Properties of G1 Continuity Conditions Between Two B-Spline Surfaces

  • Nailiang Zhao
  • Weiyin Ma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4035)


This paper addresses some properties of G 1 continuity conditions between two B-spline surfaces with arbitrary degrees and generally structured knots. Key issues addressed in the paper include necessary G 1 continuity conditions between two B-spline surfaces, general connecting functions, continuities of the general connecting functions, and intrinsic conditions of the general connecting functions along the common boundary. In general, one may use piecewise polynomial functions, i.e. B-spline functions, as connecting functions for G 1 connection of two B-spline surfaces. Based on the work reported in this paper, some recent results in literature using linear connecting functions are special cases of the general connecting functions reported in this paper. In case that the connecting functions are global linear functions along the common boundary commonly used in literature, the common boundary degenerates as a Bézier curve for proper G 1 connection. Several examples for connecting two uniform biquadratic B-spline surfaces with G 1 continuity are also presented to demonstrate the results.


Continuity Condition Piecewise Linear Common Boundary Arbitrary Degree NURBS Surface 


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  1. 1.
    Piegl, L., Tiller, W.: The NURBS Book. Springer, New York (1997)Google Scholar
  2. 2.
    Milroy, M.J., Bradley, C., Vickers, G.W., Weir, D.J.: G1 continuity of B-spline surface patches in reverse engineering. Computer Aided Design 27(6), 471–478 (1995)MATHCrossRefGoogle Scholar
  3. 3.
    Ma, W., Leung, P.C., Cheung, E.H.M., Lang, S.Y.T., Cui, H.: Smooth multiple surface fitting for reverse engineering. In: Proceedings of the 32nd CIRP International Seminar on Manufacturing Systems, Katholieke Universteit Leuven, Belgium, pp. 53–62 (1999)Google Scholar
  4. 4.
    Eck, M., Hoppe, H.: Automatic reconstruction of B-spline surfaces of arbitrary topology type. Computer Graphics 30(8), 325–334 (1996)Google Scholar
  5. 5.
    Piegl, L.A., Tiller, W.: Filling n-sided regions with NURBS patches. The Visual Computer 15(2), 77–89 (1999)MATHCrossRefGoogle Scholar
  6. 6.
    Shi, X., Yu, P., Wang, T.: G1 continuous conditions of biquartic B-spline surfaces. Journal of Computational and Applied Mathematics 144(1-2), 251–262 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Shi, X., Wang, T., Yu, P.: A practical construction of G1 smooth biquintic B-spline surfaces over arbitrary topology. Computer Aided Design 36(5), 413–424 (2004)CrossRefMATHGoogle Scholar
  8. 8.
    Shi, X., Wang, T., Wu, P., Liu, F.: Reconstruction of convergent G1 smooth B-spline surfaces. Computer Aided Geometric Design 21(9), 893–913 (2004)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Che, X., Liang, X., Li, Q.: G1 continuity conditions of adjacent NURBS surfaces. Computer Aided Geometric Design 22(4), 285–298 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Che, X., Liang, X.: G1 continuity conditions of B-spline surfaces. Northeastern Math. J. 18, 343–352 (2002)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nailiang Zhao
    • 1
  • Weiyin Ma
    • 2
  1. 1.Hangzhou Dianzi UniversityHangzhou, ZhejiangP.R. China
  2. 2.City University of Hong KongKowloon, Hong Kong SARChina

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