Canal surfaces with rational contour curves and blends bypassing the obstacles

Bizzarri, Michal; Lávička, Miroslav; Vršek, Jan

doi:10.1016/j.cad.2015.03.002

Cited by 10 publications

(6 citation statements)

References 22 publications

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“…The curve m(t) is called the spine curve and r(t) the radius function of S. Next, each sphere F (t) contributes to the canal surface by the so called characteristic circle C(t), which is the intersection of the sphere F (t) and the plane dF (t)/dt. In Bizzarri and Lávička (2013a); Bizzarri et al (2015Bizzarri et al ( , 2016 special curves on canal surfaces providing the whole family of rational canal surfaces sharing the same silhouette were investigated. These curves, along which the surface normals are parallel with a prescribed plane, are suitable in special cases for computing rational parameterizations of canal surfaces.…”

Section: Contour Curves On Canal Surfacesmentioning

confidence: 99%

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Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Bizzarri

Lávička

Vršek

2017

Computer Aided Geometric Design

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In this paper we study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained, in general. We focus especially on such cases in which one can identify a rational mapping which is a double cover of a rational curve. Hence, we deal with rational, elliptic or hyperelliptic curves that are birational to plane curves in the Weierstrass form and thus they are square-root parameterizable. We design a simple algorithm for computing an approximate (piecewise) rational parametrization using topological graphs of the Weierstrass curves. Predictable shapes reflecting a number of real roots of a univariate polynomial and a possibility to approximate easily the branches separately play a crucial role in the approximation algorithm. Our goal is not to give a comprehensive list of all such operations but to present at least selected interesting cases originated in geometric modelling and to show a unifying feature of the formulated method. We demonstrate our algorithm on a number of examples.

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Section: Contour Curves On Canal Surfacesmentioning

confidence: 99%

“…As derived in Bizzarri et al (2015), for the canal surface with the rational spine curve m(t) and the rational radius function r(t), the contour curve can be parameterized as…”

Section: Contour Curves On Canal Surfacesmentioning

confidence: 99%

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Bizzarri

Lávička

Vršek

2017

Computer Aided Geometric Design

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“…Some related studies on contours, silhouettes and their applications can be found e.g. in [3,4,11,16] Remark 1.1. If is a developable surface, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Hence we 1 The dashed arrow emphasizes the fact that the mapping need not to be defined for every point of the surface, but only on its dense subset modify the definition slightly. The isophote s a; is the closure of the set n p P sm X @@pA ¡ aA 2 2 @a ¡ aA@@pA ¡ @pAA a H o ; (4) where a a @a 1 ; a 2 ; a 3 A P P 2 R , is a given direction, a ros is the angle and x ¡ y a i x i y i . Hence s a; is a set of points on the surfaces where the normal direction forms angles ¦ with the direction a.…”

Section: Introductionmentioning

confidence: 99%

Contour curves and isophotes on rational ruled surfaces

Vršek

2018

Computer Aided Geometric Design

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The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the field of applied geometry. An isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector. Choosing an angle equal to =2 we obtain a special instance of a isophotethe so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes. Hence we will provide a formula for their genus. Moreover we will show that the only surfaces with a rational generic contour are just rational ruled surfaces and a one particular class of cubic surfaces. In addition we will deal with the reconstruction of ruled surfaces from their contours and silhouettes.

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“…To provide good tamper localization accuracy can be designed a topology verification for drawings the piping isometric that is in robust versus local similarity displacement and invariant to the elongation process on pipes (Su et al, 2015). The study by Bizzarri et al (2015) presented an algebraic condition to give the guarantees that a canal surface possesses rational generalized contour, then, Moritani et al (2018) proposed a new method to built a piping system through cylinder-based registration and model fitting of laser-scanned point clouds. On the other hand, to multiple pipes having arbitrary posses and free form cross-section, Zhou & Qian (2013) used the constructive G 1 pipes connection.…”

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confidence: 99%

Pipes Construction Based on the Pipes' Centre Curve Shapes

Kusno¹

2019

JMR

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Parts of pipeline need a long piece, a short segment, and various inflate-deflate models. They require as well the thickness and curvature of the pipes. The objective of this paper is to obtain some formulas for modeling the long pipe, the short tube, and various inflate-deflate pipe patches. Relating to the purposes, we use their cross-section, longitudinal section, and center curves of the pipe parts. The methods are, the using of the polar coordinates and of the real functions, to define the cross and longitudinal section of the pipe patches, respectively. Then, we calculate three orthonormal vectors that are determined by the tangent vectors of the pipe center curves and two unit vectors that are perpendicular to the tangent vectors. After that, we evaluate the formulas to model the long pipes and the short pipes, both inflate-deflate and thickness shapes. The results show that, using its center curves of the pipe, it is handy to design the long and short pipes, multiple thicknesses, various volume fluctuations of the pipes, and useful to model the inflate-deflate pipe parts.There were presented some methods to define the pipeline models. Lü & Pottmann (1996) introduced an algorithm for computing the pipe surfaces of rational spine curve (center curve of pipe) that can admit a rational parameterization. Maekawa, et al. (1998) investigated the pipe surface intersection both local and global self-intersection. Then, using the unification of two cylinders of revolution defined by different radii, Malaček & Šibrava (2006) constructed the circular surfaces. Research by Moon (2009) studied the curvature effect of pipes spin curve. He used the radial distance function form to compensate for the distortion of the pipes. Meanwhile, we can present the geometric properties of canal surfaces in which their spine curves are in the form of a unit circle and a straight line (Öztürk et al., 2010). Also, we can evaluate the substantial form of transitional tube parts whose cross sections have a polygonal shape and made of materials that cannot be creased or prolonged (Obradovic et al., 2012). This geometrical theory is based on the classical line geometry, and it can be straightforwardly implemented to the conception of a computer algorithm that generates the middle surfaces between two polygons. Fok et al. (2012) introduced the method for reconstruction of disjoint surfaces with a single equation from sectional data. Using a parametric design system of modular Pipe-Z and its parametrization can be found a trefoil, a gure-eight knot, and a pentafoil (Zawidski & Nishinari, 2013). To provide good tamper localization accuracy can be designed a topology verification for drawings the piping isometric that is in robust versus local similarity displacement and invariant to the elongation process on pipes (Su et al., 2015). The study by Bizzarri et al. (2015) presented an algebraic condition to give the guarantees that a canal surface possesses rational generalized contour, then, Moritani et al. (2018) proposed a new method to built ...

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Canal surfaces with rational contour curves and blends bypassing the obstacles

Cited by 10 publications

References 22 publications

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Contour curves and isophotes on rational ruled surfaces

Pipes Construction Based on the Pipes' Centre Curve Shapes

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Canal surfaces with rational contour curves and blends bypassing the obstacles

Cited by 10 publications

References 22 publications

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Piecewise rational approximation of square-root parameterizable curves using the Weierstrass form

Contour curves and isophotes on rational ruled surfaces

Pipes Construction Based on the Pipes&#39; Centre Curve Shapes

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Pipes Construction Based on the Pipes' Centre Curve Shapes