Abstract
In this paper, we establish new family of odd-point ternary non-stationary approximating subdivision schemes by introducing the \(sine\) function in the Lagrange identities. It is to be observed that the limiting conic sections, generated by the proposed schemes compare to the existing non-stationary approximating schemes, have less deviation from being an exact conic sections. Moreover, proposed family of 3-point ternary schemes with fewer initial control points produces better limiting conic sections than other existing schemes. Furthermore, the proposed family of schemes is non-stationary counterpart of the stationary scheme of Hassan and Dodgson (Ternary and three-point univariate subdivision schemes. Nashboro Press, Brentwood, 2003), Lian (Appl Appl Math 3:176–187, 2008), Siddiqi and Rehan (Int J Comput Math 87:1709–1715, 2009), Siddiqi and Rehan (Appl Math Comput 216:970–982, 2010), Mustafa et al. (Am J Comput Math 1:111–118, 2011), Aslam et al. (J Appl Math 2011:13, 2011) and Conti and Romani (J Math Anal Appl 407:443–456, 2013).
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Ashraf, P., Mustafa, G.: A generalized non-stationary 4-point b-ary approximating scheme. Br. J. Math. Comput. Sci. 4, 104–119 (2014)
Aslam, M., Mustafa, G., Ghaffar, A.: (2n\(-1\))-point ternary approximating and interpolating subdivision schemes. J. Appl. Math. 2011, 13 (2011). Article ID 832630
Conti, C., Romani, L.: Dual univariate m-ary subdivision schemes of de-Rham-type. J. Math. Anal. Appl. 407, 443–456 (2013)
Daniel, S., Shunmugaraj, P.: Some interpolating non-stationary subdivision schemes. Int. Symp. Comput. Sci. Soc. (2011). doi:10.1109/ISCCS.2011.110
Daniel, S., Shunmugaraj, P.: Some non-stationary subdivision schemes. Geom. Model. Imag. (GMAI’07) (2007). doi:10.1109/GMAI.2007.30
Daniel, S., Shunmugaraj, P.: Three point stationary and non-stationary subdivision schemes. In: 3rd International Conference on Geometric Modeling and Imaging (2008). doi:10.1109/GMAI.2008.13
Daniel, S., Shunmugaraj, P.: An approximating \(C^2\) non-stationary subdivision schemes. Comput. Aided Geom. Des. Int. J. 26, 810–821 (2009)
Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)
Dyn, N., Levin, D.: Analysis of asymptotically equivalent binary subdivision schemes. J. Math. Anal. Appl. 193, 594–621 (1995)
Hassan, M.F., Dodgson, N.A.: Ternary and three-point univariate subdivision schemes. In: Cohen, A., Marrien, J.L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 199–208. Nashboro Press, Brentwood (2003)
Lian, J.-A.: On a-ary subdivision for curve design: 3-point and 5-point interpolatory schemes. Appl. Appl. Math. 3, 176–187 (2008)
Ko, K.P., Lee, B.-G., Joon Yoon, G.: A ternary 4-point approximating subdivision scheme. Appl. Math. Comput. 190, 1563–1573 (2007)
Mustafa, G., Ghaffar, A., Khan, F.: The odd-point ternary approximating schemes. Am. J. Comput. Math. 1, 111–118 (2011). doi:10.4236/ajcm.2011.12011
Mustafa, G., Bari, M.: A new class of odd-point ternary non-stationary schemes. Br. J. Math. Comput. Sci. 4, 133–152 (2014)
Siddiqi, S.S., Rehan, K.: A ternary three point scheme for curve designing. Int. J. Comput. Math. 87, 1709–1715 (2009)
Siddiqi, S.S., Rehan, K.: Modified form of binary and ternary 3-point subdivision scheme. Appl. Math. Comput. 216, 970–982 (2010)
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This work is supported by NRPU (P. No. 3183) and Indigenous Ph.D. Scholarship Scheme of Higher Education Commission (HEC) Pakistan.
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Mustafa, G., Bari, M. A new class of odd-point ternary non-stationary approximating schemes. SeMA 68, 29–51 (2015). https://doi.org/10.1007/s40324-015-0031-3
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DOI: https://doi.org/10.1007/s40324-015-0031-3