Research article

The inequalities for the analysis of a class of ternary refinement schemes

  • Received: 04 July 2020 Accepted: 13 September 2020 Published: 24 September 2020
  • MSC : 65D17, 65D07, 65D05

  • The ternary refinement schemes are the generalized version of the binary refinement schemes. This class of the schemes produce the smooth curves with the less number of refinement steps as compared to the binary class of schemes. In this paper, we present the inequalities for the analysis of a class of ternary refinement schemes. There are three simple algebraic expressions in each inequality. Further these algebraic expressions contain only the coefficients used in the refinement rules of the ternary schemes.

    Citation: Ghulam Mustafa, Syeda Tehmina Ejaz, Dumitru Baleanu, Yu-Ming Chu. The inequalities for the analysis of a class of ternary refinement schemes[J]. AIMS Mathematics, 2020, 5(6): 7582-7604. doi: 10.3934/math.2020485

    Related Papers:

  • The ternary refinement schemes are the generalized version of the binary refinement schemes. This class of the schemes produce the smooth curves with the less number of refinement steps as compared to the binary class of schemes. In this paper, we present the inequalities for the analysis of a class of ternary refinement schemes. There are three simple algebraic expressions in each inequality. Further these algebraic expressions contain only the coefficients used in the refinement rules of the ternary schemes.


    加载中


    [1] G. M. Chaikin, An algorithm for high speed curve generation, Comput. Graphics Image Process., 3 (1974), 346-349.
    [2] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx., Springer, Boston, MA, 5 (1989), 49-68.
    [3] G. de Rham, Sur une courbe plane, J. Math. Pures Appl., 35 (1956), 25-42.
    [4] N. Dyn, J. A. Gregory, D. Levin, 4-point interpolatory subdivision scheme for curve design, Comput. Aided Geom. Des., 4 (1987), 257-268.
    [5] N. Dyn, J. A. Gregory, D. Levin, Analysis of uniform binary subdivision scheme for curve design, Constr. Approx., 7 (1991), 127-147.
    [6] N. Dyn, Analysis of convergence and smoothness by formalism of Laurent polynomials, A. Iske, E. Quak, M. S. Floater, Tutorials on Multiresolution in Geometric Modelling, Eds., Springer, (2002), 51-68 (chapter 3).
    [7] G. Farin, Curves and surfaces for CAGD: A practical guide, Academic Press, 2002.
    [8] M. F. Hassan, N. A. Dodgson, Ternary and three-point univariate subdivision schemes, Tec. Rep. No. 520, University of Cambridge Computer Laboratory, 2001. Available from: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-520.pdf.
    [9] M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson, et al. An interpolating 4-point C2 ternary stationary subdivision scheme, Comput. Aided Geom. Des., 19 (2002), 1-18.
    [10] F. Khan, G. Mustafa, Ternary six-point interpolating subdivision scheme, Lobachevskii J. Math., 29 (2008), 153-163.
    [11] C. A. Micchelli, H. Prautzch, Uniform refinement of curves, Linear Algebra Appl., 114 (1989), 841-870.
    [12] G. Mustafa, M. Zahid, Numerical algorithm for analysis of n-ary subdivision schemes, Appl. Appl. Math., 8 (2013), 614-630.
    [13] G. Mustafa, R. Hameed, D. Baleanu, et al. A class of refinement schemes with two shape control parameters, IEEE ACCESS, 8 (2020), 98316-98329.
    [14] R. Qu, Recursive subdivision algorithms for curve and surface design, Ph.D Thesis, Department of Mathematics and Statistics, Brunei University, Uxbridge, Middlesex, Britain, 1990.
    [15] M. Sabin, Analysis and design of univariate subdivision schemes, Geometry and Computing, Springer, ISBN 978-3-642-13647-4, 6 (2010).
    [16] S. S. Siddiqi, K. Rehan, Modified form of binary and ternary 3-point subdivission schemes, Appl. Math. Comput., 216 (2010), 970-982.
    [17] H. Zheng, M. Hu, G. Peng, Ternary even symmetric 2n-point subdivision, Int. Conf. Comput. Intell. Software Eng., IEEE, 978 (2009), 4244-4507.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2221) PDF downloads(85) Cited by(1)

Article outline

Figures and Tables

Figures(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog