Full Text:   <2038>

Summary:  <1735>

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: 6234

Citations:  Bibtex RefMan EndNote GB/T7714


Xiao-juan DUAN


-   Go to

Article info.
Open peer comments

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


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.

@article{title="Degree elevation of unified and extended spline curves",
author="Xiao-juan Duan, Guo-zhao Wang",
journal="Journal of Zhejiang University Science C",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Degree elevation of unified and extended spline curves
%A Xiao-juan Duan
%A Guo-zhao Wang
%J Journal of Zhejiang University SCIENCE C
%V 15
%N 12
%P 1098-1105
%@ 1869-1951
%D 2014
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.C1400076

T1 - Degree elevation of unified and extended spline curves
A1 - Xiao-juan Duan
A1 - Guo-zhao Wang
J0 - Journal of Zhejiang University Science C
VL - 15
IS - 12
SP - 1098
EP - 1105
%@ 1869-1951
Y1 - 2014
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.C1400076

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.


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。

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Barry, P.J., Goldman, R.N., 1988. A recursive proof of a B-spline identity for degree elevation. Comput. Aided Geom. Des., 5(2):173-175.

[2]Cao, J., Wang, G.Z., 2011. Non-uniform B-spline curves with multiple shape parameters. J. Zhejiang Univ.-Sci. C (Comput. & Electron.), 12(10):800-808.

[3]Cohen, E., Lyche, T., Riesenfeld, R., 1980. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Comput. Graph. Image Process., 14(2):87-111.

[4]Cohen, E., Lyche, T., Schumaker, L.L., 1985. Algorithms for degree-raising of splines. ACM Trans. Graph., 4(3):171-181.

[5]Forrest, A.R., 1972. Interactive interpolation and approximation by B’ezier polynomials. Comput. J., 15(1):71-79.

[6]Li, Y.J., Wang, G.Z., 2005. Two kinds of B-basis of the algebraic hyperbolic space. J. Zhejiang Univ.-Sci., 6A(7):750-759.

[7]Lu, Y.G., Wang, G.Z., Yang, X.N., 2002. Uniform hyperbolic polynomial B-spline curves. Comput. Aided Geom. Des., 19(6):379-393.

[8]Min, C.Y., Wang, G.Z., 2004. C-B-spline basis is B-basis. J. Zhejiang Univ. (Sci. Ed.), 31(2):148-150 (in Chinese).

[9]Prautzsch, H., 1984. Degree elevation of B-spline curves. Comput. Aided Geom. Des., 1(2):193-198.

[10]Shen, W.Q., Wang, G.Z., 2010. Changeable degree spline basis functions. J. Comput. Appl. Math., 234(8):2516-2529.

[11]Shen, W.Q., Wang, G.Z., Yin, P., 2013. Explicit representations of changeable degree spline basis functions. J. Comput. Appl. Math., 238:39-50.

[12]Wang, G.Z., Deng, C.Y., 2007. On the degree elevation of B-spline curves and corner cutting. Comput. Aided Geom. Des., 24(2):90-98.

[13]Wang, G.Z., Fang, M.E., 2008. Unified and extended form of three types of splines. J. Comput. Appl. Math., 216(2):498-508.

[14]Wang, G.Z., Chen, Q.Y., Zhou, M.H., 2004. NUAT B-spline curves. Comput. Aided Geom. Des., 21(2):193-205.

[15]Xu, G., Mourrain, B., Duvigneau, R., et al., 2011. Parameterization of computational domain in isogeometric analysis: methods and comparison. Comput. Method Appl. Mech. Eng., 200(23-24):2021-2031.

[16]Xu, G., Mourrain, B., Duvigneau, R., et al., 2013. Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications. Comput.-Aided Des., 45(2):395-404.

[17]Zhang, B., Wang, G.Z., 2008. Degree elevation of algebraic hyperbolic B-spline curves and corner cutting based on bi-order spline. Chin. J. Comput., 31(6):1056-1062 (in Chinese).

[18]Zhang, J.W., 1996. C-curves: an extension of cubic curves. Comput. Aided Geom. Des., 13(3):199-217.

[19]Zhang, J.W., 1997. Two different forms of C-B-splines. Comput. Aided Geom. Des., 14(1):31-41.

[20]Zhu, P., Wang, G.Z., Yu, J.J., 2010. Degree elevation operator and geometric construction of C-B-spline curves. Sci. China Inform. Sci., 53(9):1753-1764.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2024 Journal of Zhejiang University-SCIENCE