New cubic rational basis with tension shape parameters

  • Yuan-peng Zhu
  • Xu-li HanEmail author


By using the blossom approach, we construct four new cubic rational Bernsteinlike basis functions with two shape parameters, which form a normalized B-basis and include the cubic Bernstein basis and the cubic Said-Ball basis as special cases. Based on the new basis, we propose a class of C 2 continuous cubic rational B-spline-like basis functions with two local shape parameters, which includes the cubic non-uniform B-spline basis as a special case. Their totally positive property is proved. In addition, we extend the cubic rational Bernsteinlike basis to a triangular domain which has three shape parameters and includes the cubic triangular Bernstein-Bézier basis and the cubic triangular Said-Ball basis as special cases. The G 1 continuous conditions are deduced for the joining of two patches. The shape parameters in the bases serve as tension parameters and play a foreseeable adjusting role on generating curves and patches.


Bernstein basis Said-Ball basis tension shape parameter totally positive property B-spline basis Bernstein-Bézier basis 

MR Subject Classification

65D07 65D17 


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Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.School of Mathematics and StatisticsCentral South UniversityChangshaChina

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