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The <i>m</i>-Point Quaternary Approximating Subdivision Schemes
Department of Mathematics, University of the Punjab, Lahore, Pakistan
Centre for Undergraduate Studies, University of the Punjab, Lahore, Pakistan
- 1 Department of Mathematics, University of the Punjab, Lahore, Pakistan
- 2 Centre for Undergraduate Studies, University of the Punjab, Lahore, Pakistan
American Journal of Computational Mathematics·Volume 03 (2013)·Pages 6–10·Published 30 April 2013·DOI10.4236/ajcm.2013.31A002
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Abstract
In this article, the objective is to introduce an algorithm to produce the quaternary m -point (for any integer m>1 ) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.
KeywordsCox-De Boor Recursion FormulaQuaternaryApproximating Subdivision SchemesConvergence and Smoothness
- G. M. Chaikin, “An Algorithm for High Speed Curve Generation,” Graph cuts in Computer Vision, Vol. 3, No. 4, 1974, pp. 346-349.
- N. Dyn, J. A. Gregory and D. Levin, “A 4-Points Interpolatory Subdivision Scheme for Curve Design,” Computer Aided Geometric Design, Vol. 4, No. 4, 1987, pp. 257-268. doi:10.1016/0167-8396(87)90001-X
- N. Dyn, “Tutorials on Multresolution in Geometric Modelling,” In: A. Iske, E. Quak and M. S. Floater, Eds., Summer School Lectures Notes Series: Mathematics and Visualization, Springer, 1995, ISBN: 3-540-43639-1.
- S. S. Siddiqi and M. Younis, “Construction of m-Point Approximating Subdivision Schemes,” Applied Mathematics Letters, Vol. 26, No. 3, 2013, pp. 337-343. doi:10.1016/j.aml.2012.09.016
- C. Beccari, G. Casciola and L. Romani, “A Non-Stationary Uniform Tension Controlled Interpolating 4-Point scheme Reproducing Conics,” Computer Aided Geometric Design, Vol. 24, No. 1, 2007, pp. 1-9. doi:10.1016/j.cagd.2006.10.003
- M. F. Hassan and N. A. Dodgson, “Ternary and Three Point Univariate Subdivision Schemes,” In: A. Cohen, J.-L. Merrien and L. L. Schumaker, Eds., Curve and Surface Fitting: Sant-Malo, Nashboro Press, Brentwood, 2003, pp. 199-208.
- M. F. Hassan, I. P. Ivrissimtzis, N. A. Dodgson and M. A. Sabin, “An Interpolating 4-Point Ternary Stationary Subdivision Scheme,” Computer Aided Geometric Design, Vol. 19, No. 1, 2002, pp. 1-18. doi:10.1016/S0167-8396(01)00084-X
- K. P. Ko, B.-G. Lee and G. J. Yoon, “A Ternary 4-Point Approximating Subdivision Scheme,” Applied Mathematics and Computation, Vol. 190, No. 2, 2007, pp. 1563-1573. doi:10.1016/j.amc.2007.02.032
- G. Mustafa, A. Ghaffar and F. Khan, “The Odd-Point Ternary Approximating Schemes,” American Journal of Computational Mathematics, Vol. 1, No. 2, 2011, pp. 111-118.
- S. R. Buss, “3-D Computer Graphics A Mathematical Introduction with OpenGL,” 1st Edition, Cambridge University Press, New York, 2003. doi:10.1017/CBO9780511804991
- Y. Tang, K. P. Ko and B. G. Lee, “A New Proof of the Smoothness of 4-Point Deslauriers-Dubuc Scheme,” Journal of Applied Mathematics and Computing Vol. 18, No. 1-2, 2005, pp. 553-562.
- C. Conti and K. Hormann, “Polynomial Reproduction for Univariate Subdivision Schemes of any Arity,” Journal of Approximation Theory, Vol. 163, No. 4, 2011, pp. 413-437. doi:10.1016/j.jat.2010.11.002