A new 4-point

Computer Aided Geometric Design (CAGD) is a branch of applied mathematics concerned with algorithms for the design of smooth curves/surfaces. One common approach to the design of curves/surfaces related to CAGD is the subdivision scheme. It is an algorithm to generate smooth curves and surfaces as a sequence of successively refined control polygons. At each refinement level, new points are added into the existing polygon and the original points remain existed or discarded in all subsequent sequences of control polygons. The number of points inserted at level

Now a days, the variety of subdivision schemes investigated; our interest is in the direction of quaternary schemes. The goal of this paper is to construct 4-point quaternary subdivision scheme having the higher smoothness and approximation order but smaller support than existing 4-point binary and ternary schemes.

Here we present a 4-point quaternary approximating subdivision scheme. A polygon

A general compact form of univariate quaternary subdivision scheme

For the analysis of subdivision scheme with mask

Let

Theorem

Let

If

Apply Theorem

Corollary

The approximation order of a convergent subdivision scheme

This section is devoted for analysis of 4-point quaternary approximating subdivision scheme by using Laurent polynomial method. The following result shows that scheme is

The 4-point quaternary approximating subdivision scheme (

For the given mask of proposed scheme

From the above discussion, we conclude that our scheme is

(a) Graph against Höder exponent and parameter

When dealing with open initial polygon

Subdivision rule (

In this section, we discuss approximation order and support of basic limit function of 4-point quaternary approximating scheme.

Here we show that the approximation order of proposed scheme is five. The following lemma based on the technique of Sabin [

The proposed 4-point quaternary subdivision scheme reproduces all the cubic polynomials for

We carry out this result by taking our origin the middle of an original span with ordinate

If

If

If

If

The theorem is an easy consequences of Lemma

A 4-point quaternary approximating subdivision scheme has approximation order 5.

The basic function of a subdivision scheme is the limit function of proposed scheme for the following data:

The effect of parameter on the shape of the basic limit function/limit curve of the proposed scheme. Doted lines show control polygons; whereas solid lines indicate basic limit functions/curves. (a) Here,

Basic limit functions

Close curves

Open curves

The basic limit function

Since the basic function is the limit function of the scheme for the data (

In Table

Comparison of proposed 4-point scheme with other 4-point schemes.

Scheme | Type | Approximation order | Support | |
---|---|---|---|---|

Binary 4-point [ | Interpolating | 4 | 6 | 1 |

Binary 4-point [ | Interpolating | 4 | 6 | 1 |

Binary 4-point [ | Approximating | 4 | 7 | 2 |

Ternary 4-point [ | Interpolating | 3 | 5 | 2 |

Ternary 4-point [ | Interpolating | 3 | 5 | 2 |

Ternary 4-point [ | Approximating | 4 | 5.5 | 2 |

Proposed scheme | Approximating | 5 | 5 | 3 |

In Figures

Support width of for binary, ternary, and proposed 4-point approximating schemes has been shown in (a), (b), and (c), respectively.

Binary 4-point [

Ternary 4-point [

Proposed

It shows the comparison at 1st, 2nd, and 3rd level of binary, ternary, and quaternary 4-point approximating schemes. Dotted lines indicate initial control polygons, whereas dashed, thin solid, and bold solid continuous curves are generated by binary, ternary, and quaternary schemes, respectively.

Level 1

Level 2

Level 3

The authors are pleased to acknowledge the anonymous referees whose precious and enthusiastic comments made this manuscript more constructive. First author pays special thanks to Professor Deng Jian Song (University of Science and Technology of China) for his continuous research assistance. This work is supported by the Indigenous PhD scholarship scheme of Higher Education Commission Pakistan.