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A Computational Method for Subdivision Depth of Ternary Schemes

Author

Listed:
  • Faheem Khan

    (Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan)

  • Ghulam Mustafa

    (Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan)

  • Aamir Shahzad

    (Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan)

  • Dumitru Baleanu

    (Department of Mathematics, Cankaya University, Ankara 06530, Turkey
    Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan)

  • Maysaa M. Al-Qurashi

    (Department of Mathematics, King Saud University, Riyadh 11495, Saudi Arabia)

Abstract

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k -th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

Suggested Citation

  • Faheem Khan & Ghulam Mustafa & Aamir Shahzad & Dumitru Baleanu & Maysaa M. Al-Qurashi, 2020. "A Computational Method for Subdivision Depth of Ternary Schemes," Mathematics, MDPI, vol. 8(5), pages 1-22, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:817-:d:359622
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    References listed on IDEAS

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    1. Deng, Chongyang & Jin, Wenbiao & Li, Yajuan & Xu, Huixia, 2017. "A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon," Applied Mathematics and Computation, Elsevier, vol. 304(C), pages 10-19.
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    Cited by:

    1. Aamir Shahzad & Faheem Khan & Abdul Ghaffar & Shao-Wen Yao & Mustafa Inc & Shafqat Ali, 2021. "A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes," Mathematics, MDPI, vol. 9(8), pages 1-20, April.
    2. Samsul Ariffin Abdul Karim & Faheem Khan & Ghulam Mustafa & Aamir Shahzad & Muhammad Asghar, 2023. "An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes," Mathematics, MDPI, vol. 11(11), pages 1-12, May.

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    1. Aamir Shahzad & Faheem Khan & Abdul Ghaffar & Shao-Wen Yao & Mustafa Inc & Shafqat Ali, 2021. "A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes," Mathematics, MDPI, vol. 9(8), pages 1-20, April.
    2. Samsul Ariffin Abdul Karim & Faheem Khan & Ghulam Mustafa & Aamir Shahzad & Muhammad Asghar, 2023. "An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes," Mathematics, MDPI, vol. 11(11), pages 1-12, May.

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