Determining knots by optimizing the bending and stretching energies

Fan Zhang, Xue ying Qin, Xue mei Li, Fu hua Cheng

Research output: Contribution to journalArticlepeer-review


For a given set of data points in the plane, a new method is presented for computing a parameter value (knot) for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.

Original languageEnglish
Pages (from-to)53-67
Number of pages15
JournalApplied Mathematics
Issue number1
StatePublished - Mar 1 2017

Bibliographical note

Funding Information:
Supported by the National Natural Science Foundation of China (61602277, 61672327, 61472227) and the Shandong Provincial Natural Science Foundation, China (ZR2016FQ12).

Publisher Copyright:
© 2017, Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg.


  • interpolation
  • knot
  • optimizing energy
  • quadratic polynomial

ASJC Scopus subject areas

  • Applied Mathematics


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