Changeable degree spline basis functions

Shen, Wanqiang; Wang, Guozhao

doi:10.1016/j.cam.2010.03.015

Cited by 27 publications

(24 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Zhu

Han

Liu

2015

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Wang et al presented piecewise algebraic and trigonometric uniform B-spline curves with a shape parameter of degree k (k ≤ 2) [10,11]. Shen and Wang extended B-spline basis functions to changeable degree spline (CD-spline) basis functions, each of which may consist of polynomials of different degrees on its support interval [12]. Yin and Tan introduced trigonometric polynomial uniform B-splines curves with multiple shape parameters [13].…”

Section: Introductionmentioning

confidence: 99%

The construction of λμ-B-spline curves and its application to rotational surfaces

Qin

et al. 2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…Multi degree splines (MD-splines, for short) are piecewise functions comprised of polynomial segments of different degrees. They were proposed in the seminal paper [12] and they have been a subject of study in several recent works [10,[13][14][15][16]18]. A more general setting was considered in [6], where the section spaces belong to the class of Extended Chebyshev spaces and are not constrained to be of the same dimension.…”

Section: Introductionmentioning

confidence: 99%

On multi-degree splines

Beccari

Casciola

Morigi

2017

Computer Aided Geometric Design

View full text Add to dashboard Cite

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multidegree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to Bézier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines.

show abstract

Changeable degree spline basis functions

Cited by 27 publications

References 7 publications

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

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

The construction of λμ-B-spline curves and its application to rotational surfaces

On multi-degree splines

Contact Info

Product

Resources

About