%0 Journal Article
%T A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability
%A Ghulam Mustafa
%A Jayyada Irum
%A Mehwish Bari
%J ISRN Computational Mathematics
%D 2012
%R 10.5402/2012/924839
%X A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included. 1. Introduction Geometric modeling plays a significant role to cover up the gap between computer and industry. It has a pivotal importance in the fields of aircraft manufacturing, automobile industry, and general product design. One of the most important tools of computer aided geometric design is ¡°Subdivision.¡± Subdivision is a well flourished field. It is a process of taking unrefined shape and to polish it up to produce another shape that is more visually tempting. Due to the comprehensibility and simplicity of this method, it is used in the fields of 3D geometrical measurement, computer graphics, computer animation, and computer aided geometric design. In 1986, Dubuc [1] presented a interpolation through an iterative scheme. Dyn et al. [2] introduced a 4-point interpolating subdivision scheme for curve design. Later on, Deslauriers and Dubuc [3] introduced a symmetric iterative interpolation process. Weissman [4] also offered a 6-point interpolating scheme in 1990. In 2002, Hassan et al. [5, 6] gave ternary three-point and 4-point interpolatory schemes. Further analysis of ternary three-point univariate scheme was given in technical report by Hassan and Dodgson [7] in 2004. Dyn [8] has given the analysis of the convergence and smoothness of interpolating and approximating schemes by Laurent¡¯s polynomial method. In 2007, Beccari et al. [9] presented an interpolating 4-point ternary nonstationary scheme with tension control. They also offered a nonstationary uniform tension controlled interpolating 4-point scheme reproducing conics [10] in 2007. Ko [11] in his Ph.D. thesis presented a detailed study on subdivision scheme. Zheng et al. [12] presented the method to find the differentiability of a four-point ternary scheme. Lian [13] extended 3-point and 5-point interpolating schemes into -ary subdivision scheme for curve design. Conti et al. [14] derived symmetric subdivision masks of the Hurwitz type to the interpolating scheme masks. In this paper, we present a new 5-point ternary interpolating subdivision scheme with one parameter. 2. Preliminaries Let , , denote a sequence of points
%U http://www.hindawi.com/journals/isrn.computational.mathematics/2012/924839/