1. Introduction
Due to the use of techniques that are simple and easy to handle, the subdivision method is widely used for generating smooth curves from original data points, and this method plays an important role in computer-aided geometric design, computer-aided design, and image processing.
According to the relationship of the original points and the limit curves, the two categories we are going to divide them into are interpolation subdivision [
1,
2,
3,
4,
5,
6] and approximation subdivision [
7,
8,
9,
10]. The limit curve of the interpolation subdivision scheme is through all original points, which can protect the shape of the original control polygon. In 2002, Hassan et al. [
2] presented a ternary 4-point interpolating subdivision scheme that generates a
limiting curve for the tension parameter satisfied with
. In 2010 and 2012, Mustafa et al. [
3,
4] introduced 6-point and 5-point ternary interpolating schemes with a shape parameter in succession, and proved the limit curves are
or
and continue for a certain range of parameter
. In 2012 and 2013, Siddiqi and Rehan [
5,
6] proposed two schemes of ternary 4-point interpolating subdivision in which the limiting curve is
or
continuous. The above schemes have the following common characteristics: the masks in the scheme are all simple linear combinations of parameters, and when the limit curve is
or
continuous, the selection range of parameters is small, so it is impossible to know what will happen to the limit curves at other infinite intervals.
Compared with interpolation subdivision, the limit curve of the approximation subdivision scheme did not continue through the original points, but this scheme had less support and the limit curve had higher smoothness. In 2004, Hassan and Dodgson [
7] derived a ternary 3-point approximating subdivision scheme that generates the
limiting curve. In 2007, Ko et al. [
8] introduced an improved ternary 4-point approximating subdivision scheme derived from cubic polynomial interpolation, and used similar methods to generalize ternary
point approximating subdivision schemes. In 2012, Ghaffar and Mustafa [
9] investigated a general formula to generate the family of an even-point ternary approximating subdivision scheme with a shape parameter. In 2015, Rehan and Siddiqi [
10] proposed a ternary 4-point approximating subdivision scheme that generates the limiting curve of
continuity. These experimental results show that when the continuity was higher, the limiting curve of the approximation scheme deviated further from the original points.
This motivated us to present a ternary interpolation scheme with high smoothness and more degrees of freedom for the curve design. The proposed scheme not only provides the mask of 4-point schemes, but also generalizes and unifies several well-known schemes.
Furthermore, the subdivision scheme is important not only in the geometric design of smooth curves, but also in the construction of irregular shapes. Zheng et al. [
11,
12] proved that the limit curves generated by binary 4-point and ternary 3-point interpolation subdivision schemes are fractals. Siddiqi et al. [
13,
14] described the fractal behavior of ternary 4-point interpolation subdivision schemes.
In this paper, the fractal behavior of the ternary 4-point rational interpolation subdivision scheme is investigated and analyzed. Through examples, it was found that when the parameter selection is close to the singular point, the limit curve pattern is more turbulent. As the parameter value becomes far from the singular point, the limit curve becomes smoother and finally reaches and continuity.
2. Preliminaries
A general ternary subdivision scheme
S with the initial values
recursively defines new discrete values as follows:
where the set
of coefficients is called the mask of the scheme. A necessary condition for uniform convergence of the subdivision scheme (
1) is that
The
Z-transform of the mask a of subdivision scheme can be given as
which is called the symbol or Laurent polynomial.
A subdivision scheme is said to be uniformly convergent if for every initial data
, there is a continuous function
f such that for any closed interval
As a result,
f is regarded as the limit function of the subdivision scheme, and is denoted
.
In 2002, Hassan et al. [
2] provided a sufficient and necessary condition for a uniform convergent subdivision scheme. Firstly, they used matrix formalism to derive necessary conditions for a scheme to be
based on the eigenvalues of the subdivision matrix. If the limiting curve is
continuity, the eigenvalues
satisfy:
Secondly, a subdivision scheme S is uniform convergent if and only if there is an integer , such that . Then, the subdivision scheme is uniform convergent.
This paper is organized as follows. In
Section 3, a ternary 4-point rational interpolating subdivision scheme is presented. The continuity analyses are in
Section 4. In
Section 5, the fractal behavior of subdivision schemes is introduced. In
Section 6, examples are considered to demonstrate the role of the parameter. Conclusions are drawn in
Section 7.
5. Fractal Behavior
In this paper, the fractal behavior of a ternary 4-point interpolation subdivision scheme is developed and analyzed.
The original data points are
. Let
be the set of control points at level
k, and
satisfies the scheme (
5). We need to analyze the effect of the parameter
on the sum of the length of all the small edges between two arbitrary fixed control points
and
after
k subdivision steps. For simplicity, we only analyze the effect between two initial points
and
.
According to the subdivision scheme (
5), it is known that
, where
and:
We defined the three edge vectors as:
Let
,
and
, and we can get
. Since
,
can be written as:
Equations (
13) and (
14) are non-homogeneous difference equations to be solved simultaneously. Since
gives the special solution:
where
,
, and
Since
and
, it follows that
Using Equations (
15), (
16), and (
17), yields:
or
and the corresponding characteristic equation is:
When
, the roots of (
21) are
,
, and the solution of Equation (
16) is
where
are the linear combinations of
.
Similarly, the solution of Equation (
17) is:
where
and
are the linear combinations of
.
The solution of Equation (
18) is:
where
are the linear combinations of
.
Theorem 3. If and , then the limit curve of the subdivision scheme (5) is a fractal curve. Proof. From Equations (
22), (
23), and (
24), it might be concluded by induction that
small edge vectors between the two initial control points
and
after
k subdivision steps can be expressed as
For
and
, then
Let
refer to the length of a vector
and
. Then, we have
Hence, so far as the initial points and are concerned, the sum of the lengths of all the small edges tends to infinity as k approaches infinity. Therefore, the limit curve of ternary 4-point scheme is a fractal curve when and , and this parameter is valued near the point . □
We give the comparison of the range for continuity and fractal behavior of proposed 4-point ternary scheme with other existing ternary schemes in
Table 1.