Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A B-spline curve extension algorithm

B 样条曲线延伸算法


B-spline curve extension is an important operation in computer aided design systems. In this paper, we present a new extension algorithm for B-spline curves. The algorithm uses curve unclamping to generate a uniform B-spline curve segment from the original curve and gradually extends the segment to pass through every target point. Algorithms of uniform B-spline curves are used such that our algorithm has a low time cost and can easily handle arbitrary-order derivative constraints at the target points. Generalization for non-uniform rational B-spline curve extension is also discussed, and examples show the efficiency of our method.



B 样条曲线延伸是计算机辅助造型系统中的一个重要操作。本文中我们提出一种 B 样条曲线延伸的新算法。算法使用曲线开支化将输入转化为均匀 B 样条曲线, 然后将其逐步延伸到各个目标点处。我们借助均匀 B 样条的求值方法加速曲线延伸操作, 使得算法拥有很高的时间效率, 并可以处理目标点处的任意阶导数约束。我们还进一步拓展算法, 以处理 NURBS 曲线延伸问题。在实验部分我们给出一些构造实例, 以验证算法有效性。

This is a preview of subscription content, log in to check access.


  1. 1

    Hu S M, Tai C L, Zhang S H. An extension algorithm for B-splines by curve unclamping. Comput Aid Des, 2002, 34: 415–419

  2. 2

    Liu Y J, Qiu R Q, Liang X H. NURBS curve blending using extension. J Zhejiang Univ Sci A, 2009, 10: 570–576

  3. 3

    Chen X D, Ma W. Geometric point interpolation method in R3 space with tangent directional constraint. Comput Aid Des, 2012, 44: 1217–1228

  4. 4

    Chen X D, Ma W, Paul J C. Cubic B-spline curve approximation by curve unclamping. Comput Aid Des, 2010, 42: 523–534

  5. 5

    Shetty S, White P. Curvature-continuous extensions for rational B-spline curves and surfaces. Comput Aid Des, 1991, 23: 484–491

  6. 6

    Zhou Y F, Zhang C M, Gao S S. Extension of B-spline curves with G2 continuity. In: Proceedings of Advances in Visual Computing. Berlin: Springer, 2008. 1096–1105

  7. 7

    Fan H, Zhang C M, Li J J. Extension algorithm for B-splines with GC2-continuous (in Chinese). Chin J Comput, 2005, 28: 933–938

  8. 8

    Mo G L, Zhao Y N. A new extension algorithm for cubic B-splines based on minimal strain energy. J Zhejiang Univ Sci A, 2006, 7: 2043–2049

  9. 9

    Xu G, Wang G Z. Extensions of uniform cubic B-spline curve with local shape parameters (in Chinese). J Comput Res Develop, 2007, 44: 1032–1037

  10. 10

    Xu J. Smooth B-spline curves extension with ordered points constraint. Adv Mater Res, 2011, 311: 1439–1445

  11. 11

    Zhang T, Wang X, Jiang Q, et al. G2-continuity extension algorithm for disk B-spline curve. In: Proceedings of Computer-Aided Design and Computer Graphics (CAD/Graphics), Guangzhou, 2013. 413–414

  12. 12

    Jiang Q, Wu Z, Zhang T, et al. An extension algorithm for ball B-spline curves with G2 continuity. In: Proceedings of International Conference on Cyberworlds, Yokohama, 2013. 252–258

  13. 13

    Piegl L A, Tiller W. The NURBS Book. Berlin: Springer-Verlag, 1995

  14. 14

    Mortenson M E. Geometric Modeling. New York: Wiley, 1985

Download references

Author information

Correspondence to Hejin Gu.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lu, Y., Shi, K., Yong, J. et al. A B-spline curve extension algorithm. Sci. China Inf. Sci. 59, 32103 (2016).

Download citation


  • curve extension
  • B-spline/NURBS
  • unclamping
  • clamping
  • uniform


  • 曲线延伸
  • B 样条/有理B样条
  • 闭支曲线
  • 开支曲线
  • 均匀节点