The 4-Point α-Ary Approximating Subdivision Scheme
Affiliation(s)
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan. Dept. of Computer Science & Technology, Tsinghua University, Beijing 100084, P. R. China.
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan. Dept. of Computer Science & Technology, Tsinghua University, Beijing 100084, P. R. China.
ABSTRACT
A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.
Cite this paper
A. Ghaffar, G. Mustafa and K. Qin, "The 4-Point α-Ary Approximating Subdivision Scheme," Open Journal of Applied Sciences, Vol. 3 No. 1, 2013, pp. 106-111. doi: 10.4236/ojapps.2013.31B1022.
A. Ghaffar, G. Mustafa and K. Qin, "The 4-Point α-Ary Approximating Subdivision Scheme," Open Journal of Applied Sciences, Vol. 3 No. 1, 2013, pp. 106-111. doi: 10.4236/ojapps.2013.31B1022.
References
[1] G. de Rham, “Un Peude Mathematiques a Proposed Une Courbe Plane,” Revwede Mathematiques Elementry II, Oevred Completes, 1947, pp. 678-689. [2] G. M. Chaikin, “An Algorithm for High-Speed Curve Generation,” Computer Graphics and Image Processing, Vol. 3, No. 4, 1974, pp. 346-349. doi:10.1016/0146-664X(74)90028-8 [3] J. -A. Lian, “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. [4] J. -A. Lian, “On A-ary Subdivision for Curve Design: II. 3-point and 5-point Interpolatory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 3, No. 2, 2008, pp. 176-187. [5] J. -A. Lian, “On a-ary Subdivision for Curve design: III. 2m-point and (2m+1) point Interpolatory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 4, No. 2, 2009, pp. 434-444. [6] G. Mustafa and F. Khan, “A New 4-point Quaternary Approximating Subdivision Scheme,” Abstract and Applied Analysis, Vol. 2009, Article ID 301967, 14 pages. [7] G. Mustafa and A. R. Najma , “The Mask of (2b + 4)-point n-ary Subdivision Scheme,” Computing, Vol. 90, No. 1-2, 2010, pp. 1-14. doi:10.1007/s00607-010-0108-x [8] A. Ghaffar, G. Mustafa and K. Qin, “Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity,” Open Journal of Applied Sciences, Vol. 2, No. 4B, 2012, pp. 48-52. doi:10.4236/ojapps.2012.24B012 [9] M. F. Hassan and N. A. Dodgson, “Ternary and Three-point Univariate Subdivision Schemes,” In: A. Cohen, J. L. Marrien, L. L. Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo2002, Nashboro Press, Brentwood, 2003, pp. 199-208. [10] N. Dyn, “Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials,” In: A.Iske, E. Quak, M. S Floater (Eds), Tutorials on Multiresolution in Geometric Modelling, Springer, 2002, pp. 51-68. doi:10.1007/978-3-662-04388-2_3 [11] G. Mustafa, F. Khan and A. Ghaffar, “The m-Point Approximating Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 30, No. 2, 2009, pp. 138-145. doi:10.1134/S1995080209020061 [12] N. Dyn, M. S. Floater and K. Horman, “A Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extension,” In Mathematical Methods for Curves and Surfaces: Tromso 2004, M. Daehlen, K. Morken, and L. L. Schumaker (eds.), 2005, pp. 145-156. [13] H. Zheng, M. Hu and G. Peng, “P-ary Subdivision Generalizing B-splines,” Second International Conference: On Computer and Electrical Engineering, 2009, pp. 214-218. doi:10.1109/ICCEE.2009.204 [14] A. Ghaffar and G. Mustafa, “A Family of Even-Point ternary Approximating Schemes,” ISRN Applied Mathematics, Vol. 2012, 2012, Article ID 197383, 14 pages. [15] H. Zheng, M. Hu and G. Peng, “Ternary Even Symmetric 2n-point Subdivision,” International Conference on: Computational Intelligence and Software Engineering,2009, pp. 1-4.doi:10.1109/CISE.2009.5363033 [16] K. P. Ko, B. -G. Lee and G. Joon Yoon. “A Ternary 4-point Approximating Subdivision Scheme,” Applied Mathematics and Computation, Vol. 190, 2007 [17] K. P. Ko, “A Quatnary Approximating 4-point Subdivision Scheme,” J. KSIAM, Vol. 13, No. 4, 2009, PP. 307-341. [18] C. Beccari, G. Casciola and L. Romani, “A Non-stationary Uniform Tension Controlled Interpolating 4-point Scheme Reproducing Conics,” Computer Aided Geo-metric Design, Vol. 24, No. 1, 2007, pp. 1-9. [19] G. Casciola and L. Romani. “An Interpolating 4-point c2 Ternarynon-stationary Subdivision Scheme with Tension Control,” Computer Aided Geometric Design, Vol. 24, No. 2, 2007, pp. 210-219. [20] G. Casciola and L. Romani, “Shape Controlled Interpolatory Ternary Subdivision,” Applied Mathematics and Computation, Vol. 215, No. 1, 2009, pp. 916-927. [21] G. Deslauriers and S. Dubic, “Symmetric Iterative Interpolation Process,” Constractive Approximation, Vol. 5, No. 1, 1989, pp. 49-68. doi:10.1007/BF01889598 [22] M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson and M. A. Sabin, “An Interpolating 4-points C2 Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, 2002, pp. 1-18.
[1] G. de Rham, “Un Peude Mathematiques a Proposed Une Courbe Plane,” Revwede Mathematiques Elementry II, Oevred Completes, 1947, pp. 678-689. [2] G. M. Chaikin, “An Algorithm for High-Speed Curve Generation,” Computer Graphics and Image Processing, Vol. 3, No. 4, 1974, pp. 346-349. doi:10.1016/0146-664X(74)90028-8 [3] J. -A. Lian, “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. [4] J. -A. Lian, “On A-ary Subdivision for Curve Design: II. 3-point and 5-point Interpolatory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 3, No. 2, 2008, pp. 176-187. [5] J. -A. Lian, “On a-ary Subdivision for Curve design: III. 2m-point and (2m+1) point Interpolatory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 4, No. 2, 2009, pp. 434-444. [6] G. Mustafa and F. Khan, “A New 4-point Quaternary Approximating Subdivision Scheme,” Abstract and Applied Analysis, Vol. 2009, Article ID 301967, 14 pages. [7] G. Mustafa and A. R. Najma , “The Mask of (2b + 4)-point n-ary Subdivision Scheme,” Computing, Vol. 90, No. 1-2, 2010, pp. 1-14. doi:10.1007/s00607-010-0108-x [8] A. Ghaffar, G. Mustafa and K. Qin, “Unification and Application of 3-point Approximating Subdivision Schemes of Varying Arity,” Open Journal of Applied Sciences, Vol. 2, No. 4B, 2012, pp. 48-52. doi:10.4236/ojapps.2012.24B012 [9] M. F. Hassan and N. A. Dodgson, “Ternary and Three-point Univariate Subdivision Schemes,” In: A. Cohen, J. L. Marrien, L. L. Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo2002, Nashboro Press, Brentwood, 2003, pp. 199-208. [10] N. Dyn, “Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials,” In: A.Iske, E. Quak, M. S Floater (Eds), Tutorials on Multiresolution in Geometric Modelling, Springer, 2002, pp. 51-68. doi:10.1007/978-3-662-04388-2_3 [11] G. Mustafa, F. Khan and A. Ghaffar, “The m-Point Approximating Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 30, No. 2, 2009, pp. 138-145. doi:10.1134/S1995080209020061 [12] N. Dyn, M. S. Floater and K. Horman, “A Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extension,” In Mathematical Methods for Curves and Surfaces: Tromso 2004, M. Daehlen, K. Morken, and L. L. Schumaker (eds.), 2005, pp. 145-156. [13] H. Zheng, M. Hu and G. Peng, “P-ary Subdivision Generalizing B-splines,” Second International Conference: On Computer and Electrical Engineering, 2009, pp. 214-218. doi:10.1109/ICCEE.2009.204 [14] A. Ghaffar and G. Mustafa, “A Family of Even-Point ternary Approximating Schemes,” ISRN Applied Mathematics, Vol. 2012, 2012, Article ID 197383, 14 pages. [15] H. Zheng, M. Hu and G. Peng, “Ternary Even Symmetric 2n-point Subdivision,” International Conference on: Computational Intelligence and Software Engineering,2009, pp. 1-4.doi:10.1109/CISE.2009.5363033 [16] K. P. Ko, B. -G. Lee and G. Joon Yoon. “A Ternary 4-point Approximating Subdivision Scheme,” Applied Mathematics and Computation, Vol. 190, 2007 [17] K. P. Ko, “A Quatnary Approximating 4-point Subdivision Scheme,” J. KSIAM, Vol. 13, No. 4, 2009, PP. 307-341. [18] C. Beccari, G. Casciola and L. Romani, “A Non-stationary Uniform Tension Controlled Interpolating 4-point Scheme Reproducing Conics,” Computer Aided Geo-metric Design, Vol. 24, No. 1, 2007, pp. 1-9. [19] G. Casciola and L. Romani. “An Interpolating 4-point c2 Ternarynon-stationary Subdivision Scheme with Tension Control,” Computer Aided Geometric Design, Vol. 24, No. 2, 2007, pp. 210-219. [20] G. Casciola and L. Romani, “Shape Controlled Interpolatory Ternary Subdivision,” Applied Mathematics and Computation, Vol. 215, No. 1, 2009, pp. 916-927. [21] G. Deslauriers and S. Dubic, “Symmetric Iterative Interpolation Process,” Constractive Approximation, Vol. 5, No. 1, 1989, pp. 49-68. doi:10.1007/BF01889598 [22] M. F. Hassan, I. P. Ivrissimitzis, N. A. Dodgson and M. A. Sabin, “An Interpolating 4-points C2 Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, 2002, pp. 1-18.