Taiwanese Journal of Mathematics

A Non-stationary Combined Ternary 5-point Subdivision Scheme with $C^{4}$ Continuity

Zeze Zhang, Hongchan Zheng, Weijie Song, and Baoxing Zhang

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Open access

Abstract

In this paper, a family of non-stationary combined ternary $5$-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order $4$, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate $C^{2}$ interpolating limit curves and $C^{4}$ approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order $5$.

Article information

Source
Taiwanese J. Math., Advance publication (2020), 23 pages.

Dates
First available in Project Euclid: 24 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1585036818

Digital Object Identifier
doi:10.11650/tjm/200303

Subjects
Primary: 65D17: Computer aided design (modeling of curves and surfaces) [See also 68U07] 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 39B12: Iteration theory, iterative and composite equations [See also 26A18, 30D05, 37-XX]

Keywords
ternary non-stationary subdivision combined scheme convergence exponential polynomial generation/reproduction approximation order

Citation

Zhang, Zeze; Zheng, Hongchan; Song, Weijie; Zhang, Baoxing. A Non-stationary Combined Ternary 5-point Subdivision Scheme with $C^{4}$ Continuity. Taiwanese J. Math., advance publication, 24 March 2020. doi:10.11650/tjm/200303. https://projecteuclid.org/euclid.twjm/1585036818


Export citation

References

  • C. Beccari, G. Casciola and L. Romani, Shape controlled interpolatory ternary subdivision, Appl. Math. Comput. 215 (2009), no. 3, 916–927.
  • D. Burkhart, B. Hamann and G. Umlauf, Iso-geometric finite element analysis based on Catmull-Clark: subdivision solids, Comput. Graph. Forum 29 (2010), no. 5, 1575–1584.
  • M. Charina, C. Conti, N. Guglielmi and V. Protasov, Limits of level and parameter dependent subdivision schemes: A matrix approach, Appl. Math. Comput. 272 (2016), part 1, 20–27.
  • M. Charina, C. Conti and L. Romani, Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix, Numer. Math. 127 (2014), no. 2, 223–254.
  • S. W. Choi, B.-G. Lee, Y. J. Lee and J. Yoon, Stationary subdivision schemes reproducing polynomials, Comput. Aided Geom. Design 23 (2006), no. 4, 351–360.
  • C. Conti, N. Dyn, C. Manni and M.-L. Mazure, Convergence of univariate non-stationary subdivision schemes via asymptotic similarity, Comput. Aided Geom. Design 37 (2015), 1–8.
  • C. Conti, L. Gori and F. Pitolli, Totally positive functions through nonstationary subdivision schemes, J. Comput. Appl. Math. 200 (2007), no. 1, 255–265.
  • C. Conti, L. Romani and J. Yoon, Approximation order and approximate sum rules in subdivision, J. Approx. Theory 207 (2016), 380–401.
  • M. Fang, W. Ma and G. Wang, A generalized curve subdivision scheme of arbitrary order with a tension parameter, Comput. Aided Geom. Design 27 (2010), no. 9, 720–733.
  • L. Gori and F. Pitolli, Nonstationary subdivision schemes and totally positive refinable functions, in: Approximation Theory XII, San Antonio 2007, 169–170, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2008.
  • M. F. Hassan and N. A. Dodgson, Ternary and three-point univariate subdivision schemes, in: Curve and Surface Fitting (Saint-Malo, 2002), 199–208, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2003.
  • M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson and M. A. Sabin, An interpolating $4$-point $C^2$ ternary stationary subdivision scheme, Comput. Aided Geom. Design 19 (2002), no. 1, 1–18.
  • A. Levin, Interpolating nets of curves by smooth subdivision surfaces, Proceedings of the 26th annual conference on Computer graphics and interactive techniques, (1999), 57–64.
  • J. Maillot and J. Stam, A unified subdivision scheme for polygonal modeling, Comput. Graph. Forum 20 (2001), no. 3, 471–479.
  • P. Novara and L. Romani, Complete characterization of the regions of $C^2$ and $C^3$ convergence of combined ternary $4$-point subdivision schemes, Appl. Math. Lett. 62 (2016), 84–91.
  • ––––, On the interpolating $5$-point ternary subdivision scheme: A revised proof of convexity-preservation and an application-oriented extension, Math. Comput. Simulation 147 (2018), 194–209.
  • J. Pan, S. Lin and X. Luo, A combined approximating and interpolating subdivision scheme with $C^2$ continuity, Appl. Math. Lett. 25 (2012), no. 12, 2140–2146.
  • K. Rehan and M. A. Sabri, A combined ternary $4$-point subdivision scheme, Appl. Math. Comput. 276 (2016), 278–283.
  • K. Rehan and S. S. Siddiqi, A family of ternary subdivision schemes for curves, Appl. Math. Comput. 270 (2015), 114–123.
  • ––––, A combined binary $6$-point subdivision scheme, Appl. Math. Comput. 270 (2015), 130–135.
  • L. Romani, From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms, J. Comput. Appl. Math. 224 (2009), no. 1, 383–396.
  • S. S. Siddiqi and K. Rehan, Modified form of binary and ternary $3$-point subdivision schemes, Appl. Math. Comput. 216 (2010), no. 3, 970–982.
  • L. Zhang, H. Ma, S. Tang and J. Tan, A combined approximating and interpolating ternary $4$-point subdivision scheme, J. Comput. Appl. Math. 349 (2019), 563–578.
  • H. Zheng, M. Hu and G. Peng, Constructing $2n-1$ point ternary interpolatory subdivision schemes by using variation of constants, 2009 International Conference on Computational Intelligence and Software Engineering, (2009), 1–4.
  • H. Zheng and B. Zhang, A non-stationary combined subdivision scheme generating exponential polynomials, Appl. Math. Comput. 313 (2017), 209–221.