This paper is concerned with the regular dynamic loop-like subdivision scheme based on the extension of Box-Spline method. The purpose of the addressed problem is to prove the convergence of the subdivision scheme and verify the

It is well known that the subdivision method is a powerful tool in the fields of free-form surface modeling and tensor product surfaces for a long time. Starting from an arbitrary initial control mesh, subdivision schemes can produce a sequence of finer and finer meshes converging to a originally described surface. If the sequence of control nets converges in a certain sense, such procedure can be used to generate surfaces. The subdivision operations are efficient and could be well applied to arbitrary topology polygon meshes. In addition, there exist adequate theoretical tools for analyzing its convergence and continuity. Therefore, the subdivision method has become a standard technique in both academic and industrial communities.

In recent decades, there has been a tremendous progress in scheme construction [

Vertex updating rule: for every original vertex, a new vertex is calculated by using the suitable coefficients for 1-neighbor control points as shown in Figures

Edge splitting rule: for every edge in the original mesh, a new vertex is calculated by using the masks as shown in Figure

Face splitting rule: every triangle in the original mesh produces six new vertices, three from original vertices and the others from original edges. These six vertices are constructed into four new triangles.

Masks for loop scheme with

The common feature of these methods lies in their parameters which are fixed in each step of subdivision operation, which is called as stationary subdivision. Unfortunately, since the shape of a stationary subdivision surface is totally determined by control meshes, it is not convenient to add further control except mesh modification. Hence, some nonstationary subdivision schemes should be introduced; for example, the authors in [

On the other hand, the convergence is an important topic when studying subdivision surfaces. Doo and Sabin [

Motivated by the above discussion, we will investigate the subdivision method for triangle mesh which can represent a revolving surface exactly. Firstly, a new subdivision scheme based on Box-Splines is proposed, and the subdivision matrix is constructed. Secondly, the characteristic spectrum of global subdivision matrix is analyzed, and the detailed analysis for

The main contribution of this paper is summarized as follows. (1) A novel subdivision scheme with a stronger modeling ability is proposed based on the new extended Box-Splines. (2) The

In this section, we will introduce the regular dynamic loop-like subdivision scheme. Similar to the loop scheme, after one step of subdivision, the number of irregular vertices is fixed. If the mesh is further subdivided, irregular vertices will be isolated. In other words, each face contains at most one irregular vertex. In the following, we will assume that the sufficient subdivision steps have finished to generate the local subdivision structure just as shown in Figure

The subdivision stencil for continuity analysis.

For L-level neighbors,

Similar to the loop subdivision schemes [

The subdivision matrix has the following properties.

The sum of elements of each row is equal to 1.

The matrix is cyclic symmetry, that is,

Based on the property (2), (

Based on the

It is known that the characteristic spectrum of

For the block circulant matrix

Let

Denote

Then, we have the following Lemma.

The matrices

The above lemma is easy to be verified (see [

The eigenvalues of the subdivision matrix

Let

Hence, the eigenvalues of the submatrix

Then, we can obtain the eigenvalues of

In this section, we mainly focus on the convergence of the subdivision case around irregular vertices. At the regular vertices, the generated surfaces have similar properties as the

In this subsection, we will analyze the convergence of regular dynamic loop-like subdivision scheme. The subdivision process can be expressed as

Define

When

If subdivision step

Under an identical transformation to

First of all, let

By (

Secondly, let

For

Denote

Using the standard results in a numerical analysis, we get

Then, by (

Similarly, we have

By Theorem

Substituting (

Then we get

Firstly, we need to prove the following lemma.

Let

Since

The eigenvalues of

Since

Denote

(1) If

By Lemma

(2) If

If the eigenvalues of matrix

Since each row sum of

(1) We prove the (

Let

In order to prove that the absolute value of

(a) For the inequality (

(b) For the inequality (

When

When

To Sum up, we have

(c) For the inequality (

From

Then, the eigenvalues of

(2) We verify (

Since

Now, the proof of Theorem

If the kernel function

Let

Since

Let

Given any two points whose coordinates are

Based on the above result, we analyze the continuity of loop-like subdivision surface.

If the kernel function

We will verify that there exists a common tangent plane around vertex

Given any three points whose coordinates are

According to [

The regularity and injectivity of this map can be judged from the triangulation obtained from the control points by some steps subdivision [

Visualization of characteristic maps: (a)–(d) control nets from the eigenvectors of

In this section, we show several examples of subdivision surface generated by our schemes. Figure

Loop-like subdivision examples.

Control net pattern and compensation mask for the revolving surface.

Modeling examples of the revolving part.

This paper has presented a regular dynamic loop-like subdivision scheme that can be regarded as an extension of the

In the present paper, we only consider the continuity of the isotropic schemes, and some more general cases should be further investigated. Moreover, only numerical discussions are presented to demonstrate the

The research is partially supported by the National Natural Science Foundation of China under Grant nos. 61272297, 11226146, the Doctoral Fund of Ministry of Education of China under Grant no. 20090450613, and the Natural Science Foundation of the Jiangsu Higher Education Institutions under Grant no. 12KJB120002.