›› 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.

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