On the intersection with revolution and canal surfaces

Fioravanti, Mario; González-Vega, Laureano; Necula, Ioana

doi:10.1007/978-3-540-33275-6_11

Cited by 7 publications

(3 citation statements)

References 11 publications

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“…In order to find the implicit equation of a surface one may use resultants, a method which can be very appropriate when dealing with revolution, canal or ruled surfaces ( [5,6]), but using in practice the presentation through the Bézout matrix Bez(p, q) when computing such a resultants.…”

Section: The Surface-to-surface Intersection Problem By Using Implicimentioning

confidence: 99%

Mixing symbolic and numerical approaches for the surface‐to‐surface intersection problem

Fioravanti

González-Vega

2007

Proc Appl Math and Mech

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This paper shows how the efficiency of the current methodologies applied to the surface-to-surface intersection problem can be improved by combining an algebraic/symbolic framework with efficient and robust numerical techniques. The algebraic/symbolic framework is used to translate the computation of resultants, subresultants, discriminants, etc. to one or several generalized eigenvalue problems and SVD computations. The framework requires only the values of the involved polynomials at some set of points, and it will guide the numerical computations, providing thus a certificate of the topological correctness of the output.

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Section: The Surface-to-surface Intersection Problem By Using Implicimentioning

confidence: 99%

Mixing symbolic and numerical approaches for the surface‐to‐surface intersection problem

Fioravanti

González-Vega

2007

Proc Appl Math and Mech

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“…instead of those in equation (18). As in section 3.1.1, we must avoid to use nodes that anihilate R(t).…”

Section: Computing the Self-intersections Points Of γmentioning

confidence: 99%

“…If we want to intersect S with a space curve C given by the parameterization (P (t)/D(t), Q(t)/D(t), R(t)/D(t)) (with P (t), Q(t), R(t) and D(t) polynomials also known by values) then it is enough to consider the equations (see [18])…”

Section: Rational Casementioning

confidence: 99%

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

Diaz–Toca

Fioravanti

González-Vega

et al. 2013

Computer Aided Geometric Design

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The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice.We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation by the evaluation of the considered parameterizations in a set of points and its use, in order to deal with the considered geometric problems, is translated into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so called "Polynomial Algebra by Values" approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.

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Geometric Modeling and Algebraic Geometry

Jttler¹,

Piene²

2008

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On the intersection with revolution and canal surfaces

Cited by 7 publications

References 11 publications

Mixing symbolic and numerical approaches for the surface‐to‐surface intersection problem

Mixing symbolic and numerical approaches for the surface‐to‐surface intersection problem

Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

Geometric Modeling and Algebraic Geometry

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