1,721,087 research outputs found
Polynomial cubic splines with tension properties
No full textIn this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems
A geometric approach for Hermite subdivision
No full textAbstract We present a non-stationary, non-uniform scheme for two-point Hermite subdivision. The novelty of this approach relies on a geometric interpretation of the subdivision steps—related to generalized Bernstein bases—which permits to overcome the usually unavoidable analytical difficulties. The main advantages consist in extra smoothness conditions, which in turn produce highly regular limit curves, and in an elegant structure of the subdivision—described by three de Casteljau type matrices.
As a by-product, the scheme is inherently shape preserving
Effortless quasi-interpolation in hierarchical spaces
No full textWe present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework
Dimension of Tchebycheffian spline spaces over planar T-meshes: The conformality method
No full textWe present the conformality method for studying the dimension of Tchebycheffian spline spaces over T-meshes. This is a Tchebycheffian extension of the smoothing cofactor-conformality method elaborated in the literature for polynomial spline spaces. We apply this method to obtain a dimension formula which is equivalent to the one recently obtained by the so-called homological approach
Optimizing domain parameterization in isogeometric analysis based on Powell-Sabin splines
No full textWe address the problem of constructing a high-quality parameterization of a given planar physical domain, defined by means of a finite set of boundary curves. We look for a geometry map represented in terms of Powell-Sabin B-splines. Powell-Sabin splines are C1 quadratic splines defined on a triangulation, and thus the parameter domain can be any polygon.
The geometry map is generated by the following three-step procedure. First, the shape of the parameter domain and a corresponding triangulation are determined, in such a way that its number of corners matches the number of corners of the physical domain. Second, the boundary control points related to the Powell-Sabin B-spline representation are chosen so that they parameterize the boundary curve of the physical domain. Third, the remaining inner control points are obtained by solving a nimble optimization problem based on the Winslow functional.
The proposed domain parameterization procedure is illustrated numerically in the context of isogeometric Galerkin discretizations based on Powell-Sabin splines. It turns out that the flexibility rising from the generality of the parameter domain has a beneficial effect on the quality of the parameterization and also on the accuracy of the computed approximate solution
Shape Control in Powell-Sabin Quasi-Interpolation
No full textIn this paper we discuss the construction and we analyze the properties of quasi-interpolants based on an extension of C-1 Powell-Sabin quadratic splines over an arbitrary triangulation of a planar domain. These quasi-interpolants possess parameters which allow to control their shape avoiding oscillations and inflections extraneous to the behaviour of the data
Geometric Construction of Generalized Cubic Splines
No full textWe study the Bezier-like, geometric properties of four dimensional
spaces of the form span < u_k;P_1; v_k >, where u_k and v_k are subject
to general conditions, and describe the geometric construction of the
corresponding spline spaces
A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations
No full textWe propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods
On Constrained Nonlinear Hermite Subdivision
No full textWe determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187-200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonniere (Constr. Approx. 19:279-298, [2003]) and Pelosi and Sablonniere (C^1 GP Hermite Interpolants Generated by a Subdivision Scheme, Prepublication IRMAR 06-23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which produce smoother limits, are discussed
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