Geometric Modelling of a Family of 3-point Quaternary Subdivision Schemes Rζ
DOI:
https://doi.org/10.21015/vtm.v12i1.1868Abstract
Computer-aided geometric design combines mathematical concepts and computing skills that smooth curves through subdivision schemes. Subdivision schemes perform smoothing by turning the control polygon into a limit curve under a refinement rule, a prime example of which is improving the signal-to-noise ratio in modern devices. Because of the importance and location of subdivision schemes, mathematicians use them in CAD, computer graphics and advanced simulation methods. In this research, a family of 3-point Quaternary approximating subdivision scheme $R_{\zeta}$ is presented with its properties and analysis, including necessary conditions for convergence, Laurent polynomial, degree of the generation and polynomial reproduction, continuity analysis, Hölder regularity, and limit stencils. The visual performance of the proposed scheme is also presented to highlight the importance of this research and to validate the scheme.
References
S. M. Hussain, A. U. Rehman, D. Baleanu, K. S. Nisar, A. Ghaffar, and S. A. A. Karim, "Generalized 5-point approximating subdivision scheme of varying arity," *Mathematics*, vol. 8, no. 4, p. 474, 2020.
U. Reif and M. A. Sabin, "Old problems and new challenges in subdivision," *Journal of Computational and Applied Mathematics*, vol. 349, pp. 523–531, 2019.
Y. Liu, H. Shou, and K. Ji, "Review of subdivision schemes and their applications," *Recent Patents on Engineering*, vol. 16, no. 4, pp. 50–62, 2022.
M. Asghar and G. Mustafa, "A family of binary approximating subdivision schemes based on binomial distribution," *Mehran University Research Journal of Engineering & Technology*, vol. 38, no. 4, pp. 1087–1100, 2019.
M. Asghar, M. J. Iqbal, and G. Mustafa, "A family of high continuity subdivision schemes based on probability distribution," *Mehran University Research Journal of Engineering & Technology*, vol. 38, no. 2, pp. 389–398, 2019.
L. Zhang, H. Ma, S. Tang, and J. Tan, "A combined approximating and interpolating ternary 4-point subdivision scheme," *Journal of Computational and Applied Mathematics*, vol. 349, pp. 563–578, 2019.
S. A. A. Karim, F. Khan, G. Mustafa, A. Shahzad, and M. Asghar, "An efficient computational approach for computing subdivision depth of non-stationary binary subdivision schemes," *Mathematics*, vol. 11, no. 11, p. 2449, 2023.
H. Yang, K. Kim, and J. Yoon, "A family of C^2 four-point stationary subdivision schemes with fourth-order accuracy and shape-preserving properties," *Journal of Computational and Applied Mathematics*, p. 115843, 2024.
P. Ashraf, B. Nawaz, D. Baleanu, K. S. Nisar, A. Ghaffar, M. A. A. Khan, and S. Akram, "Analysis of geometric properties of ternary four-point rational interpolating subdivision scheme," *Mathematics*, vol. 8, no. 3, p. 338, 2020.
A. Shahzad, F. Khan, A. Ghaffar, G. Mustafa, K. S. Nisar, and D. Baleanu, "A novel numerical algorithm to estimate the subdivision depth of binary subdivision schemes," *Symmetry*, vol. 12, no. 1, p. 66, 2020.
G. de Rham, "Sur une courbe plane," *Journal de Mathématiques Pures et Appliquées*, vol. 35, pp. 25–42, 1956.
G. M. Chaikin, "An algorithm for high-speed curve generation," *Computer Graphics and Image Processing*, vol. 3, no. 4, pp. 346–349, 1974.
S. S. Siddiqi and N. Ahmad, "An approximating C^4 stationary subdivision scheme," *European Journal of Scientific Research*, vol. 15, no. 1, pp. 97–102, 2006.
S. S. Siddiqi and K. Rehan, "Modified form of binary and ternary 3-point subdivision schemes," *Applied Mathematics and Computation*, vol. 216, no. 3, pp. 970–982, 2010.
M. Aslam, G. Mustafa, and A. Ghaffar, "(2n-1)-point ternary approximating and interpolating subdivision schemes," *Journal of Applied Mathematics*, vol. 2011, no. 1, p. 832630, 2011.
G. Mustafa, P. Ashraf, and M. Aslam, "Binary univariate dual and primal subdivision schemes," *SeMA Journal*, vol. 65, pp. 23–35, 2014.
G. Mustafa, M. Asghar, S. Ali, A. Qamar, and J. B. Liu, "The family of multiparameter quaternary subdivision schemes," *Journal of Mathematics*, vol. 2021, no. 1, p. 4732464, 2021.
Y. Liu, H. Shou, and K. Ji, "Review of subdivision schemes and their applications," *Recent Patents on Engineering*, vol. 16, no. 4, pp. 50–62, 2022.
A. Viscardi, "Optimized dual interpolating subdivision schemes," *Applied Mathematics and Computation*, vol. 458, p. 128215, 2023.
U. Gulzar, M. J. Iqbal, I. Soomro, and M. A. Wassan, "Geometric modelling of a family of 4-point ternary approximating subdivision scheme u_φ with visual performance," *VFAST Transactions on Mathematics*, vol. 12, no. 1, pp. 290–310, 2024.
R. Mustafa, M. T. Iqbal, G. Mustafa, and S. A. A. Karim, "Advancing curve and surface images modeling with two-parameters polynomial-based quaternary subdivision schemes," *ITM Web of Conferences*, vol. 63, p. 01013, 2024.
N. Dyn, "Analysis of convergence and smoothness by the formalism of Laurent polynomials," in *Tutorials on Multiresolution in Geometric Modelling: Summer School Lecture Notes*, pp. 51–68, 2002.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC-By) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
This work is licensed under a Creative Commons Attribution License CC BY
