Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2324893, 9 pages

https://doi.org/10.1155/2018/2324893

## Six-Point Subdivision Schemes with Cubic Precision

^{1}School of Mathematics, Hefei University of Technology, Hefei 230009, China^{2}School of Computer and Information, Hefei University of Technology, Hefei 230009, China

Correspondence should be addressed to Zhi Liu

Received 10 July 2017; Revised 5 November 2017; Accepted 22 November 2017; Published 3 January 2018

Academic Editor: Dan Simon

Copyright © 2018 Jun Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic* B*-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic* B*-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the* B*-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin’s family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness.

#### 1. Introduction

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. [1]. In 1989, it was extended to the 6-point binary interpolatory scheme by Weissman [2]. Most approximating schemes were developed from splines. Two of the most famous approximating schemes are Chaikin’s algorithm [3] and cubic* B*-spline refinement algorithm [4], which actually generate uniform quadratic and cubic* B*-spline curves with continuity and continuity, respectively.

The deep connection between interpolatory schemes and approximating schemes has been studied in many literatures [5–15]. In 2001, Maillot and Stam [5] introduced a push-back operation which is applied at each round of approximating refinement to progressively interpolate the control vertices. In 2007, Li and Ma [6] observed a relation between 4-point interpolatory subdivision and cubic* B*-spline curve refinement, and, motivated by this relation, they proposed a universal method for constructing interpolatory subdivision through the addition of weighted averaging operations to the mask of approximating subdivision. In 2008, Lin et al. [7] found another relation between 4-point interpolatory subdivision and cubic* B*-spline refinement and constructed interpolatory subdivision from approximating subdivision based on the relation. The deep connection between interpolatory and approximating schemes was also studied in [8–12] which exploited the generating functions of approximating subdivision and interpolatory subdivision. In 2012, Pan et al. [13] provided a combined ternary approximating and interpolatory subdivision scheme with continuity. Li and Zheng [14] constructed interpolatory subdivision from primal approximating subdivision with a new observation of the link between interpolatory and approximating subdivision. In 2013, Luo and Qi [15] made some theoretical analysis from the generation polynomial perspective and constructed some new interpolatory schemes from approximating schemes.

Our work is motivated by a new observation about the 4-point interpolatory subdivision and the quintic* B*-spline curve refinement. The observation gives us heuristics to construct combined subdivision schemes from existing subdivision schemes. The idea is to construct the counterparts of cubic and quintic* B*-spline refinements and make the relations between the 6-point interpolatory subdivision and the counterparts of cubic and quintic* B*-spline refinements similar to those between the 4-point interpolatory subdivision and the cubic, quintic* B*-spline curve refinements. Since the 6-point interpolatory subdivision from which the new subdivision scheme is deduced has good properties such as high smoothness and high accuracy, we are interested in studying which properties of the new subdivision scheme are better than their counterparts.

The new family of 6-point combined subdivision schemes is defined as follows:

(1) is called the 6-point combined interpolatory and approximating binary subdivision scheme. If , (1) generates 6-point interpolatory subdivision; otherwise, (1) produces approximating subdivision. It is proved that when suitably setting the tension parameter, all schemes from (1) are able to generate curves with continuity and reproduce cubic polynomials, whereas the* B*-spline refinements attain only linear precision. Moreover, we also make a comparison of properties between our family and famous Hormann-Sabin’s family [16] which has the same cubic precision.

#### 2. Preliminaries

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 , the set of control points at level are recursively defined by the following binary subdivision rules: The finite set is called* mask*. The iterative algorithm based on the repeated application of (2) is termed* subdivision scheme* and is denoted by . The symbol of the scheme is defined as .

Theorem 1 (see [17]). *Let a binary subdivision scheme be convergent. Then the mask satisfies *

Theorem 2 (see [17]). *Let subdivision scheme with mask satisfy (3). Then there exists a subdivision scheme (first-order divided difference scheme of ) with the property where and . The symbol of is . Generally, if (the th-order divided difference scheme of ) exists with mask , then the symbol of is .*

Theorem 3 (see [17]). *(a) Let subdivision scheme have mask , and its th-order divided difference scheme exists with mask satisfying If there exists an integer , such that , then the subdivision scheme is continuous, where In particular, when ,**(b) Let with being contractive (i.e., maps any initial data to zero). Then, is convergent and continuous.*

Theorem 4 (see [18, 19]). *Let denote the space of all univariate polynomials with real coefficients up to degree . Then a univariate subdivision scheme *(i)*generates if and only if *(ii)*reproduces with respect to the parametrization with and denoting the subdivision level, if and only if it generates and *

*3. Construction of the New Family*

*This section first explains a new observation about the relation between 4-point interpolatory subdivision and quintic B-spline refinement. Then, a new family of 6-point combined subdivision schemes is deduced.*

*3.1. A New Observation*

*Given an initial control polygon with vertices , as shown in Figure 1, the rules of 4-point interpolatory subdivision for generating level vertices are and quintic B-spline refinement for generating level vertices is *