A Unified Interpolating Subdivision Scheme for Curves / Surfaces by Using Newton Interpolating Polynomial

This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.


Introduction
Subdivision schemes have become important in recent years because of giving a specific and proficient way to describe smooth curve/surfaces.It is an algorithm method to generate smooth curve/surfaces as a sequence of successively refined polyhedral meshes.Their beauty lies in the elegant mathematical formulation and simple implementation.The flexibility and simplicity of subdivision schemes become more appropriate in computer and industrial applications.
There are two general classes of subdivision scheme namely interpolating and approximating.If the limit curve/ surface approximates the initial control polygon and that subdivision, the newly generated control points are not in the limit curve/surface, the scheme is said to be approximating.It is called interpolating if after subdivision, the control points are interpolated on the limit curve/ surface.Among interpolating subdivision scheme 4-point interpolating scheme [1] was one of the initial scheme.Nowadays spacious mixture of interpolating scheme [2][3][4][5][6][7][8] has been anticipated in the literature with different shape parameters.
In 1978, Catmull-Clark [9] and Doo-Sabin [10] first introduced subdivision surface schemes, which generalised the tensor product of bicubic and biquadratic Bsplines respectively.After that, Kobbelt [11] gave the tensor product of the curve case and he generalized the four-point interpolatory subdivision scheme for curve to the surface by using tensor product.
The proposed work gives a new idea in finding subdivision rules for curves and surfaces using Newton interpolating polynomial.The proposed method is simple and avoids complex computation when deriving subdivision rules.Since higher arity subdivision schemes have high approximation order and lower support than their counterpart of lower arity schemes.Therefore researchers are focusing in introducing higher arity schemes (i.e., ternary, quaternary, ..., n-ary).This paper presents a general formula for (2m + 2)-point n-ary interpolating rules for curves.Since the subdivision schemes for surface design have gained more popularity in computer animation industries.So, a new approach for regular quad meshes using 2-dimensional Newton interpolating formula is also the part of this paper.
In the following section, there is presented a brief introduction about the preliminary concepts used in this work.In Sections 3 and 4, new formula for interpolating subdivision schemes is given for curves and surfaces by using Newton interpolating polynomial.In Section 5, application of the subdivision schemes is also accessible.A few remarks and conclusions are given in Section 6.

Preliminaries
Given a sequence of control points , , 1 where the upper index 0 k  indicates the subdivision level.An n-ary subdivision curve is defined by where m > 0 and , 0 1, 0,1,..., 1. 2) The set of coefficients   , j 0 , 0,1,..., 1 called subdivision mask.In the limit the process (2.1) defines an infinite set of points in The sequence of control points is connected, in a natural way, with the diadic mesh points The process then defines a scheme whereby and replaces the values and at the mesh points S and while is inserted at the new mesh points Labeling of old and new points is shown in Figure 1, which illustrates subdivision scheme (2.1). Let m be the space of all polynomials of degree wh er e m i s no n-n eg a t iv e in t eg er .I f In general, the coefficient of the Newton form of polynomial is called divided difference, the divided difference 0 n is a symmetric function, hence can be found by following method, [x ,..., x ], and can originate by the subsequent way, Ν (x)

Construction of the Subdivision Scheme for Univariate Case
This section gives the construction of (2m + 2)-point binary and ternary interpolating schemes.Then by induction, a general formula for (2m + 2)-point n-ary interpolating subdivision scheme is formulated for curve case.

(2m + 2)-Point Binary Interpolating Scheme
To construct the rules for binary 2-point interpolating scheme, consider be the Fundamental Newton polynomial to the node points {0, 1}.The Newton polynomial replicate linear polynomial P in the way that taking m = 0 in (2.3), we achieve where j is divided difference can be calculated by (2.4), and by setting in (2.5).This implies that with following Gamma function Now, to construct the desired 2-point ternary subdivision scheme, let Since we want to construct uniform and stationary scheme reproducing polynomials up to a fixed degree, it is sufficient to consider the case i = 0 with subdivision level k = 0.This implies that Now as an affine combination of 2-point we suppose that at (k + 1)th level, the point the desired binary 2-point interpolating subdivision scheme is given by In composite form (3.3) can be written as where Continuing in the same way for m = 1 in (2.3), where be the Newton polynomial to the node points {−1, 0, 1, 2} then we have the following compact form of 4-point binary subdivision scheme Consequently, we can generate a general form of (2m + 2)-point binary interpolating scheme, which is of the following form and subdivision level 0. k 

(2m + 2)-Point Ternary Interpolating Scheme
To construct the rules for ternary 4-point interpolating scheme, consider x   be the Newton polynomial to the node points {−1, 0, 1, 2}.The Newton polynomial reproduces cubic polynomial P in the way that taking m = 1 in (2.3), we achieve .
Now by using (2.4) and (2.5) in (3.5), we have Now to construct the 4-point ternary subdivision scheme, take   we attain the following iterative rules for ternary 4-point interpolating subdivision scheme, In composite form, the above rules can be written as Accordingly, general formula for (2m + 2)-point ternary interpolating scheme is given by Copyright © 2013 SciRes.

OJAppS
where and subdivision level 0. k 

(2m + 2)-Point n-Ary Interpolating Subdivision Scheme (Generalization)
Now there is presented a general formula for (2m + 2)point n-ary (i.e.binary, ternary and so on) interpolating subdivision scheme by using Newton interpolating polynomial.This new formula will be helpful to drive interpolating subdivision rule plainly and quickly.The general formula for (2m + 2)-point n-ary interpolating subdivision scheme has the following form where 0 and indicates 0 k  the subdivision level, where n stand for n-ary scheme.

Tensor Product of (2m + 2)-Point n-Ary Interpolating Subdivision Scheme
Given a sequence of control points , , where the upper index indicates the subdivision level.An n-ary subdivision surface scheme in the tensor product form is defined by where  

) defines an infinite set of points in
The sequence of values is related, in a natu- ral way, with the diadic mesh points , , , The process then defines a scheme whereby 1 , replaces the value at the mesh point are inserted at the new mesh points 1 , old and new points is shown in Figure 2 which illustrates subdivision scheme (4.1).
Here, we present a general formula for tensor product of (2m+2)-point n-ary interpolating scheme in the following form, where Copyright © 2013 SciRes.OJAppS Here, 27 81 indicate the subdivision level and .
k 4.1.In the fo at som he bivariate interp es come from our pr ).  For obtaining ivision scheme, sub in ( , 2 n n  Remar llowing, it is to be noted th e of t olating subdivision schem  For obtaining tensor product of ternary 4-point interpolating scheme, taking the values       Remark 4.2.It can be loosely say that the support is the tensor product of the supports of the two regions, just as one can loosely say that Kobbelt subdivision scheme for surface [11] is the generalization of the tensor prod ct 4-point DD subdivision scheme [12]. Lemma 4.2.[15] Given initial control polygon let the values be desubdivision p gether with (2.2), then if a scheme is derived from a tensor product, then the level of continuity can be determined between pieces by reference to the underlying basis functio continuity as their cou

Conclusion
This work gives a variety of subdivision schemes for the univariate and bivariate cases by using Newton's interpolating formula.The work presented here is a new approach to the subdivision rules, which reduce the com putational cost.Most of the well-known subdivision schemes are the special cases of the proposed work (3.13) and (4.2).

Figure 2 .
Figure 2. Solid lines show one face of coarse polygons whereas dotted lines are refined polygons.(a)-(c) can be obtained by subdividing one face into four, nine and sixteen new faces by using (4.1) for n = 2, 3, 4 respectively.

Lemma 4 . 1 .
[14] Given initial control polygon let the values be desubdivision p gether with (2.2), then the schemes derived by tensor product naturally get four-sided support regions.
ns, i.e., all the tensor product schemes have the same nterparts.
This section is devoted for the visual performance of curves/surfaces.It is illustrated by some examples, obtained from the proposed work (3.13) and (4.2).The stepwise subdivision effects are shown in Figures3 and 4 .