CLC number: TP391.7
On-line Access: 2024-08-27
Received: 2023-10-17
Revision Accepted: 2024-05-08
Crosschecked: 2014-11-13
Cited: 2
Clicked: 6942
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",
volume="15",
number="12",
pages="1098-1105",
year="2014",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.C1400076"
}
%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
TY - JOUR
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
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.
[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.
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