On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Bastl, Bohumír; Jüttler, Bert; Lávička, Miroslav; Schulz, Tino; Šı́r, Zbyněk

doi:10.1007/s11786-014-0192-y

Cited by 11 publications

(12 citation statements)

References 20 publications

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“…The parameter S, referred to as the Eulerian coordinate, identifies a section along C which consists of all points whose reference position is on the plane perpendicular to the reference curve at S. Without loss of generality, the parametrization of the reference curve is assumed to be natural or arc-length, so that its tangent is a unit vector. The constraint surface, considered frictionless and undeformable, is the normal ringed surface [23] generated by sweeping a circle of radius Q(S) centred on the reference curve and in the normal plane to C .…”

Section: Lagrangian Formulation (A) Problem Definitionmentioning

confidence: 99%

Eulerian formulation of elastic rods

2016

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In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface. This reformulation hinges on describing the rod-deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this reparametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two-points boundary-value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment.

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Section: Lagrangian Formulation (A) Problem Definitionmentioning

confidence: 99%

Eulerian formulation of elastic rods

2016

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“…Proof. Let (s(t), R(t)) be a proper parameterization of M 2 , the algorithms presented in [3,4] allow to construct a proper parameterization x(t, s) of the resulting canal surface (not necessarily with real coefficients) such that for a fixed t 0 ∈ C the parametric curve x(t 0 , s) is a characteristic circle determined by the sphere (x − s(t 0 )) 2 = R(t 0 ). The desired mapping ξ is then the restriction of the rational map s • x −1 : X S to the section H ⊂ X .…”

Section: Plane Sections Of Rational Canal Surfacesmentioning

confidence: 99%

“…Let (s(t), R(t)) be a parameterization of M 2 , then X is generated by the characteristic circles Σ(t) = Σ ′ (t) = 0, c.f. (4). We will derive the conditions under which C 0 , the characteristic circle Σ(t 0 ) = Σ ′ (t 0 ) = 0, contains the point q corresponding to p. Assume without loss of generality s(t 0 ) ∈ N p X .…”

Section: Case Ii: G(h) =mentioning

confidence: 99%

“…(9), can be expressed as s • φ, which is a proper rational parameterization of S. The computation of the squared radius easily follows and we obtain a parameterization of M 2 . Finally, using [4] we compute from M 2 a parameterization x(t, s) of the associated canal surface and if f (x(t, s)) = 0 we can conclude that X is a canal surface with the squared medial axis transform M 2 .…”

Section: Reconstruction Of the Squared Mat From The Coherent Mappingmentioning

confidence: 99%

“…The class of rational canal surfaces with a rational spine curve and a rational radius function is a proper subset of the class of rational canal surfaces -all canal surfaces with a rational spine curve and rational squared radius function admit a rational parameterization, cf. [3,4]. The reverse problem (a rational parametric description of a curve or a surface is given, find the corresponding implicit equation) is called the implicitization problem.…”

Section: Introduction and Related Workmentioning

confidence: 99%

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Recognizing implicitly given rational canal surfaces

Vršek

Lávička

2016

Journal of Symbolic Computation

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It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In this paper we will investigate in more detail the case of canal surfaces that are often used in geometric modelling, Computer-Aided Design and technical practice (e.g. as blending surfaces smoothly joining two parts with circular ends). It is known that if the squared medial axis transform is a rational curve then so is also the corresponding surface. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rational canal surface or not. Our goal is to formulate a simple and efficient algorithm whose input is a polynomial with the coefficients from some subfield of R and the output is the answer whether the surface is a rational canal surface. In the affirmative case we also compute a rational parameterization of the squared medial axis transform which can be then used for finding a rational parameterization of the implicitly given canal surface.

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Surfaces that are covered by two pencils of circles

Lubbes

2021

Math. Z.

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On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Cited by 11 publications

References 20 publications

Eulerian formulation of elastic rods

Eulerian formulation of elastic rods

Recognizing implicitly given rational canal surfaces

Surfaces that are covered by two pencils of circles

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