Abstract
Unlike a Bézier curve, a spline curve is hard to be obtained through geometric corner cutting on control polygons because the degree elevation operator is difficult to be obtained and geometric convergence is hard to be achieved. In order to obtain geometric construction algorithm on C-B-splines, firstly we construct the degree elevation operator by using bi-order splines in this paper. Secondly we can obtain a control polygon sequence by degree elevation based on the degree elevation operator derived from a C-B-spline curve. Finally, we prove that this polygon sequence will converge to initial C-B-spline curve. This geometric construction algorithm possesses strong geometric intuition. It is also simple, stable and suitable for hardware to perform. This algorithm is important for CAD modeling systems, since many common engineering curves such as ellipse, helix, etc. can be represented explicitly by C-B-splines.
Similar content being viewed by others
References
Zhang J W. Two different forms of C-B-splines. Comput Aid Geom Des, 1997, 14:31–41
Zhang J W. C-curves: an extension of cubic curves. Comput Aid Geom Des, 1996, 13:199–217
Wang G Z, Chen Q Y, Zhou M H. NUAT B-spline curves. Comput Aid Geom Des, 2004, 21:193–205
Min C Y, Wang G Z. C-B-spline basis is B-basis (in Chinese). J Zhejiang Univ (Sci Ed), 2004, 31:148–150
Mainar E, Peña J M, Sanchez-Reyes J. Shape preserving alternatives to the rational Bézier model. Comput Aid Geom Des, 2001, 18:37–60
Zeng T J, Wang W M, Zhang J W. Study on surface of revolution modeling with C-B-splines (in Chinese). J Eng Graph, 2004, 25:104–108
Yang Q M, Wang G Z. Inflection points and singularities on C-curves. Comput Aid Geom Des, 2004, 21:207–213
Mainar E, Peña J M. A basis of C-Bézier splines with optimal properties. Comput Aid Geom Des, 2002, 19:291–295
Hoffmann M, Juhasz I. On interpolation by spline curves with shape parameters. In: Proceedings of Geometric Modeling and Processing. Hangzhou: Zhejiang University, 2008. 205–214
Hoffmann M, Li Y J, Wang G Z. Paths of C-Bézier and C-B-spline curves. Comput Aid Geom Des, 2006, 23:463–475
Lin X, Luo G M, Zhang J W. Catmull-Clark subdivision surfaces modeling with C-B-splines (in Chinese). J Imag Graph, 2002, 7(A):876–881
Zeng T J, Luo G M, Zhang J W. Shape modification of Catmull-Clark subdivision surface (in Chinese). J Comput Aid Des Comput Graph, 2004, 16:707–711
Zeng T J. Modeling with C-B-spline and the system development for its subdivision surfaces (in Chinese). Dissertation for the Master Degree, Hangzhou: Zhejiang University, 2004
Wang W T, Wang G Z. Trigonometric polynomial B-splines with shape parameter. Prog Nat Sci, 2004, 14:1023–1026
Jiang C J, Tang Y H. An explicit form of for higher-order C-B-spline curves and surfaces (in Chinese). Num Math J Chinese Univ, 2008, 30:26–39
Wang G J, Wang G Z, Zheng J M. Computer Aided Geometric Design (in Chinese). Beijing: High Education Press; Heidelberg: Springer, 2001. 8–9
Huang Q X, Hu S M, Ralph R M. Fast degree elevation and knot insertion for B-spline curves. Comput Aid Geom Des, 2005, 22:183–197
Liu W. A simple, efficient degree raising algorithm for B-spline curves. Comput Aid Geom Des, 1997, 14:693–698
Sun J N, Wang R H. Solutions to problems of algorithms for degree elevation of B-spline curves (in Chinese). J Dalian Univ Tech, 2003, 43:397–398
De B C. Cutting corners always works. Comput Aid Geom Des, 1987, 4:125–131
Paluszny M, Prautzsch H, Schäfer M. A geometric look at corner cutting. Comput Aid Geom Des, 1997, 14:421–447
Wang G Z, Deng C Y. On the degree elevation of B-spline curves and corner cutting. Comput Aid Geom Des, 2007, 24:90–98
Chen Q Y, Wang G Z. A class of Bézier-like curves. Comput Aid Geom Des, 2 2003, 20:29–39
Dong C S, Wang G Z. On convergence of the control polygon series of C-Bézier curves. In: Proceedings of Geometric Modeling and Processing, Beijing: Tsinghua University, 2004. 49–56
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhu, P., Wang, G. & Yu, J. Degree elevation operator and geometric construction of C-B-spline curves. Sci. China Inf. Sci. 53, 1753–1764 (2010). https://doi.org/10.1007/s11432-010-4053-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11432-010-4053-2