A Family of 4-Point n -Ary Interpolating Scheme Reproducing Conics

The n -ary subdivision schemes contrast favorably with their binary analogues because they are capable to produce limit functions with the same (or higher) smoothness but smaller support. We present an algorithm to generate the 4-point n -ary non-stationary scheme for trigonometric, hyperbolic and polynomial case with the parameter for describing curves. The performance, analysis and comparison of the 4-point ternary scheme are also presented.


Introduction
Subdivision is a method for making smooth curves/surfaces, which first emerged an addition of splines to arbitrary topological control nets.Effectiveness of subdivision algorithms, their flexibility and ease make them appropriate for many relative computer graphics applications.The schemes generating curves are considered to be the basic tools for the design of schemes generating surfaces.
A general form of univariate n-ary subdivision scheme S which maps a control polygon  of coefficients is called mask of the subdivision scheme.The set of coefficients

 
: determines the subdivision rule at level and is termed as the mask at -th level.If the mask is independent of , namely if , the subdivision scheme is called stationary otherwise it is called non-stationary.Sometimes, in computer graphics and geometric modeling, it is required to have schemes to construct circular parts or parts of conics.It seems that (linear) stationary schemes cannot generate conics and non-stationary schemes have such a capability to generate trigonometric polynomials, trigonometric splines and, in particular, circles, ellipses and so on.Such schemes are useful in computer graphics and geometric modeling.Successful efforts have been made to establish approximating and interpolating non-stationary schemes which can provide smooth curves and reproduce circle or some trigonometric curves.
The theoretical bases regarding non-stationary schemes are derived from the analysis of stationary schemes.Jena et al. [1] worked on 4-point binary non-stationary subdivision scheme for curve interpolation.Yoon [2] presented the analysis of binary non-stationary interpolating scheme based on exponential polynomials.Beccari et al.
[3] worked on 4-point binary non-stationary uniform tension controlled interpolating scheme reproducing conics.Daniel and Shunmugaraj [4] presented 4-point ternary non-stationary interpolating subdivision scheme.In this paper, we present an algorithm to construct 4-point n-ary scheme.For simplicity, we have discussed and analyzed 4-point ternary scheme.
This paper is organized as follows.Section 2 presents the construction of 4-point n-ary non-stationary interpolating subdivision schemes.As an example, 4-point ternary scheme is presented in this section.Section 3 provides the smoothness of proposed schemes.In the last section conclusion and visual performance of proposed schemes are presented.

Construction of 4-Point n-Ary Scheme
Here we suggest the following algorithm to construct the non-stationary n-ary 4-point   p i  at level k and get system of linear equations by interpolating. The data k i h p  corresponding to the abscissas , 1 ,0,1 ,  Solve the system of linear equations by any well known method to get the values of unknowns.
 Evaluate the interpolating function  

Ternary 4-Point Interpolating Scheme
Given a set of control points at level , using above algorithm, we define a unified ternary 4-point interpolating scheme that makes a new set of control points by the rule: where the parameter 1 k   can easily be updated at each subdivision step through following equation Therefore, given parameter , k  the subdivision rules are achieved by first computing 1 k   using Equation (2.2) and then by substituting depending on the choice of the parameter, we get different schemes.For and 1 in (2.1), we can generate following schemes exact for trigonometric (2.3), hyperbolic (2.4) and polynomial (2.5) respectively.
where 1 1 , after replacing sin and cos functions by sinh and cosh functions in 1 2 , , Remark 2.1.The scheme (2.3) and (2.4) can be considered as a non-stationary counterpart of the DD stationary scheme [5] i.e. scheme (2.5) because, the masks of the schemes (2.3) and (2.4) converge to the mask of scheme (2.5):

Smoothness Analysis
The subdivision scheme given in the previous subsection, the coefficients   1,2,3,4 The mask of scheme (2.3) satisfies following inequalities for sufficiently large .
The following two Lemmas are the consequence of previous Lemmas.
Proof.By (2.1), we have It can be easily verified that Lemma 3.6.
The stationary scheme   Note that This work is supported by the Indigenous Ph.D Scholarship Scheme of Higher Education Commission (HEC) Pakistan.
By Lemma 3.3, we can also show that

Conclusions
To the aim of reproducing conic sections, we introduce an algorithm for generation of 4-point n-ary interpolating scheme.In particular, we define 4-point interpolating scheme that unifies three different curves schemes which are capable of representing trigonometric, hyperbolic and polynomial functions.

Figure 1 .
Figure 1.Dotted lines indicate the initial closed and open polygons.Solid continuous curves are generated by proposed ternary interpolating scheme for trigonometric case.