Determining Knots by Minimizing the Second Derivative

  • Fan ZhangEmail author
  • Xueying Qin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8971)


In constructing a parametric curve interpolating a set of data points, one of the key problems is to specify a parameter value (node) for each data point. A new method of choosing knots is presented. For each data points, the new method constructs a quadratic polynomial curve by three adjacent data points. The node parameters of the quadratic curve are determined by minimizing the second derivative of the quadratic curve. And the knot interval between two adjacent data points is determined by two quadratic curves associated with the two adjacent data points. Experiments showed that the curves constructed using the knots by the new method generally have better interpolation precision.


Polynomial curve Determining knots Second derivative Minimizing 


  1. 1.
    Brodlie, K.W.: A review of methods for curve and function drawing. In: Brodlie, K.W. (ed.) Mathematical Methods in Computer Graphics and Design, pp. 1–37. Academic Press, London (1980)Google Scholar
  2. 2.
    Su, B., Liu, D.: Computational Geometry, p. 49. Shanghai Academic Press, Shanghai (1982). (in Chinese)Google Scholar
  3. 3.
    Li, W., Xu, S., Zheng, S., Zhao, G.: Target curvature driven fairing algorithm for planar cubic B-spline curves. Comput. Aided Geom. Des. 21(5), 499–513 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and Their Applications, p. 51. Academic Press, New York (1967)zbMATHGoogle Scholar
  5. 5.
    De Boor, C.: A practical Guide to Splines, p. 318. Springer, New York (1978)zbMATHCrossRefGoogle Scholar
  6. 6.
    Faux, I.D., Pratt, M.J.: Computational Geometry for Design and Manufacture, p. 176. Ellis Horwood, New York (1979)zbMATHGoogle Scholar
  7. 7.
    Farin, G.: Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, p. 111. Academic press, New York (1989)Google Scholar
  8. 8.
    Lee, E.T.Y.: Choosing nodes in parametric curve interpolation. CAD 21(6), 363–370 (1989)zbMATHGoogle Scholar
  9. 9.
    Zhang, C., Cheng, F., Miura, K.: A method for determining knots in parametric curve interpolation. CAGD 15, 399–416 (1998)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Zhang, C., Han, H., Cheng, F.: Determining knots by minimizing energy. J. Comput. Sci. Technol. 21(6), 261–264 (2006)zbMATHCrossRefGoogle Scholar
  11. 11.
    Hartley, P.J., Judd, C.J.: Parametrization and shape of B-spline curves for CAD. CAD 12(5), 235–238 (1980)Google Scholar
  12. 12.
    Marin, S.P.: An approach to data parametrization in parametric cubic spline interpolation problems. J. Approximation Theor. 41, 64–86 (1984)zbMATHCrossRefGoogle Scholar
  13. 13.
    Xie, H., Qin, H.: A novel optimization approach to the effective computation of NURBS knots. Int. J. Shape Model. 7(2), 199–227 (2001)CrossRefGoogle Scholar
  14. 14.
    Gotsman, C., Gu, X., Sheffer, A.: Fundamentals of spherical parameterization for 3D meshes. In: Computer Graphics Proceedings, Annual Conference Series, ACM SIGGRAPH, pp. 358–363. ACM Press, San Diego (2003)Google Scholar
  15. 15.
    Gu, X., Yau, S.-T.: Global conformal surface parameterization. In: ACM Symposium on Geometry Processing, pp. 127–137. ACM Press, San Diego (2003)Google Scholar
  16. 16.
    Floater, M.S.: Reimers M1 meshless parameterization and surface reconstruction. Comput. Aided Geom. Des. 18(2), 77–92 (2001)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyShandong UniversityJinanChina

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