International Journal of Analysis

Volume 2016, Article ID 1092476, 15 pages

http://dx.doi.org/10.1155/2016/1092476

## Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur, Pakistan

Received 23 August 2016; Revised 24 October 2016; Accepted 30 October 2016

Academic Editor: Jacques Liandrat

Copyright © 2016 Rabia Hameed and Ghulam Mustafa. 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

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.

#### 1. Introduction and Related Work

Subdivision is a process of generating curves/surfaces by iteratively refining a set of control points according to some specific refinement rules. The set of these specific rules are called the subdivision scheme. For stationary, linear, and uniform subdivision schemes, these refinement rules are same at each refinement level. So the set of new control points at next refinement level can be generated by computing the affine combination of the set of control points at previous refinement level. A general compact form of linear, uniform, and stationary binary -variate () subdivision scheme which maps a polygon to a refined polygon is defined asThe symbol of above subdivision scheme is given by the Laurent polynomialwhere is called the mask of subdivision scheme. The detailed information about refinement rules and Laurent polynomial can be found in [1–3].

Surface modeling via subdivision is very important topic in computer graphics and computer aided geometric design and it has been studied by several authors; see surveys [4–10] and references therein.

The construction of univariate subdivision schemes by different variants of Lane-Riesenfeld algorithm has been studied by few authors in [8, 11–13] and construction of bivariate subdivision schemes by a variant of Lane-Riesenfeld algorithm has been studied by Romani [8]. In fact, the Lane-Riesenfeld algorithm is the combination of two operators. One operator refines the initial points and other operator smooths these refining points for a fixed number of times. Therefore it is also named as Refining-Smooth algorithm (RS-algorithm). Moreover, more smoothing steps provide limit curves/surfaces of wider support as well as of higher smoothness.

Lane and Riesenfeld [7] proposed an algorithm for generating -degree B-spline curve, which is the simplest form of the RS-algorithm. In Lane-Riesenfeld algorithm, they combined the symbols of linear B-spline scheme and odd stencil of linear B-spline scheme as refining and smoothing operators, respectively. Therefore the symbol of th family member of Lane-Riesenfeld schemes gives symbol of -degree B-spline scheme. Their family gives continuity and linear polynomial reproduction.

In literature, there are few variants which are applied on the Lane-Riesenfeld algorithm. For example, Cashman et al. [12] also proposed a family of univariate subdivision schemes by using the RS-algorithm which is based on Dubuc Deslauriers 4-point interpolatory scheme [14]. They combined the symbols of -point interpolatory subdivision scheme and odd stencil of -point interpolatory subdivision scheme as refining and smoothing operators, respectively. Their univariate family gives cubic polynomial reproduction but by increasing smoothing stages continuity of their family may or may not be increased.

Ashraf et al. [11] proposed a family of univariate subdivision schemes by using similar technique to that used by Cashman et al. [12] on Dubuc Deslauriers 6-point interpolatory scheme [14]. They also combined the symbol and odd stencil’s symbol of Dubuc Deslauriers 6-point interpolatory scheme [14] as refining and smoothing operators, respectively. Their univariate family gives quintic polynomial reproduction but level of continuity of their family does not increase in general by increasing the smoothing stages.

Mustafa et al. [13] also proposed a family of univariate subdivision schemes by using the symbol of 4-point interpolatory subdivision scheme [15] as the refining operator and the symbol of even stencil of the 4-point approximating scheme [16] as the smoothing operator. Their univariate family gives cubic polynomial reproduction but level of continuity of their family also does not increase in general by increasing the smoothing stages.

Romani [8] proposed families of univariate and bivariate subdivision schemes by using RS-algorithm in which the refining operator is based on a perturbation of Chaikin’s corner cutting subdivision scheme [17] and the smoothing operator takes the average of two adjacent vertices as in Lane-Riesenfeld algorithm. Their univariate family gives linear polynomial reproduction and continuity of their th family member is . Moreover, if , their th family member becomes th member of Hormann and Sabin’s [16] family and it gives cubic polynomial reproduction.

There are few variants of Lane-Riesenfeld algorithm but all those algorithms started with two binary schemes, by taking the symbol of first scheme as the refining operator and the odd or even symbol of second scheme as the smoothing operator. Our approach is different in the sense that we do not use symbol of any existing subdivision scheme as refining operator and also we choose appropriate smoothing operator by ourselves.

In this paper, we propose a new RS-algorithm which is also a variant of Lane-Riesenfeld algorithm that compared with the one proposed by Romani; we modify refining and smoothing operators both. There are two main differences between our RS-algorithm and Romani’s RS-algorithm, which are as follows:(i)Both refining operators depend on Chaikin points, computed around initial points, but the difference between our refining operator and Romani’s refining operator is that Romani draws two vectors at every initial control point and changes position of both Chaikin points, which have been computed around corresponding initial point, by using these two vectors, while we draw one vector at every Chaikin point and change the position of each Chaikin point by using each corresponding vector which we have drawn on it (see Figure 1).(ii)Difference between both smoothing operators is that Romani’s smoothing operator takes average of two adjacent points while our smoothing operator takes weighted average of the four adjacent points.