Using a bihomogeneous resultant to find the singularities of rational space curves

Shi, Xiaoran; Jia, Xiaohong; Goldman, Ron

doi:10.1016/j.jsc.2012.09.005

Cited by 18 publications

(7 citation statements)

References 13 publications

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“…We adopt the following technique to solve the problem of singularity. We use the methods [46][47][48] to find all singular points on the parametric curve and the corresponding parametric value of each singular point as many as possible. Then, the hybrid second order method comes into work.…”

Section: The Improved Algorithmmentioning

confidence: 99%

“…Because of a singular point on the parametric curve, we have also added some pre-processing steps before our method. (1) Find the singular point (0,0,3) and the corresponding parametric value 0 by using the methods [21,[46][47][48]. (2) Using our method, the orthogonal projection points of test points (2,4,2) and (2,2,2) and their corresponding parameter values 0 and 0 are calculated, respectively.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

Liang

Hou

et al. 2018

Mathematics

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Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order convergence and its non-dependence on the initial value. For some special cases of divergence for the H-H-H method, we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybrid second order method for orthogonal projection onto parametric curve in an n-dimensional Euclidean space (hereafter, our method). Our method essentially utilizes hybrid iteration, so it converges faster than current methods with a second order convergence and remains independent from the initial value. We provide some numerical examples to confirm robustness and high efficiency of the method.

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Section: The Improved Algorithmmentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

Liang

Hou

et al. 2018

Mathematics

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“…Thus this matrix is a square matrix. The resultant of the three bivariate polynomials ρ, η, ω is precisely the determinant of this coefficient matrix (Shi et al, 2013). When ν 1 = ν 2 = ν 3 = ν, this resultant reduces to the Dixon resultant for three bivariate polynomials of bidegree (m, ν) (Dixon, 1908).…”

Section: Implicit Equations Of Translational Surfacesmentioning

confidence: 99%

“…In general, the bidegrees of the µ-basisP ′ ,Q ′ ,R ′ vary from case to case, so the special form of the resultant in Shi et al (2013) presented before Example 3.6 cannot be used to compute the resultant ofP ′ • x,Q ′ • x,R ′ • x. Therefore, the construction of a general resultant matrix as in Cox et al (1998), Dickenstein and Emiris (2003), Gelfand et al (1994) must be used here.…”

Section: Implicit Equations Of Translational Surfacesmentioning

confidence: 99%

Syzygies for translational surfaces

Wang

Goldman

2018

Journal of Symbolic Computation

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Implicit equationA translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three special syzygies for a translational surface from a µ-basis of one of the generating space curves, and we show how to compute the implicit equation of a translational surface from these three special syzygies. Examples are provided to illustrate our theorems and flesh out our algorithms.

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“…The are many papers and books on the theory and computation of resultants for a variety of different types of polynomial systems, e.g., [13] (see Chapters 3 and 7), [18] (see Chapters 3, 8, and 13), and [16,17,25].…”

Section: Q(s T) · X R(s T) · X)) + Deg R(st)·x (Res(p(s T) · X mentioning

confidence: 99%

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

Shen¹,

Goldman²

2017

SIAM J. Appl. Algebra Geometry

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Abstract. We investigate conditions under which the resultant of a µ-basis for a rational tensor product surface is the implicit equation of the surface without any extraneous factors. In this case, we also derive a formula for the implicit degree of the rational surface based only on the bidegree of the rational parametrization and the bidegrees of the elements of the µ-basis without any knowledge of the number or multiplicities of the base points, assuming only that all the base points are local complete intersections. We conclude that in this case the implicit degree of a rational surface of bidegree (m, n) is at most mn, so the rational surface must have at least mn base points counting multiplicity. When the resultant of a µ-basis generates extraneous factors, we show how to predict and compute these extraneous factors from either the existence of bad base points or anomalies occurring in the parametrization at infinity. Examples are provided to flesh out the theory.

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Using a bihomogeneous resultant to find the singularities of rational space curves

Cited by 18 publications

References 13 publications

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space

Syzygies for translational surfaces

Strong $\mu$-Bases for Rational Tensor Product Surfaces and Extraneous Factors Associated to Bad Base Points and Anomalies at Infinity

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