Advertisement

New cubic rational basis with tension shape parameters

  • Yuan-peng Zhu
  • Xu-li HanEmail author
Article

Abstract

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.

Keywords

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

MR Subject Classification

65D07 65D17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B A Barsky. Local control of bias and tension in Beta-splines, ACM Trans Graph, 1983, 2: 109–134.zbMATHCrossRefGoogle Scholar
  2. [2]
    B A Barsky. Computer graphics and geometric modeling using Beta-splines, Springer, Heidelberg, 1988.zbMATHGoogle Scholar
  3. [3]
    T Bosner, M Rogina. Variable degree polynomial splines are Chebyshev splines, Adv Comput Math, 2013, 38: 383–400.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J Cao, G Z Wang. An extension of Bernstein-Bézier surface over the triangular domain, Progr Natur Sci, 2007, 17: 352–357.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    J M Carnicer, E Mainar, J M Peña. Critical length for design purpose and extended Chebyshev spaces, Constr Approx, 2004, 20: 55–71.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    J M Carnicer, J M Peña. Total positive bases for shape preserving curve design and optimality of B-splines, Comput Aided Geom Design, 1994, 11: 635–656.CrossRefGoogle Scholar
  7. [7]
    J M Carnicer, J M Peña. Total positivity and optimal bases, In: Total Positivity and its Applications, Kluwer Academic, Dordrecht, 1996, 133–155.CrossRefGoogle Scholar
  8. [8]
    P Costantini. Curve and surface construction using variable degree polynomial splines, Comput Aided Geom Design, 2000, 17: 419–466.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P Costantini, P D Kaklis, C Manni. Polynomial cubic splines with tension properties, Comput Aided Geom Design, 2010, 27: 592–610.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    P Constantini, B I Kvasov, C Manni. On discrete hyperbolic tension splines, Adv Comput Math, 1999, 11: 331–354.MathSciNetCrossRefGoogle Scholar
  11. [11]
    P Costantini, T Lyche, C Manni. On a class of weak Tchebysheff systems, Numer Math, 2005, 101: 333–354.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    P Costantini, C Manni. Geometric construction of spline curves with tension properties, Comput Aided Geom Design, 2003, 20: 579–599.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    P Costantini, C Manni. A geometric approach for Hermite subdivision, Numer Math, 2010, 115: 333–369.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    P Costantini, C Manni. Curve and surface construction using Hermite subdivision schemes, J Comput Appl Math, 2010, 233: 1660–1673.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    P Costantini, F Pelosi, M Sampoli. New spline spaces with generalized tension properties, BIT, 2008, 48: 665–688.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    Q Duan, F X Bao, S T Du, E H Twizell. Local control of interpolating rational cubic spline curves, Comput Aided Design, 2009, 41: 825–829.CrossRefGoogle Scholar
  17. [17]
    G Farin. Curves and Surfaces for CAGD, 5th ed, Morgan Kaufmann, San Francisco, 2001.Google Scholar
  18. [18]
    T N TGoodman, H BSaid. Properties of generalized Ball curves and surfaces, Comput Aided Design, 1991, 23: 554–560.CrossRefGoogle Scholar
  19. [19]
    T N TGoodman, H BSaid. Shape preserving properties of the generalized Ball basis, Comput Aided Geom Design, 1991, 8: 115–121.MathSciNetCrossRefGoogle Scholar
  20. [20]
    J A Gregory, M Sarfraz. A rational cubic spline with tension, Comput Aided Geom Design, 1990, 9: 1–13.MathSciNetCrossRefGoogle Scholar
  21. [21]
    X L Han. Convexity-preserving piecewise rational quartic interpolation, SIAM J Numer Anal, 2008, 46: 920–929.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    X L Han, Y P Zhu. Curve construction based on five trigonometric blending functions, BIT, 2012, 52: 953–979.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    M Hoffmann, I Juhász, G Károlyi. A control point based curve with two exponential shape parameters, BIT, 2014, 54: 691–710.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    J Hoschek, D Lasser. Fundamentals of Computer Aided Geometric Design, A K Peters, Wellesley, MA, 1993.zbMATHGoogle Scholar
  25. [25]
    S M Hu, G Z Wang, T G Jin. Properties of two types of generalized Ball curves, Comput Aided Design, 1996, 28: 125–133.CrossRefGoogle Scholar
  26. [26]
    M L Mazure. Quasi-Chebychev splines with connexion matrices: application to variable degree polynomial splines, Comput Aided Geom Design, 2001, 18: 287–298.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    M L Mazure. Blossoms and optimal bases, Adv Comput Math, 2004, 20: 177–203.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    M L Mazure. Chebyshev spaces and Bernstein bases, Constr Approx, 2005, 22: 347–363.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    M L Mazure. On a general new class of quasi Chebyshevian splines, Numer Algorithms, 2011, 58: 399–438.zbMATHMathSciNetCrossRefGoogle Scholar
  30. [30]
    M L Mazure. On a new criterion to decide whether a spline space can be used for design, BIT, 2012, 52: 1009–1034.zbMATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    G M Nielson. A locally controllable spline with tension for interactive curve design, Comput Aided Geom Design, 1984, 1: 199–205.zbMATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    L Piegl, W Tiller. The NURBS Book, Springer, New York, 1995.zbMATHCrossRefGoogle Scholar
  33. [33]
    D Salomon, F B Schneider. The Computer Graphics Manual, Springer, New York, 2011.zbMATHCrossRefGoogle Scholar
  34. [34]
    L L Schumaker. Spline Functions: Basic Theory, 3rd edition, Cambridge University Press, 2007.CrossRefGoogle Scholar
  35. [35]
    L L Yan, J F Liang. An extension of the Bézier model, Appl Math Comput, 2011, 218: 2863–2879.zbMATHMathSciNetCrossRefGoogle Scholar
  36. [36]
    L Q Yang, X M Zeng. Bézier curves and surfaces with shape parameters, Int J Comput Math, 2009, 86: 1253–1263.zbMATHMathSciNetCrossRefGoogle Scholar
  37. [37]
    Y P Zhu, X LHan. A class of αβγ-Bernstein-Bézier basis functions over triangular domain, Appl Math Comput, 2013, 220: 446-454.MathSciNetCrossRefGoogle Scholar
  38. [38]
    Y P Zhu, X L Han. Curves and surfaces construction based on new basis with exponential functions, Acta Appl Math, 2014, 129: 183–203.MathSciNetCrossRefGoogle Scholar
  39. [38]
    Y P Zhu, X L Han, S J Liu. Curve construction based on four αβ-Bernstein-like basis functions, J Comput Appl Math, 2015, 273: 160–181.zbMATHMathSciNetCrossRefGoogle Scholar

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

Personalised recommendations