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A C1 bivariate rational cubic interpolating spline

  • 1.

    College of Applied Mathematics, Chaohu University, Chaohu 238000, China

  • 2.

    School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 3.

    Anhui Fuhuang Steel Structure, Chaohu 238076, China

Funds:  Supported by the Nation Natural Science Foundation of China(11472063), the Provincial Natural Science Research Program of Higher Education Institutions of Anhui Province(KJ2013A194, KJ2013Z230), Anhui Province Colleges and Universities Outstanding Youth Talent Support Program(gxyqZD2016285).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.03.002
More Information
  • Author Bio:

    WANG Dongyin, female, born in 1978, master/associate Prof. Research field: Applied mathematics. E-mail: chaohuwdy@163.com.

  • Corresponding author: TAO Youtian
  • Received Date: 04 April 2015
  • Accepted Date: 30 September 2015
  • Rev Recd Date: 30 September 2015
  • Publish Date: 30 March 2017
    Abstract

  • Abstract

    A bivariate rational bicubic interpolating spline(BRIS) with biquadratic denominator and six shape parameters was constructed using both function values and partial derivatives of the function as the interpolation data in a rectangular domain. The C1 continuous condition of BRIS was discussed. Some properties of BRIS such as symmetry were given. BRIS was proved to be bounded and its error was estimated. In the end, a numerical example was given to illustrate the effect of the shape parameters on the shape of BRIS surface.

    Abstract

    A bivariate rational bicubic interpolating spline(BRIS) with biquadratic denominator and six shape parameters was constructed using both function values and partial derivatives of the function as the interpolation data in a rectangular domain. The C1 continuous condition of BRIS was discussed. Some properties of BRIS such as symmetry were given. BRIS was proved to be bounded and its error was estimated. In the end, a numerical example was given to illustrate the effect of the shape parameters on the shape of BRIS surface.
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    • References(28)
  • [1]
    BZIER P. The Mathematical Basis of the UNISURF CAD System[M]. London: Butterworth, 1986.
    [2]
    BOOR C. B-form basics[M]// Geometric Modeling. Philadelphia: SIAM, 1987: 131-148.
    [3]
    CHUI C. Multivariate Splines[M]. Philadelphia: SIAM, 1988.
    [4]
    DENG J, CHEN F, FENG Y. Dimensions of spline spaces over T-meshes[J]. J Comput Appl Math, 2006, 194(2): 267-283.
    [5]
    DENG J, CHEN F, LI X. Polynomial splines over hierarchical T-meshes[J]. Graphical Models, 2008, 74(4): 76-86.
    [6]
    DIERCK P, TYTGAT B. Generating the Bézier points of BETA-spline curve[J]. Comput Aided Geom Des, 1989, 6(4): 279-291.
    [7]
    FARIN G. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide[M]. fifth ed. Menlo Park: Morgan Kaufman, 2002.
    [8]
    KONNO K, CHIYOKURA H. An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches[J]. Comput Aided Geom Des, 1996, 13(9): 825-849.
    [9]
    MLER R. Universal parametrization and interpolation on cubic surfaces[J]. Comput Aided Geom Des, 2002, 19(7): 479-502.
    [10]
    PIEGL L. On NURBS: A survey[J]. IEEE Comput Graph Appl, 1991, 11(1): 55-71.
    [11]
    SEDERBERG T, ZHENG J, BAKENOV A, et al. T-splines and T-NURCCS[J]. ACM Transactions on Graphics, 2003, 22(3): 477-484.
    [12]
    TAN J, TANG S. Composite schemes for multivariate blending rational interpolation[J]. J Comput Appl Math, 2002, 144(1-2): 263-275.
    [13]
    WANG R. Multivariate Spline Functions and Their Applications[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Kluwer Academic Publishers, 2001.
    [14]
    BAO F, SUN Q, DUAN Q. Point control of the interpolating curve with a rational cubic spline[J]. J Vis Commun Image R, 2009, 20(4): 275-280.
    [15]
    BAO F, SUN Q, PAN J, et al. A blending interpolator with value control and minimal strain energy[J]. Comput Graph, 2010, 34(2): 119-124.
    [16]
    DELBOURGO R. Shape preserving interpolation to convex data by rational functions with quadratic numerator and linear denominator[J]. IMA J Numer Anal, 1989, 9(1): 23-136.
    [17]
    DUAN Q, DJIDJELI K, PRICE W, et al. The approximation properties of some rational cubic splines[J]. Int J Comput Math, 1999, 72(2): 155-166.
    [18]
    DUAN Q, BAO F, DU S, et al. Local control of interpolating rational cubic spline curves[J]. Comput Aided Des, 2009, 41(11): 825-829.
    [19]
    GREGORY J,SARFRAZ M, YUEN P. Interactive curve design using C2 rational splines[J]. Comput Graph, 1994, 18(2): 153-159.
    [20]
    HAN X. Convexity preserving piecewise rational quartic interpolation[J]. SIAM J Numer Anal, 2008, 46(2): 920-929.
    [21]
    SARFRAZ M. A C2 rational cubic spline which has linear denominator and shape control[J]. Ann Univ Sci Budapest, 1994, 37: 53-62.
    [22]
    ABBAS M, MAJID A, ALI J. Positivity-preserving rational bi-cubic spline interpolation for 3D positive data[J]. Appl Math Comput, 2014, 234: 460-476
    [23]
    DUAN Q, WANG L, TWIZELL E. A new bivariate rational interpolation based on function values[J]. Inf Sci, 2004, 166(1-4): 181-191.
    [24]
    DUAN Q, WANG L, TWIZELL E. A new weighted rational cubic interpolation and its approximation[J]. Appl Math Comput, 2005, 168(2): 990-1003.
    [25]
    DUAN Q, ZHANG Y, TWIZELL E. A bivariate rational interpolation and the properties[J]. Appl Math Comput, 2006, 179(1): 190-199.
    [26]
    DUAN Q, LI S, BAO F, et al. Hermite interpolation by piecewise rational surface[J]. Appl Math Comput, 2008, 198(1): 59-72.
    [27]
    HUSSAIN M, SARFRAZ M. Positivity-preserving interpolation of positive data by rational cubics[J]. J Comput Appl Math, 2008, 218(2): 446-458.
    [28]
    ZHANG Y, DUAN Q, TWIZELL E. Convexity control of a bivariate rational interpolating spline surfaces[J]. Comput Graph, 2007, 31(5): 679-687.
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    [1]
    BZIER P. The Mathematical Basis of the UNISURF CAD System[M]. London: Butterworth, 1986.
    [2]
    BOOR C. B-form basics[M]// Geometric Modeling. Philadelphia: SIAM, 1987: 131-148.
    [3]
    CHUI C. Multivariate Splines[M]. Philadelphia: SIAM, 1988.
    [4]
    DENG J, CHEN F, FENG Y. Dimensions of spline spaces over T-meshes[J]. J Comput Appl Math, 2006, 194(2): 267-283.
    [5]
    DENG J, CHEN F, LI X. Polynomial splines over hierarchical T-meshes[J]. Graphical Models, 2008, 74(4): 76-86.
    [6]
    DIERCK P, TYTGAT B. Generating the Bézier points of BETA-spline curve[J]. Comput Aided Geom Des, 1989, 6(4): 279-291.
    [7]
    FARIN G. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide[M]. fifth ed. Menlo Park: Morgan Kaufman, 2002.
    [8]
    KONNO K, CHIYOKURA H. An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches[J]. Comput Aided Geom Des, 1996, 13(9): 825-849.
    [9]
    MLER R. Universal parametrization and interpolation on cubic surfaces[J]. Comput Aided Geom Des, 2002, 19(7): 479-502.
    [10]
    PIEGL L. On NURBS: A survey[J]. IEEE Comput Graph Appl, 1991, 11(1): 55-71.
    [11]
    SEDERBERG T, ZHENG J, BAKENOV A, et al. T-splines and T-NURCCS[J]. ACM Transactions on Graphics, 2003, 22(3): 477-484.
    [12]
    TAN J, TANG S. Composite schemes for multivariate blending rational interpolation[J]. J Comput Appl Math, 2002, 144(1-2): 263-275.
    [13]
    WANG R. Multivariate Spline Functions and Their Applications[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Kluwer Academic Publishers, 2001.
    [14]
    BAO F, SUN Q, DUAN Q. Point control of the interpolating curve with a rational cubic spline[J]. J Vis Commun Image R, 2009, 20(4): 275-280.
    [15]
    BAO F, SUN Q, PAN J, et al. A blending interpolator with value control and minimal strain energy[J]. Comput Graph, 2010, 34(2): 119-124.
    [16]
    DELBOURGO R. Shape preserving interpolation to convex data by rational functions with quadratic numerator and linear denominator[J]. IMA J Numer Anal, 1989, 9(1): 23-136.
    [17]
    DUAN Q, DJIDJELI K, PRICE W, et al. The approximation properties of some rational cubic splines[J]. Int J Comput Math, 1999, 72(2): 155-166.
    [18]
    DUAN Q, BAO F, DU S, et al. Local control of interpolating rational cubic spline curves[J]. Comput Aided Des, 2009, 41(11): 825-829.
    [19]
    GREGORY J,SARFRAZ M, YUEN P. Interactive curve design using C2 rational splines[J]. Comput Graph, 1994, 18(2): 153-159.
    [20]
    HAN X. Convexity preserving piecewise rational quartic interpolation[J]. SIAM J Numer Anal, 2008, 46(2): 920-929.
    [21]
    SARFRAZ M. A C2 rational cubic spline which has linear denominator and shape control[J]. Ann Univ Sci Budapest, 1994, 37: 53-62.
    [22]
    ABBAS M, MAJID A, ALI J. Positivity-preserving rational bi-cubic spline interpolation for 3D positive data[J]. Appl Math Comput, 2014, 234: 460-476
    [23]
    DUAN Q, WANG L, TWIZELL E. A new bivariate rational interpolation based on function values[J]. Inf Sci, 2004, 166(1-4): 181-191.
    [24]
    DUAN Q, WANG L, TWIZELL E. A new weighted rational cubic interpolation and its approximation[J]. Appl Math Comput, 2005, 168(2): 990-1003.
    [25]
    DUAN Q, ZHANG Y, TWIZELL E. A bivariate rational interpolation and the properties[J]. Appl Math Comput, 2006, 179(1): 190-199.
    [26]
    DUAN Q, LI S, BAO F, et al. Hermite interpolation by piecewise rational surface[J]. Appl Math Comput, 2008, 198(1): 59-72.
    [27]
    HUSSAIN M, SARFRAZ M. Positivity-preserving interpolation of positive data by rational cubics[J]. J Comput Appl Math, 2008, 218(2): 446-458.
    [28]
    ZHANG Y, DUAN Q, TWIZELL E. Convexity control of a bivariate rational interpolating spline surfaces[J]. Comput Graph, 2007, 31(5): 679-687.
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