Interference analysis of conics and quadrics

Wang, WP; Krasauskas, Rimvydas

doi:10.1090/conm/334/05973

Cited by 11 publications

(10 citation statements)

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“…In related topics, a simple condition in terms of the number of negative real roots of the characteristic equation f (λ) is given by Wang et al (2001) for the separation of two ellipsoids. Similar algebraic conditions are obtained in Wang and Krasauskas (2004) for characterizing non-degenerate configurations formed by two ellipses in 2D and ellipsoids in 3D.…”

Section: Related Worksupporting

confidence: 57%

“…For arrangement computation, it is an interesting problem to classify all possible partitions of R 3 that can be formed by two ellipsoids. It is also possible to apply the results here to derive efficient algebraic conditions for collision detection between various types of quadric surfaces, such as cones and cylinders, following the framework in Wang and Krasauskas (2004).…”

Section: Discussionmentioning

confidence: 99%

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Using signature sequences to classify intersection curves of two quadrics

Wang²,

Mourrain

et al. 2009

Computer Aided Geometric Design

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We present a method that uses signature sequences to classify the intersection curve of two quadrics (QSIC) or, equivalently, quadric pencils in PR 3 (3D real projective space), in terms of the shape, topological properties, and algebraic properties of the QSIC. Specifically, for a QSIC we consider its singularity, reducibility, the number of its components, and the degree of each irreducible component, etc. There are in total 35 different types of non-degenerate quadric pencils. For each of the 35 types of QSICs given by these nondegenerate pencils, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of the quadric pencil. We show how to compute a signature sequence with rational arithmetic and use it to determine the type of the intersection curve of any two quadrics which form a non-degenerate pencil. As an example of application, we discuss how to apply our results to collision detection of cones in 3D affine space.

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Section: Related Worksupporting

confidence: 57%

Section: Discussionmentioning

confidence: 99%

Using signature sequences to classify intersection curves of two quadrics

Wang²,

Mourrain

et al. 2009

Computer Aided Geometric Design

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“…For the nongeneric classes, this provides a precise answer to a question formulated in [20]. We plan also to develop this point in a forthcoming paper with B. Mourrain.…”

Section: Final Remarksmentioning

confidence: 83%

Equations, inequations and inequalities characterizing the configurations of two real projective conics

Briand

2007

AAECC

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Abstract. Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.MSC2000: 13A50 (invariant theory), 13J30 (real algebra).

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“…As a consequence, λ 2 and λ 3 are both positive or negative. If λ 2 and λ 3 are negative then, according to (14), l 2 < 0, and according to (15), l 1 < 0. A sharper result than the converse is also true: if λ 2 and λ 3 are positive then, l 2 < 0 or l 1 < 0.…”

Section: The Seven Possible Root Patterns For P (λ)mentioning

confidence: 99%

“…A survey on the obtained results using this approach can be found in [14]. The origin of this approach can be traced back to the development of the aforementioned classic tests [8].…”

Section: Introductionmentioning

confidence: 99%

New algebraic conditions for the identification of the relative position of two coplanar ellipses

Alberich-Carramiñana

Elizalde

Thomas

2017

Computer Aided Geometric Design

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The identification of the relative position of two real coplanar ellipses can be reduced to the identification of the nature of the singular conics in the pencil they define and, in general, their location with respect to these singular conics in the pencil. This latter problem reduces to find the relative location of the roots of univariate polynomials. Since it is usually desired that all generated expression are algebraic to simplify further analysis, including the case in which the ellipses undergone temporal variations, all recent methods available in the literature rely mathematical tools such as Sturm-Habicht sequences or subresultant sequences. This paper presents an alternative based on more elementary tools which results in a binary decision tree to classify the relative location of two ellipses in 12 different classes. The decision at each node is taken based on the sign of a set of algebraic/rational expressions on the ellipses coefficients, the most complex of them being third and second order polynomial discriminants.

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Interference analysis of conics and quadrics

Cited by 11 publications

References 0 publications

Using signature sequences to classify intersection curves of two quadrics

Using signature sequences to classify intersection curves of two quadrics

Equations, inequations and inequalities characterizing the configurations of two real projective conics

New algebraic conditions for the identification of the relative position of two coplanar ellipses

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