This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic
Subdivision is an efficient method for generating curves and surfaces in computer aided geometric design. In general, subdivision schemes can be divided into two categories: interpolatory schemes and approximating schemes. Interpolatory schemes get better shape control while approximating schemes have better smoothness. The most well-known interpolatory subdivision scheme is the classical 4-point binary scheme proposed by Dyn et al. [
The deep connection between interpolatory schemes and approximating schemes has been studied in many literatures [
Our work is motivated by a new observation about the 4-point interpolatory subdivision and the quintic
The new family of 6-point combined subdivision schemes is defined as follows:
(
In this section, we recall some fundamental definitions and results that are necessary to the development of the subsequent results.
Given a set of initial control points
Let a binary subdivision scheme
Let subdivision scheme
(a) Let subdivision scheme
(b) Let
Let generates reproduces
This section first explains a new observation about the relation between 4-point interpolatory subdivision and quintic
Given an initial control polygon with vertices
The relation between 4-point interpolatory subdivision and
The relation between 4-point scheme and quintic
The relation between 4-point scheme and cubic
Denote by
A new observation is
We further found that though 6-point interpolatory subdivision is also constructed from polynomial interpolation just like 4-point interpolatory subdivision, analogous connection does not exist between 6-point interpolatory subdivision and quintic
As is shown in [
Suppose the new subdivision have the following rule:
The mask and symbol of subdivision (
Denote the family of subdivision (
Some limit curves generated by
The scheme
The symbol of
When
Hence, by Theorem
Polynomial generation and polynomial reproduction are desirable properties because any convergent subdivision scheme that reproduces polynomials of degree
Let
The subdivision scheme
The symbol of
Then,
Moreover, when
If applying the parameter shift
To consider the reproduction degree of the subdivision scheme
Hence, using Theorem
As the new family of subdivision schemes
Comparison between properties of cubic
Scheme | Support | Continuity | Generation degree | Reproduction degree |
---|---|---|---|---|
4-p interpolatory scheme | 6 | 1 | 3 | 3 |
Cubic |
4 | 2 | 3 | 1 |
Quintic |
6 | 4 | 5 | 1 |
6-p interpolatory scheme | 10 | 2 | 5 | 5 |
|
10 | 3 | 3 | 1 |
|
10 | 4 | 5 | 3 |
|
10 | 4 | 7 | 3 |
Comparison between properties of Hormann-Sabin’s family
Scheme | Support | Continuity | Generation degree | Reproduction degree |
---|---|---|---|---|
|
6 | 1 | 3 | 3 |
|
7 | 2 | 4 | 3 |
|
8 | 3 | 5 | 3 |
|
9 | 4 | 6 | 3 |
|
10 | 5 | 7 | 3 |
|
11 | 6 | 8 | 3 |
|
|
|
|
3 |
|
||||
|
10 | 4 | 5 | 3 |
|
||||
|
10 | 4 | 7 | 3 |
In this paper, we present a new family of 6-point combined subdivision schemes which provides the representation of wide variety of shapes and a subfamily of subdivision schemes with high smoothness and cubic precision. All these properties are required in many applications, such as computer aided geometric design and geometric modeling. The subfamily
The polynomial reproduction property of
Comparison of limit curves (the blue curves) generated by (a) 4-p interpolatory scheme, cubic
Comparison of limit curves generated by
The authors declare that they have no conflicts of interest.
This work is supported by the National Natural Science Foundation of China under Grant nos. 61472466 and 61070227, the NSFC-Guangdong Joint Foundation Key Project under Grant no. U1135003, and the Fundamental Research Funds for the Central Universities under Grant no. JZ2015HGXJ0175.