A Family of 2 n -Point Ternary Non-Stationary Interpolating Subdivision Scheme

This article offers 2 n -point ternary non-stationary interpolating subdivision schemes, with the tension parameter, by using Lagrange identities. By choosing the suitable value of tension parameter, we can get different limit curves according to our own choice. Tightness or looseness of the limit curve depends upon the increment or decline the value of tension parameter. The proposed schemes are the counter part of some existing parametric and non-parametric stationary schemes. The main purpose of this article is to reproduce conics and the proposed schemes reproduce conics very well such that circle, ellipse, parabola and hyperbola. We also establish a deviation error formula which is useful to calculate the maximum deviation of limit curve from the original limit curve. The presentation and of the proposed schemes are verified by closed and open figures. The given table shows the less deviation of the limit curves by proposed scheme as compare to the existing scheme. Graphical representation of deviation error is also presented and it shows that as the number of control points increases, the deviation error decreases.

The important schemes for applications should allow controlling the shape of the limit curve and being capable of reproducing families of curves widely used in Computer Graphics, such as conic sections and polynomials. Initially, stationary subdivision schemes are established but they do not have the capability to produce conics. Later on, the work on non-stationary schemes grows rapidly which can produce conics. Jena et al. [8] proposed 4-point binary non-stationary interpolating scheme. This scheme reproduces elements of the linear space spanned by {1, sin(αx), cos(αx)}. A non-stationary uniform tension controlled interpolating 4-point scheme with a single tension parameter having C 1 continuity was proposed by Beccari et. al. [9]. A 4point ternary interpolating non-stationary scheme spanned by{1, sin(αx), cos(αx)} was proposed by Daniel and Shunmugaraj [10]. Bari and Mustafa [11] proposed a family of 4-point n-ary interpolating scheme. They also worked on odd-point non-stationary interpolating subdivision scheme [12]. Conti et. al. [13] introduced a new equivalence notion between non-stationary subdivision schemes, termed asymptotic similarity, which is weaker than asymptotic equivalence. Novara and Romani [14] defined the building blocks to obtain new families of non-stationary subdivision schemes. Mustafa and Ashraf [15] presented a family of 4-point odd-ary interpolating non-stationary schemes. The common criteria to evaluate the quality of a subdivision scheme are smoothness and shape preserving properties. The idea is to construct a 2n-point (for any integer n>2) ternary interpolating scheme with the ability that the masks of the proposed schemes with suitable tension parameter converge to stationary schemes and preserve the shape of initial polygon due to interpolating behavior. Bari [16] discuss the non-stationary work.
In this paper, Section 2 presents some results which are useful to generate a class of non-stationary ternary interpolating schemes. We proposed 2n-point nonstationary ternary interpolating schemes in Section 3, providing the user with a tension parameter that, when increased within its range of definition, can generate continuous limit curves. It also provides the convergence of proposed interpolating schemes; such schemes repair the draw backs of its stationary analogue [1][2]12] which does not give the possibility to appreciate significant modification, such that the limit curve of stationary subdivision scheme is determined completely by its initial control mesh. So it is not suitable to alter the shape by the scheme itself. Furthermore, a stationary subdivision scheme can't produce conics, which are useful in different applications. Moreover, the limit curves formed by proposed schemes are more accurate because of interpolating behavior of schemes. In particular if the initial control points are equidistant and lie on a circle, the proposed schemes generate circle.
Other conics such that ellipse, parabola and hyperbola are formed by taking the initial data points from their parametric equation and in the result after applying proposed schemes, the limit curve will be ellipse, parabola and hyperbola respectively.

PRELIMINARIES
A ternary univariate subdivision scheme is defined in terms of a mask consisting of a finite set of non-zero The scheme, in compact form, is given by a subdivision rule: 3 1 If the mask a k is independent of k the subdivision scheme k a S corresponding to the mask a k is called stationary otherwise it is called non-stationary.

2N-POINT TERNARY INTERPOLATING SCHEME
In this section, we present general explicit formulae to construct the mask of 2n-point ternary non-stationary interpolating subdivision scheme.    Next, we will prove that the scheme converges and is C 2 Now we introduce the normalized scheme (corresponding to Equation (11)).
The normalized scheme is defined as follows: Note that the sum of coefficients of normalized scheme is equal to one.
Note that the sum of coefficients of normalized scheme is equal to one.

Remark-1:
The general form for the weights of normalized schemes for n>2 can be written as: We present the proof of (i) and the proof of (ii), (iii) and (iv) are similar.
From Lemma-1, we get following lemma.

Lemma-3:
For scheme Equation (13) following inequalities also hold: The proofs of (iii), and (iv) are similar.
From Lemma-3, we get following lemma.

Lemma-4.
For scheme Equation (13) and are independent of k.

Remark-2. From (i-iv) of Lemma-4, we observe that
This means that the mask of the scheme Equation (13) with converge to the mask of the scheme [2].
Similarly, for , in Equations (9-10) and by proving/ using similar inequalities like in Lemma-1 and Lemma-3, we get non-stationary counter part of stationary schemes of [1] respectively.
Proof. We claim that and scheme S a of [2] (also see the Remark-2) From (i) of Lemma-4, it follows that: . Now we will discuss the continuity of 6-point scheme Equation (17). For this first we will prove the following lemmas. Proof of these lemmas is similar to the proof of Lemmas-1-4.
From Lemma-7, we get following lemma.

Lemma-8:
For scheme Equation (17)  and from Lemma-6, following inequalities hold: in Fig 1(a-d) are taken by the parametric equation of ellipse, parabola and hyperbola respectively and limit curve in