Explicit representations of changeable degree spline basis functions

Shen, Wanqiang; Wang, Guozhao; Yin, Ping

doi:10.1016/j.cam.2012.08.017

Cited by 19 publications

(17 citation statements)

References 17 publications

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“…Multi-degree splines were introduced in [34] for degrees (1,2,3) and (1, n). In a more general framework, basis functions for multi-degree splines were constructed in [37,36,38] by means of recursive integral relations involving global quantities that [19] observed as being difficult to compute. Instead, [19] presented a recursive geometric algorithm for computing multi-degree spline curves.…”

Section: Related Literaturementioning

confidence: 99%

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Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Toshniwal

Speleers

Hiemstra

et al. 2017

Computer Methods in Applied Mechanics and Engineering

101

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We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions yield C k smooth polar splines for any k ≥ 0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k ∈ {0, 1, 2} are presented. Optimal approximation behavior is observed numerically, and examples of free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.

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Section: Related Literaturementioning

confidence: 99%

“…• We approach multi-degree splines as piecewise-NURBS, a more general viewpoint than the piecewise-polynomial approach in [34,37,36,38,19].…”

Section: Our Contributionsmentioning

confidence: 99%

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Toshniwal

Speleers

Hiemstra

et al. 2017

Computer Methods in Applied Mechanics and Engineering

101

View full text Add to dashboard Cite

show abstract

“…The variable degree polynomial splines have been widely used for constructing shape preserving interpolation and approximation splines; see [25][26][27]. In [28], based on some truncated polynomial functions, the explicit representations of changeable degree spline basis functions were given. In [29], a kind of five trigonometric blending functions with two exponential shape parameters and was proposed in the space spanned by span{1, sin [30], a generalization of these five trigonometric blending functions was presented.…”

Section: Introductionmentioning

confidence: 99%

A Class of Trigonometric Bernstein‐Type Basis Functions with Four Shape Parameters

Zhu

Liu²

2019

Mathematical Problems in Engineering

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In this work, a family of four new trigonometric Bernstein-type basis functions with four shape parameters is constructed, which form a normalized basis with optimal total positivity. Based on the new basis functions, a kind of trigonometric Bézier-type curves with four shape parameters, analogous to the cubic Bézier curves, is constructed. With appropriate choices of control points and shape parameters, the resulting trigonometric Bézier-type curves can represent exactly any arc of an ellipse or parabola. The four shape parameters have tension control roles on adjusting the shape of resulting curves. Moreover, a new corner cutting algorithm is also proposed for calculating the trigonometric Bézier-type curves stably and efficiently.

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“…For modifications of the curve form, there are some works on the practical methods for generating curves by using tension shape ∩ FC 2k+3 (k ∈ Z + ) continuous by specifying some values of the shape parameters. For the problems of shape preserving interpolation and approximation, the variable degree polynomial spline basis shows great potential applications [23][24][25][26][27][28][29]. In 2000, Costantini [23] constructed a class of variable degree polynomial spline bases in the space span{1, t, (1 − t) p , t q }, where p, q are two arbitrary integers greater than or equal to 3 and serve as tension shape parameters.…”

Section: Introductionmentioning

confidence: 99%

Curve construction based on four αβ-Bernstein-like basis functions

Zhu

Han

Liu

2015

Journal of Computational and Applied Mathematics

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MSC: 65D07 65D17Keywords: Said-Ball basis Bernstein basis Quasi Extended Chebyshev space B-spline curve Shape parameter Totally positive property a b s t r a c t Four new αβ-Bernstein-like basis functions with two exponential shape parameters, are constructed in this paper, which include the cubic Said-Ball basis functions and the cubic Bernstein basis functions. Within the general framework of Quasi Extended Chebyshev space, we prove that the proposed αβ-Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding αβ-Bézier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar αβ-Bézier-like curve having single or double inflection points, a loop or a cusp, or be locally or globally convex in terms of the relative position of its control polygons' side vectors. Based on the new proposed αβ-Bernstein-like basis, a class of αβ-B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated αβ-B-splinelike curves have C 2 continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C 2 ∩ FC k+3 (k ∈ Z + ) continuous for particular choice of shape parameters. The exponential shape parameters serve as tension shape parameters and play a predictable adjusting role on generating curves.

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Explicit representations of changeable degree spline basis functions

Cited by 19 publications

References 17 publications

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

A Class of Trigonometric Bernstein‐Type Basis Functions with Four Shape Parameters

Curve construction based on four αβ-Bernstein-like basis functions

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