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Fractal properties of interpolatory subdivision schemes and their application in fractal generation

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  • Zheng, Hongchan
  • Ye, Zhenglin
  • Lei, Youming
  • Liu, Xiaodong

Abstract

In this paper, we present a novel method to analyze the fractal properties of the 4-point binary and the 3-point ternary interpolatory subdivision schemes. The relationship between the parameter and the fractal behavior of the limit curve of the two schemes is obtained, respectively. As an application of the obtained results, the generation of fractal curves and surfaces is discussed. Many examples show that the results presented in this paper offer a direct means for a fast generation of fractals.

Suggested Citation

  • Zheng, Hongchan & Ye, Zhenglin & Lei, Youming & Liu, Xiaodong, 2007. "Fractal properties of interpolatory subdivision schemes and their application in fractal generation," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 113-123.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:1:p:113-123
    DOI: 10.1016/j.chaos.2005.10.075
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    References listed on IDEAS

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    1. Berezowski, Marek, 2005. "Analysis of dynamic solutions of complex delay differential equation exemplified by the extended Mandelbrot’s equation," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1399-1404.
    2. Chung, K.W. & Ma, H.M., 2005. "Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1145-1158.
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    Cited by:

    1. Siddiqi, Shahid S. & Salam, Wardat us & Rehan, Kashif, 2015. "A new non-stationary binary 6-point subdivision scheme," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1227-1239.
    2. Calabrò, F. & Corbo Esposito, A. & Mantica, G. & Radice, T., 2016. "Refinable functions, functionals, and iterated function systems," Applied Mathematics and Computation, Elsevier, vol. 272(P1), pages 199-207.
    3. Siddiqi, Shahid S. & us Salam, Wardat & Butt, Nadeem Ahmad, 2015. "Fractal behavior of ternary 4-point interpolatory subdivision scheme with tension parameter," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 148-158.
    4. Siddiqi, Shahid S. & Salam, Wardat us & Rehan, Kashif, 2016. "Construction of binary four and five point non-stationary subdivision schemes from hyperbolic B-splines," Applied Mathematics and Computation, Elsevier, vol. 280(C), pages 30-38.
    5. Siddiqi, Shahid S. & Salam, Wardat us & Rehan, Kashif, 2015. "Binary 3-point and 4-point non-stationary subdivision schemes using hyperbolic function," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 120-129.

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