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CLC number: TP391.7

On-line Access: 2014-12-05

Received: 2014-03-06

Revision Accepted: 2014-08-25

Crosschecked: 2014-11-13

Cited: 2

Clicked: 4218

Citations:  Bibtex RefMan EndNote GB/T7714

 ORCID:

Xiao-juan DUAN

http://orcid.org/0000-0002-7833-6626

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Journal of Zhejiang University SCIENCE C 2014 Vol.15 No.12 P.1098-1105

http://doi.org/10.1631/jzus.C1400076


Degree elevation of unified and extended spline curves


Author(s):  Xiao-juan Duan, Guo-zhao Wang

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   heng-juan0721@163.com, wanggz@zju.edu.cn

Key Words:  Degree elevation, Unified and extended splines (UE-splines), Bi-order UE-splines, Corner cutting, Geometric explanation


Xiao-juan Duan, Guo-zhao Wang. Degree elevation of unified and extended spline curves[J]. Journal of Zhejiang University Science C, 2014, 15(12): 1098-1105.

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Abstract: 
unified and extended splines (UE-splines), which unify and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis functions which are defined by the integral definition of splines. Then some important properties of bi-order UE-splines are given, especially for the transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.

UE样条曲线的升阶

针对UE样条悬而未决的升阶问题,给出UE样条的升阶方法,并揭示此方法的几何意义。 引入一种新的样条基函数-双阶UE样条基函数。在原始节点向量中逐个插入互异节点,将UE样条函数按区间逐段升阶,最终使UE样条在整个定义域内达到升阶效果,并给出这种升阶方法的几何意义。 由于曲线在节点处的连续性保持不变,低阶的UE样条曲线可由高阶UE样条曲线表示。首先,引入一种新的样条基函数-双阶UE样条基函数。这种样条基在整个节点区间有两种阶数。其中,前一段节点区间的次数比后一段节点区间的次数高1次(图1)。然后,通过往节点向量中插入节点,双阶UE样条基的某特定区间次数升高1次,从而得到双阶UE样条在节点插入前后的基函数关系(图2)继而得到节点插入前后双阶UE样条函数控制顶点之间的关系。通过逐个插入互异节点,可使UE样条逐段升阶。最后,根据节点插入前后的新旧控制顶点关系,证明UE样条的升阶可以理解为其控制多边形的割角过程(图3、4)。 通过在节点向量中逐个插入互异节点,解决了UE样条的升阶问题,并证明了UE样条的升阶可以解释为其控制多边形的割角过程。
升阶;UE样条;双阶UE样条;割角;几何解释

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