

Computational Science — ICCS 2001

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“…Macaulay's (1916) formulation expresses the resultant as a ratio of two determinants. Bajaj et al (1988) and Chionh (1990) use such procedures to compute the resultant of three parametric equations for implicitizing. In Canny (1990), Canny computed the resultant by perturbing the equations when determinants were zero, which introduces an additional variable and increases the symbolic complexity of the resulting expression.…”

Section: Introductionmentioning

confidence: 99%

“…However, their technique is valid only for star-shaped curves. Many of these and other methods of implicitization, such as multivariate resultants (Chionh and Goldman, 1992) are surveyed in Hoffmann (1993).…”

Section: Introductionmentioning

confidence: 99%

“…The problem of implicitizing parametric surfaces without any base points can be done using Groebner bases or resultants as shown in Buchberger (1989) and Manocha and Canny (1992b), although it is fairly complex in practice. Several other procedures have been devised to implicitize surfaces with base points (Hoffmann, 1989;Chionh, 1990;Canny, 1992a, 1992b).…”

Section: Introductionmentioning

confidence: 99%

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Implicitization of parametric curves by matrix annihilation

Yalcin¹,

Unel²,

Wolovich³

Proceedings. International Conference on Image Processing

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Abstract. Object recognition is a central problem in computer vision. When objects are defined by boundary curves, they can be represented either explicitly or implicitly. Implicit polynomial (IP) equations have long been known to offer certain advantages over more traditional parametric methods. However, the lack of general procedures for obtaining IP models of higher degree has prevented their general use in many practical applications. In most cases today, parametric equations are used to model curves and surfaces. One such parametric representation, elliptic Fourier Descriptors (EFD), has been widely used to represent 2D and 3D curves, as well as 3D surfaces. Although EFDs can represent nearly all curves, it is often convenient to have an implicit algebraic description F(x, y) = 0, for several reasons. Algebraic curves and surfaces have proven very useful in many model-based applications. Various algebraic and geometric invariants obtained from these implicit models have been studied rather extensively, since implicit polynomials are well-suited to computer vision tasks, especially for single computation pose estimation, shape tracking, 3D surface estimation from multiple images and efficient geometric indexing into large pictorial databases. In this paper, we present a new non-symbolic implicitization technique called the matrix annihilation method, for converting parametric Fourier representations to algebraic (implicit polynomial) representations, thereby benefiting from the features of both.

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Using multivariate resultants to find the implicit equation of a rational surface

Cited by 28 publications

References 8 publications

Tracing index of rational curve parametrizations

Tracing index of rational curve parametrizations

Computational Methods for Geometric Processing. Applications to Industry

Implicitization of parametric curves by matrix annihilation

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