Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T06:55:36.074Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized and Unified Families of Interpolating Subdivision Schemes

Published online by Cambridge University Press:  28 May 2015

Ghulam Mustafa ,
Pakeeza Ashraf  and
Jiansong Deng
Ghulam Mustafa*
Affiliation:
Department of Mathematics, The Islamia University of Bahawalpur, Pakistan
Pakeeza Ashraf*
Affiliation:
University of Science and Technology of China, P.R. China
Jiansong Deng*
Affiliation:
University of Science and Technology of China, P.R. China
*
Corresponding author.Email address:[email protected], [email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

We present generalized and unified families of (2n)-point and (2n — 1)-point p-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. Moreover error bounds between limit curves and control polygons of schemes are also calculated. It has been observed that error bounds decrease when complexity of the scheme decrease and vice versa. Furthermore, error bounds decrease with the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme.

Keywords

65D1765D0765D05Interpolating subdivision schemeeven-ary schemesodd-ary schemesLagrange polynomialparameterserror boundstensor product
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 2 , May 2014 , pp. 193 - 213
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aslam, M., Mustafa, G., and Ghaffar, A., (2n-1)-point ternary aproximating and interpolating subdivision schemes, Journal of Applied Mathematics, (2011), Article ID 832630, 12 pages.Google Scholar
[2] Deslauriers, G. and Dubuc, S., Symmetric iterative interpolation processes, Constructive Approximation, vol. 5 (1989), pp. 49–68.Google Scholar
[3] Dyn, N., Levin, D. and Gregory, J., A 4-point interpolatory subdivision scheme for curve design, Computer Aided Geometric Design, vol. 4, no. 4 (1987), pp. 257–268.Google Scholar
[4] Dyn, N. and Levin, D., Subdivision scheme in the geometric modling, Acta Numerica, vol. 11 (2002), pp. 73–144.Google Scholar
[5] Hassan, M. F. and Dodgson, N. A., Ternary and three-point univariate subdivision schemes, in: Cohen, A., Marrien, J.L., Schumaker, L.L. (Eds.), Curve and Surface Fitting: Sant-Malo 2002, Nashboro Press, Brentwood, (2003), pp. 199–208.Google Scholar
[6] Hassan, M. F., Ivrissimitzis, I. P., Dodgson, N. A. and Sabin, M. A., An interpolating 4-point C2 ternary stationary subdivision scheme, Computer Aided Geometric Design, vol. 19 (2002), pp. 1–18.Google Scholar
[7] Zhang, H. and Wang, G., Semi-stationary subdivision operators in geometric modeling, Progress of Natural Science, vol. 12 (2002), pp. 772–776.Google Scholar
[8] Lian, Jian-Ao, On a-ary subdivision for curve design: I. 4-point and 6-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, vol. 3, no. 1 (2008), pp. 18–29.Google Scholar
[9] Lian, Jian-Ao, On a-ary subdivision for curve design: I. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, vol. 3, no. 2 (2008), pp. 176–187.Google Scholar
[10] Lian, Jian-Ao, On a-ary subdivision for curve design: I. 2m-point and (2m + 1)-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, vol. 4, no. 1 (2009), pp. 434–444.Google Scholar
[11] Ko, K. P., Lee, B. G. and Yoon, G. J., A study on the mask of interpolatory symmetric subdivision schemes, Applied Mathematics and Computation, vol. 187 (2007), pp. 609–621.CrossRefGoogle Scholar
[12] Mustafa, G. and Ashraf, P., A new 6-point ternary interpolating subdivision scheme and its differentiability, Journal of Information and Computing Science, vol. 5, no. 3 (2010), 199–210.Google Scholar
[13] Mustafa, G. and Khan, F., A new 4-point C3 quaternary approximating subdivision scheme, Abstract and Applied Analysis, (2009), Article ID: 301967, 14 pages.Google Scholar
[14] Mustafa, G. and Rehman, N. A., The mask of (2b + 4)-point n-ary subdivision scheme, Computing. Archieves for Scientific Computing, vol. 90 (2010), pp. 1–14.Google Scholar
[15] Mustafa, G. and Hashmi, M. S., Subdivision depth computation for n-ary subdivision curves/surfaces, The Visual Computing, vol. 26 (2010), pp. 841–851.Google Scholar
[16] Mustafa, G., Deng, J., Ashraf, P. and Rehman, N. A., The mask of odd point nary interpolating subdivision scheme, Journal of Applied Mathematics, (2012), Article ID: 205863, 20 pages.Google Scholar
[17] De RHAM, G., Sur une courbe plane, Journal de Mathmatiques Pures et Appliques, vol. 35 (1956), pp. 25–42.Google Scholar
[18] Zheng, H., Hu, M. and Peng, G., p-ary subdivision generalizing B-splines , 2009 Second International Conference on Computer and Electrical Engineering, DOI: 10.1109/IC-CEE.2009.204.Google Scholar
[19] Zheng, H., Hu, M., and Peng, G., Constructing (2n-1)-point ternary interpolatory subdivision schemes by using variation of constants, International Conference on Computational Intelligence and Software Engineering (CISE 2009), (2009), pp. 1–4.Google Scholar
[20] Zheng, H., Hu, M. and Peng, G., Ternary even symmetric 2n-point subdivision, International Conference on Computational Intelligence and Software Engineering (CISE 2009), DOI: 10.1109/CISE.2009.5363033.Google Scholar
[21] Zorin, D. and Schröder, P., A unified framework for primal/dual quadrilateral subdivision schemes, Computer Aided Geometric Design, vol. 18 (2001), pp. 429–454.CrossRefGoogle Scholar