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Published in Phillip A. Laplante, Dictionary of Computer Science, Engineering, and Technology, 2017
parametric surface a surface defined explicitly by the range of values of a parametric function. For a parametric function f (a) that depends upon the parameter vector a, the surface S can be defined formally as: S = {p : p = f (a), ∀a}.
An evolution model of parametric surface deformation using finite elements based on B-splines
Published in João Manuel, R. S. Tavares, R. M. Natal Jorge, Computational Modelling of Objects Represented in Images, 2018
Manuel González-Hidalgo, Arnau Mir, Gabriel Nicolau
A parametric surface is defined as S:Ω⊂IR2→IR3,(u,v)↦S(u,v)=(x(u,v),y(u,v),z(u,v)) with the necessary degree of differentiability (do Carmo 1976), where Ω is a bounded bidimensional subset. This surface will be B-spline if we can put it as a linear combination of bidimensional B-splines. That is, () S(x)=∑k∈ZZ2PkBk,hn(x)
Intelligent Robotic Vision Systems
Published in Spyros G. Tzafestas, Intelligent Robotic Systems, 2020
L. Van Gool, P. Wambacq, A. Oosterlinck
Surface boundary representations use lists of surfaces and the domain in which they are valid (their boundaries or intersections). The simplest boundary representation is the polyhedral approximation, for example, via triangulation. Surface patches can also be described by higher order descriptions at the cost of an increasing number of coefficients and increasing fitting complexity. There are many different types of parametric surface representations, such as Coon’s patches and tensor product composite surfaces, Β spline surfaces being probably the best known. In the computer vision literature, one seldom uses these descriptions, but rather often a limited set of surface shapes is supposed to occur. Typical examples include planes, cylinders, spheres, and cones. The so-called Gaussian sphere offers an instrument for their detection, especially when objects are convex (Horn, 1986; Shicai, 1987). Here the directions of the normals to the different patches are indicated by their intersection with the unit sphere. As such, cylinders are reduced to two points for the two planar sides and a circle for the curved envelope. Polyhedra become a set of points and cones a point and a circle of smaller radius than that of a cylinder. The Hough transform can be used to detect such features on the Gaussian sphere. Recent research has also focused on surface curves or regions that enjoy some global property without imposing precise parametric descriptions: lines of curvature, asymptotes, bounding contours, surface intersections, umbilic regions, and regions characterized by the constant signs of Gaussian and/or mean curvature. A step beyond this can then combine surface regions to spheres, surfaces of revolution, and, in general, volumes thought of as structured around spines and points. These can then, in turn, be investigated for peculiar spatial arrangements. This brings us to volumetric descriptions of objects.
Models for ground vehicle control on nonplanar surfaces
Published in Vehicle System Dynamics, 2023
Thomas Fork, H. Eric Tseng, Francesco Borrelli
First we differentiate constraints (5b) and (5c) with respect to time: and expand the time derivatives of , and : The matrix on the left-hand side of (8) is known as the second fundamental form of a parametric surface [21, Ch. 3] which we denote by . This term captures the curvature of the surface. We introduce the symbol for the matrix on the right, due to its interpretation as a Jacobian of the parametric surface as viewed in the body frame. Observe that the above relates motion on the surface (, ) to necessary angular velocity of the body (, ) to remain tangent to the surface.
Hierarchical alignment of 3D print with tool path based on microstructure
Published in Virtual and Physical Prototyping, 2022
Yifan Yang, Yutaka Ohtake, Tatsuya Yatagawa, Hiromasa Suzuki
In contrast, the predictable parts are the sidewalls generated by the smooth paths and do not contact any support structure. We call a sidewall that satisfies these two conditions a constant sidewall, which can be described as a swept surface (Salomon 2005, chap. 9), commonly used for modelling. A swept surface is a surface generated by moving a section curve along a path (Chang 2016, 94–96). In the case of a constant sidewall, we call the sidewall curve. And represents part of the path, which can be approximated as a straight line. is a planar curve and perpendicular to . Let and be the natural parameter of and for their arc length. A constant sidewall can be defined as a parametric surface:
CVT-based 3D image segmentation and quality improvement of tetrahedral/hexahedral meshes using anisotropic Giaquinta-Hildebrandt operator
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2018
Kangkang Hu, Yongjie Jessica Zhang, Guoliang Xu
Laplacian smoothing is the most commonly used mesh smoothing method which iteratively relocates a vertex to the geometric center of its neighbouring vertices. However, it also produces a shrinking effect and an oversmoothing result. Here, we develop a new GHO-based geometric flow to smooth the surface, which can preserve the concave/convex features and avoid volume shrinkage. Let be a smooth parametric surface in . Note that (u, v) can also be written as for convenience. The coefficients of the first fundamental form of S are defined as , where and . The coefficients of the second fundamental form of S are defined as , where and . Let , , and . The mean curvature and the Gaussian curvature . Let , the GHO acting on f is defined as