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Fractals, Wavelets, and their Applications pp 349–367Cite as

\(\mathcal{C}^{1}\)-Rational Cubic Fractal Interpolation Surface Using Functional Values

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 92))

Abstract

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. In the present article, we develop the \( \mathcal{C}^{1} \)-rational cubic fractal interpolation surface (FIS) as a fixed point of the Read-Bajraktarevi\( \acute {c} \) (RB) operator defined on a suitable function space. Our \( \mathcal{C}^{1} \)-rational cubic FIS is effective tool to stich surface data arranged on a rectangular grid. Our construction needs only the functional values at the grids being interpolated, therefore implementation is an easy task. We first construct the x-direction rational cubic FIFs (x-direction fractal boundary curves) to approximate the data generating function along the grid lines parallel to x-axis. Then we form a rational cubic FIS as a blending of these fractal boundary curves. An upper bound of the uniform distance between the rational cubic FIS and an original function is estimated for the convergence results. A numerical illustration is provided to explain the visual quality of our rational cubic FIS. An extra feature of this fractal surface scheme is that it allows subsequent interactive alteration of the shape of the surface by changing the scaling factors and shape parameters.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    A. K. B. Chand & N. Vijender

Authors
  1. A. K. B. Chand
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  2. N. Vijender
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Corresponding author

Correspondence to N. Vijender .

Editor information

Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Mecklenburg-Vorpomm., Germany

    Christoph Bandt

  2. Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Michael Barnsley

  3. Math Department, Boston University, Boston, Massachusetts, USA

    Robert Devaney

  4. Mathematical Institute, University of St Andrews, St Andrews, Fife, United Kingdom

    Kenneth J. Falconer

  5. Department of Mathematics and Statistics, University of Hyderabad, Hyderabad, India

    V. Kannan

  6. Department of Basic Sciences & Humanities, Rajagiri School of Engineering & Technology, Kerala, India

    Vinod Kumar P.B.

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Chand, A.K.B., Vijender, N. (2014). \(\mathcal{C}^{1}\)-Rational Cubic Fractal Interpolation Surface Using Functional Values. In: Bandt, C., Barnsley, M., Devaney, R., Falconer, K., Kannan, V., Kumar P.B., V. (eds) Fractals, Wavelets, and their Applications. Springer Proceedings in Mathematics & Statistics, vol 92. Springer, Cham. https://doi.org/10.1007/978-3-319-08105-2_22

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