A symbolic-numerical method for computing approximate parameterizations of canal surfaces

Bizzarri, Michal; Lávička, Miroslav

doi:10.1016/j.cad.2012.04.001

Cited by 10 publications

(6 citation statements)

References 34 publications

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“…Study [6] considers the problem of transition from representing the canal surface in a parametric form relative to the line of sphere centers to an implicit form of representing the surface by a system of equations. In [7], a special parameterization of the guide curve was proposed using a rational function. In [8], the Bezier curve was chosen as the guideline for the canal surface, and in [9], a hyperbolic curve was chosen.…”

Section: Literature Reviewmentioning

confidence: 99%

Construction of canal surfaces based on a specified flat curvature line

Frolov¹

2022

Development management

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The research relevance is predefined by the widespread use of elements and structures that have the shape of canal surfaces in engineering practice and the possibilities of reproducing the surface through kinematics. The research aims to develop new means of modeling canal surfaces referred to as a grid of curvature lines by introducing elements with special properties into the structural model that simplify the solution of differential equations and reduce the amount of computation. To achieve the research methods, the synthetic geometric method, methods of linear algebra, the theory of differential equations and differential geometry, as well as methods of computer geometric modeling and visualization of three-dimensional objects were used. Studies on modeling and studying the properties of channel surfaces are analyzed. The research on the problem of the surfaces and lines of curvature is considered in more detail and the conditions under which it is possible to simplify the solution of the differential equation are identified. It was proved that the condition of contact between the canal surface and the plane along a given plane curve is sufficient for this curve to be one of the curvature lines of the family of orthogonal to the generating circles. This allowed to reduce the solution of the differential equation to two quadratures. The expressions of the corresponding integrals and an algorithm for modeling the canal surface with the possibility of referring to a grid of curvature lines were obtained. The expressions that define the desired surface include the parametric equations of a given plane line; a function that determines the radii of the spheres of the family depending on the parameter of this line. A specific example of modeling a surface based on a defined formula was also considered, and images of this surface with visualization of the coordinate grid were presented. The research’s practical values are defined by the possibility of using the developed modeling tools in the design and computer-aided design of the geometry of real products containing surfaces of a smooth transition of variable radius

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Section: Literature Reviewmentioning

confidence: 99%

Construction of canal surfaces based on a specified flat curvature line

Frolov¹

2022

Development management

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“…A technique for computing rational parametrizations of canal surfaces was presented in [15]. We emphasize that although the canal surfaces with rational spine curves and rational radii always possess exact rational parametrizations, approximate parametrization techniques are also investigated in connection with them [16]. This is caused by the computational difficulty of decomposing a rational function into a sum of two squares (SOS) over the real numbers, which is a key ingredient in the parametrization algorithm from [15].…”

Section: Canal Surfacesmentioning

confidence: 99%

Skinning and blending with rational envelope surfaces

Bizzarri

Lvika

Kosinka

2017

Computer-Aided Design

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We continue the study of rational envelope (RE) surfaces. Although these surfaces are parametrized with the help of square roots, when considering an RE patch as the medial surface transform in 4D of a spatial domain it yields a rational parametrization of the domain's boundary, i.e., the envelope of the corresponding 2-parameter family of spheres. We formulate efficient algorithms for G 1 data interpolation using RE surfaces and apply the developed methods to rational skinning and blending of sets of spheres and cones/cylinders, respectively. Our results are demonstrated on several computed examples of skins and blends with rational parametrizations.

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“…3. While we are focusing on exact rational parametrizations of rolling ball blends and canal surfaces, there are also recent developments in approximative rational parametrization techniques for canal surfaces [8][9][10], and generalizations to ringed surfaces [5,7]. There is also recent research on the use of Dupin cyclides as blending surfaces, [21,22,24].…”

Section: Rolling Ball Blends Between the Natural Uadricsmentioning

confidence: 99%

“…Theorem 2. 8. Let B H be the plane/cone rolling ball blend of radius R with spine H. Then the blend is parametrized by…”

Section: Hyperbolic and Parabolic Blendsmentioning

confidence: 99%

“…( 3.11) Combining these steps, we arrive at the following algorithm for minimal bi-degree parametrizations of variable radius rolling ball blends: Algorithm 3. 8. Consider two surfaces, and a canal surface containing a rolling ball blend between them corresponding to a rational curve f (t ) = (s(t ); r (t )) ∈ 3, 1 .…”

Section: Rational Parametrizations Of Canal Surfacesmentioning

confidence: 99%

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