›› 2012, Vol. 27 ›› Issue (4): 841-850.doi: 10.1007/s11390-012-1268-2

• Regular Papers • Previous Articles     Next Articles

A Geometric Approach for Multi-Degree Spline

Xin Li1 (李新), Zhang-Jin Huang2 (黄章进), and Zhao Liu1 (刘昭)   

  1. 1. School of Mathematical Science, University of Science and Technology of China, Hefei 230026, China;
    2. School of Computer Science, University of Science and Technology of China, Hefei 230026, China
  • Received:2011-01-17 Revised:2012-05-02 Online:2012-07-05 Published:2012-07-05
  • Supported by:

    This work was supported by the National Natural Science Foundation of China under Grant Nos.11031007, 60903148, 60803066, the Chinese Universities Scientific Fund, the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry of China, and the Startup Scientific Research Foundation of Chinese Academy of Sciences.

Multi-degree spline (MD-spline for short) is a generalization of B-spline which comprises of polynomial segments of various degrees. The present paper provides a new definition for MD-spline curves in a geometric intuitive way based on an efficient and simple evaluation algorithm. MD-spline curves maintain various desirable properties of B-spline curves, such as convex hull, local support and variation diminishing properties. They can also be refined exactly with knot insertion. The continuity between two adjacent segments with different degrees is at least C1 and that between two adjacent segments of same degrees d is Cd-1. Benefited by the exact refinement algorithm, we also provide several operators for MD-spline curves, such as converting each curve segment into Bézier form, an efficient merging algorithm and a new curve subdivision scheme which allows different degrees for each segment.

[1] Sederberg T W, Zheng J M, Sewell D, Sabin M. Non-uniformrecursive subdivision surfaces. In Proc. the 25th SIGGRAPH,Orlando, USA, July 1998, pp.387-394.

[2] Peters J. Patching Catmull-Clark meshes. In Proc. the27th SIGGRAPH, New Orleans, Louisiana, USA, July 2000,pp.255-258.

[3] Kaklis P D, Pandelis D G. Convexity-preserving polynomialsplines of non-uniform degree. IMA Journal of NumericalAnalysis, 1990, 10(2): 223-234.

[4] Costantini P. Variable degree polynomial splines. In Curvesand Surfaces with Applications in CAGD, Rabut C, LeMehaute A, Schumaker L L (Eds.), Nashville: Vanderbilt UniversityPress, 1997, pp.85-94.

[5] Costantini P. Curve and surface construction using variabledegree polynomial splines. Computer Aided Geometric Design,2000, 17(5): 419-446.

[6] Wang G Z, Deng C Y. On the degree elevation of B-splinecurves and corner cutting. Computer Aided Geometric Design,2007, 24(2): 90-98.

[7] Shen W Q, Wang G Z. A basis of multi-degree splines. ComputerAided Geometric Design. 2010, 27(1): 23-35.

[8] Shen W Q, Wang G Z. Changeable degree spline basis functions.Journal of Computational and Applied Mathematics,2010, 234(8): 2516-2529.

[9] Sederberg T W, Zheng J M, Song X W. Knot intervals andmulti-degree splines. Computer Aided Geometric Design,2003, 20(7): 455-468.

[10] Lyche T, Morken K. Knot removal for parametric B-splinecurves and surfaces. Computer Aided Geometric Design,1987, 4(3): 217-230.
No related articles found!
Full text



[1] Wu Xindong;. Inductive Learning[J]. , 1993, 8(2): 22 -36 .
[2] Zhao Yu; Zhang Qiong; Xiang Hui; Shi Jiaosing; He Zhijun;. A Simplified Model for Generating 3D Realistic Sound in the Multimedia and Virtual Reality Systems[J]. , 1996, 11(4): 461 -470 .
[3] Zhong-Xuan Liu, Shi-Guo Lian, and Zhen Ren. Quaternion Diffusion for Color Image Filtering[J]. , 2006, 21(1): 126 -136 .
[4] Hui-Zhan Yi and Xue-Jun Yang. Toward the Optimal Configuration of Dynamic Voltage Scaling Points in Real-Time Applications[J]. , 2006, 21(6): 893 -900 .
[5] Bai-Lin Yang, Frederick W. B. Li, Zhi-Geng Pan, and Xun Wang. An Effective Error Resilient Packetization Scheme for Progressive Mesh Transmission over Unreliable Networks[J]. , 2008, 23(6 ): 1015 -1025 .
[6] Jie Wu. Collaborative Mobile Charging and Coverage[J]. , 2014, 29(4): 550 -561 .
[7] Fei Xia, De-Jun Jiang, Jin Xiong, Ning-Hui Sun. A Survey of Phase Change Memory Systems[J]. , 2015, 30(1): 121 -144 .
[8] Xi-Te Wang, De-Rong Shen, Mei Bai, Tie-Zheng Nie, Yue Kou, Ge Yu. An Efficient Algorithm for Distributed Outlier Detection in Large Multi-Dimensional Datasets[J]. , 2015, 30(6): 1233 -1248 .
[9] Jian Dai, Zhi-Ming Ding, Jia-Jie Xu. Context-Based Moving Object Trajectory Uncertainty Reduction and Ranking in Road Network[J]. , 2016, 31(1): 167 -184 .
[10] Feng-Feng Pan, Yin-Liang Yue, Jin Xiong. dCompaction: Speeding up Compaction of the LSM-Tree via Delayed Compaction[J]. , 2017, 32(1): 41 -54 .

ISSN 1000-9000(Print)

CN 11-2296/TP

Editorial Board
Author Guidelines
Journal of Computer Science and Technology
Institute of Computing Technology, Chinese Academy of Sciences
P.O. Box 2704, Beijing 100190 P.R. China
E-mail: jcst@ict.ac.cn
  Copyright ©2015 JCST, All Rights Reserved