Abstract

Subdivision schemes play a vital role in Computer Aided Geometric Design these days. A new univariate symmetric ternary 6-point approximating subdivision scheme has been introduced that generates the limiting curve of C⁵ continuity and its limit functions has a support on (-9, 8). The Laurent polynomial method has been used to investigate the continuity of the subdivision scheme. To determine the maximum degree of smoothness of the subdivision scheme, Holder exponent of the scheme has been calculated. The behavior of the proposed subdivision scheme has been depicted through four examples.

Keywords:

Approximating subdivision scheme, convergence and smoothness, Laurent polynomial, mask, ternary,

References

1. Chaikin, G.M., 1974. An algorithm for high speed curve generation. Comput. Vision Graph., 3(4): 346-349.
   CrossRef    
2. Hassan, M.F. and N.A. Dodgson, 2003. Ternary and Three Point Univariate Subdivision Schemes. Curve and Surface Fitting: Sant-Malo 2002 (Albert Cohen, Jean-Louis Merrien and Larry L. Schumaker, eds. Nashboro Press, Brentwood), pp: 199-208.
   
3. Hassan, M.F., I.P. Ivrissimitzis, N.A. Dodgson and M.A. Sabin, 2002. An interpolating 4-points C2 ternary stationary subdivision scheme. Comput. Aided Geom. D., 19: 1-18.
   CrossRef    
4. Ko, K.P., B.G. Lee and G.J. Yoon, 2007. A ternary 4-point approximating subdivision scheme. Appl. Math. Comput., 190: 1563-1573.
   CrossRef    
5. Rioul, O., 1992. Simple regularity criteria for subdivision schemes. SIAM J. Math. Anal., 23: 1544-1576.
   CrossRef    
6. Siddiqi, S.S. and K. Rehan, 2009. A stationary ternary C4 scheme for curve sketching. Eur. J. Sci. Res., 30(3): 380-388.
   
7. Siddiqi, S.S. and K. Rehan, 2010a. Improved binary four point subdivision scheme and new corner cutting scheme. Comput. Math. Appl., 59: 2647-2657.
   CrossRef    
8. Siddiqi, S.S. and K. Rehan, 2010b. Modified form of binary and ternary 3-point subdivision schemes. Appl. Math. Comput., 216: 970-982.
   CrossRef    
9. Siddiqi, S.S. and K. Rehan, 2010c. A ternary three point scheme for curve designing. Int. J. Comput. Math., 87(8): 1709-1715.
   CrossRef    
10. Zheng, H., Z. Ye, Z. Chen and H. Zhao, 2005. A controllable ternary interpolatory subdivision scheme. Int. J. CAD/CAM, 5(1).
    
11. Zheng, H., Z. Ye, Z. Chen and H. Zhao, 2007. Fractal range of a 3-point ternary interpolatory subdivision scheme with two parameters. Chaos, Soliton. Fract., 32: 1838-1845.
    CrossRef    

Competing interests

The authors have no competing interests.

Open Access Policy

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Copyright

The authors have no competing interests.