The Odd-Point Ternary Approximating Schemes


We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.

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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. doi: 10.4236/ajcm.2011.12011.

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The authors declare no conflicts of interest.


[1] C. Beccari, Gcasiola and L. Romani, “An Interpolating 4-Point Ternary Non-Stationary Subdivision Scheme with Tension Control,” Computer Aided Geometric Design, Vol. 24, No. 4, 2007, pp. 210-219. doi:10.1016/j.cagd.2007.02.001
[2] N. Dyn, “Interpolatory Subdivision Schemes and Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials,” In: A. Iske, E. Quak and M. S Floater, Eds., Tutorials on Multiresolu-tion in Geometric Modeling, Springer, Dordrecht, 2002, pp. 51-68 (Chapter 2 and 3).
[3] N. Dyn, K. Hormann, M. A. Sabin and Z. Shen, “Polynomial Reproduction by Symmetric Subdivision Schemes,” Journal of Approximation Theory, Vol. 155, No. 1, 2008, pp. 28-42. doi:10.1016/j.jat.2008.04.008
[4] G. Mustafa, and A. R. Na-jma, “The Mask of - Point -ary Subdivision Scheme,” Computing, Vol. 90, No. 1-2, 2010, pp. 1-14. doi:10.1007/s00607-010-0108-x
[5] M. F. Hassan, “Further Analysis of Ternary and 3-Point Univariate Subdivision Schemes,” Technical Report 599, University of Cambridge Computer Laboratory, ISSN 1476-2986, 2004.
[6] M. F. Hassan, I. P. Ivrissimitzis, 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
[7] K. Hormann and M. A. Sabin, “A Family of Subdivision Schemes with Cubic Pre-cision,” Computer Aided Geometric Design, Vol. 25, No. 1, 2008, pp. 41-52. doi:10.1016/j.cagd.2007.04.002
[8] J.-A. Lian, “On a-ary Subdivision for Curve Design: I. 4-Point and 6-Point Interpo-latory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 3, No. 1, 2008, pp. 18-29.
[9] J.-A. Lian, “On a-ary Subdivision for Curve De-sign: II. 3-Point and 5-Point Interpolatory Schemes,” Applica-tions and Applied Mathematics: An International Journal, Vol. 3, No. 2, 2008, pp. 176-187.
[10] J.-A. Lian, “On a-ary Subdi-vision for Curve Design: III. -Point and -Point Interpolatory Schemes,” Applications and Applied Mathematics: An International Journal, Vol. 4, No. 2, 2009, pp. 434-444.
[11] F. Khan and G. Mustafa, “Ternary Six-Point Interpolating Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 29, No. 3, 2008, pp. 153-163.
[12] K. P. Ko, B.-G. Lee and G. 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
[13] A. Levin, “Polynomial Generation and Quasi-Interpolation in Stationary and Non-Uniform Subdivision Schemes,” Computer Aided Geo-metric Design, Vol. 20, No. 1, 2003, pp. 41-60. doi:10.1016/S0167-8396(03)00006-2
[14] D. Levin, “Using Laurent Polynomial Representation for the Analysis of Non-Uniform Binary Subdivision Schemes,” Advances in Computational Mathematics, Vol. 11, No. 1, 1999, pp. 41-54. doi:10.1023/A:1018907522165
[15] G. Mustafa and F. Khan, “A New 4-Point Quaternary Approximating Subdivision Scheme,” Abstract and Applied Analysis, Vol. 2009, 2009, Article ID 301967. doi:10.1155/2009/301967
[16] M. Sabin, “Eigenanalysis and Artifacts of Subdivision Curves and Surfaces, Tutorials on Multiresolution in Geometric Modeling,” In: A. Isle, E. Quak and M. S. Floater, Eds., Springer, Berlin, 2002, pp. 69-92.
[17] S. S. Siddiqi and K. Rehan, “Modified Form of Binary and Ternary 3-Point Subdivision Scheme,” Applied Mathematics and Computation, Vol. 216, No. 3, 2010, pp. 970-982. doi:10.1016/j.amc.2010.01.115
[18] Y. Tang, K. P. Ko and B.-G. Lee, “A New Proof of Smoothness of 4-Point Deslauriers-Dubic Scheme,” Journal of Applied Mathematics and Computing, Vol. 18, No. , 2005, pp. 553-562.

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