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Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

Author

Listed:
  • Pakeeza Ashraf

    (Department of Mathematics, Government Sadiq College Women University, Bahawalpur 63100, Pakistan)

  • Bushra Nawaz

    (Department of Mathematics, Government Sadiq College Women University, Bahawalpur 63100, Pakistan)

  • Dumitru Baleanu

    (Department of Mathematics, Cankaya University, 06790 Ankara, Turkey
    Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
    Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan)

  • Kottakkaran Sooppy Nisar

    (Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia)

  • Abdul Ghaffar

    (Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
    Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam)

  • Muhammad Aqeel Ahmed Khan

    (Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan)

  • Saima Akram

    (Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan, Multan 66000, Pakistan)

Abstract

Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.

Suggested Citation

  • Pakeeza Ashraf & Bushra Nawaz & Dumitru Baleanu & Kottakkaran Sooppy Nisar & Abdul Ghaffar & Muhammad Aqeel Ahmed Khan & Saima Akram, 2020. "Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme," Mathematics, MDPI, vol. 8(3), pages 1-19, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:338-:d:328230
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    References listed on IDEAS

    as
    1. Pitolli, Francesca, 2014. "Ternary shape-preserving subdivision schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 106(C), pages 185-194.
    2. Novara, Paola & Romani, Lucia, 2018. "On the interpolating 5-point ternary subdivision scheme: A revised proof of convexity-preservation and an application-oriented extension," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 147(C), pages 194-209.
    3. Siddiqi, Shahid S. & Noreen, Tayyaba, 2015. "Convexity preservation of six point C2 interpolating subdivision scheme," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 936-944.
    4. Ghulam Mustafa & Jiansong Deng & Pakeeza Ashraf & Najma Abdul Rehman, 2012. "The Mask of Odd Points -Ary Interpolating Subdivision Scheme," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-20, November.
    Full references (including those not matched with items on IDEAS)

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