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This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2*N* + 4)-point n-ary interpolating curve scheme for *N* ≥ 0 and *n* ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.

There are two general classes of subdivision schemes, namely, approximating and interpolating schemes. The limit curve of an approximating scheme usually does not pass through the control points of control polygon. As the level of refinement increases, the polygon usually shrinks towards the final limit curve. The interpolating schemes are more attractive than approximating schemes because of their interpolation property. All vertices in the control polygon are located on the limit curve of the interpolation scheme, which facilitates and simplifies the graphics algorithms and engineering designs.

Lian generalized the classical binary 4-point and 6-point interpolatory subdivision schemes to a-ary setting for any integer a ≥ 3. After that, the a-ary 3-point and 5-point interpolatory subdivision schemes for curve design for arbitrary odd integer a ≥ 3 [3,4] were introduced. After that, Lian [

The rest of the paper is organized as follows. Section 2 gives some preliminaries results and a new relation for (2N + 4)-point n-ary interpolating curve scheme for closed and open polygon to access main result. Section 3 presents the construction for general formula of the surface case using Lagrange interpolating polynomial, and some characteristics are also discussed. In Section 4, we also give some numerical examples for the visual performance of the proposed work. This work also provides some special cases of the classical subdivision schemes.

Let

where the set

Let

for which

and

where,

Using all the above mentioned identities Ko [

where

and

The free parameters

where

Here, n stands for n-ary subdivision scheme (i.e. n = 2(binary), 3(Ternary), 4(quaternary)···),

Here,

Setting

Here,

Setting

where,

Setting

With

where

When dealing with open initial polygon

where

Example: If an open polygon is refined by using the 6-point ternary interpolating subdivision scheme using (2.10), then two auxiliary points

where

Then, for

For,

Given a set of control points

where,

Given initial values

Let

where

and

Here, some important results for the formulation of required form of tensor product scheme can be verified using (2.3). That is for each

The mask of a subdivision scheme shows the contribution of a single original vertex to each new, subdivided vertex. To find the mask of a scheme, we need to find all ways to get from the origin to each point in the grid. For the tensor product scheme, this is simply the tensor product of the univariate case.

Lemma 3.1. [

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 Doo-Sabin is the generalization of the tensor product of two Chaikin constructions.

Lemma 3.2. [

The general formula which generates the mask

and

where

Here, n, m stands for n-ary, m-ary subdivision schemes respectively (i.e n, m = 2(binary), 3(ternary), 4(quaternary)···),

The free parameter

where

where

As each mask

The tensor product of

where,

Taking

Example: Consider the tensor product of the 4-point DD interpolating subdivision scheme, while DD scheme can be calculated using the result (2.5) mentioned in Section 2. The Laurent polynomial of the scheme is given as

This implies

Since,

Using the result obtained above for the tensor product of interpolatory scheme (3.8), tensor product of 4-point DD scheme can be calculated directly. Since the DD scheme has

then formula (3.8) attains the form,

As each mask

As

Example: A simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [

The modified form of Dyn scheme can be evaluated by setting the value of n = 2, N = υ = 0, and (where) in (2.10) and (2.11), the following refinement rules are obtained

They used the simple tensor product as the basis for the modification of refinement rules of irregular quadrilateral nets. Since it is interpolating scheme, so

Finally, the face point

Instead of taking the tensor product the above rules can be directly obtained by substituting

then the formula (3.8) acquires the form,

At each mask

Using the result (3.4) and (3.5) the constants

Example: Using the results for the interpolating curve subdivision schemes (2.10) the 4-point interpolatory scheme [

Put

Also, from (3.9)

Taking

After calculating the mask from (3.4) and (3.5) and substituting all the results in equation (3.18) following 4-point ternary interpolating tensor product scheme is obtained

Here, the performances of some of the schemes which are deduced from the proposed formula are shown.