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

Authors: Rabia Hameed; Ghulam Mustafa
Journal: International Journal of Analysis
 (2016)

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## 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.


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## Body

## 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 binarym-variate (m=1,2) subdivision scheme Sa which maps a polygon fk-1=fαk-1,α∈Zm to a refined polygon fk=fαk,α∈Zm is defined as(1)fαk=∑β∈Zmaα-2βfβk-1,α∈Zm.The symbol of above subdivision scheme is given by the Laurent polynomial(2)az=∑α∈Zmaαzα,z∈C∖0m,where a=aα,α∈Zm 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 n+1-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 nth family member of Lane-Riesenfeld schemes gives symbol of n+1-degree B-spline scheme. Their family gives Cn 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 4-point interpolatory subdivision scheme and odd stencil of 4-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 nth family member is Cn. Moreover, if ω=1/16, their nth family member becomes n+3th 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 Figure1).(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.Figure 1
The geometrical difference between our refining stage and Romani’s refining stage.
(a)
Our refining stage (b)
Romani’s refining stageOur refining and smoothing operators are different from the operators proposed by Romani. The framework proposed in this paper provides families of schemes where all members are approximating while the first member of Romani’s family is interpolatory and all the other members are approximating. Moreover, the flexibility of our shape parameter allows us to adjust the value of shape parameter locally according to the geometry of initial polygon.The article is organized as follows. In Section2, we present the framework for the construction of univariate and bivariate families of subdivision schemes and their graphical and mathematical comparisons with Romani’s families. In Section 3, we present some basic properties of these families of subdivision schemes that include discussion on basic limit functions, analysis of continuity, polynomial generation, and reproduction of the schemes. In Section 4, applications of some members of the family and comparisons with Romani schemes are presented. Conclusions are given in Section 5.


## 2. Construction and Comparison

The Refining-Smooth subdivision scheme is the scheme which consists of two operators. One operator refines the initial points while the other operator smooths the refined points for the required number of times. If we denote refining operator byR, smoothing operator by S, and the subdivision operator by Sal, then (3)fk=Salfk-1,where (4)fk-1,1=Rfk-1,fk=fk-1,n+1=Sfk-1,n,n=1,2,…,l.In fact, refining operatorR maps a polygon fk-1={fαk-1,α∈Zm} to a refined polygon fk-1,1={fαk-1,1,α∈Zm}, where m=1,2; that is, (5)fαk-1,1=∑β∈Zmrα-2βfβk-1,α∈Zm,and smoothing operator S maps a polygon fk-1,n={fαk-1,n,α∈Zm} to a refined polygon fk-1,n+1=fαk-1,n+1,α∈Zm which is defined as (6)fαk-1,n+1=∑β∈Zmsβfβk-1,n,α∈Zm,n=1,2,…,l. The symbols of refining and smoothing operators are (7)rz=∑α∈Zmrαzα,sz=∑α∈Zmsαzα,z∈C∖0m, with (8)r1=2m,s1=1,where1∈Rm.In RS-algorithm, combination of one refining operator and l smoothing operators is called the subdivision operator. Moreover, we denote proposed univariate and bivariate families of subdivision schemes by Sal and Sbl, respectively.The construction procedure for our families of schemes is given below.


### 2.1. Univariate Family

Here we denote the Chaikin points byC2ik-1,1 and C2i+1k-1,1 that are computed around the point fik-1,0=fik-1 as shown in Figure 1. In this figure, black solid circles show control points at (k-1)th subdivision level, blue crosses show the Chaikin points, and blue solid circles show refining points. Red vectors are involved in the insertion of even point f2ik-1,1 and green vectors are involved in the insertion of odd point f2i+1k-1,1. In this figure, we have shown the refinement of only two Chaikin points which have been inserted around point fik-1.In our framework, we compute vectorsd2ik-1,1 and d2i+1k-1,1 by taking the linear combinations of two vectors, as shown in Figure 1(a):(9)d2ik-1,1=l+1C2ik-1,1-fi+1k-1+l-1C2ik-1,1-fi-1k-1,d2i+1k-1,1=l+1C2i+1k-1,1-fi-1k-1+l-1C2i+1k-1,1-fi+1k-1,where in Figure 1(b) vectors are computed as (10)d2ik-1,1=l+3C2ik-1,1-C2i+1k-1,1+l-12fik-1-fi-1k-1,d2i+1k-1,1=l+3C2i+1k-1,1-C2ik-1,1+l-12fik-1-fi+1k-1, with l∈N.In our framework, we compute refining point by adding each Chaikin point to the corresponding vector; that is,(11)f2ik-1,1=C2ik-1,1+ωd2ik-1,1,f2i+1k-1,1=C2i+1k-1,1+ωd2i+1k-1,1,where ω∈R. While in Romani’s framework refining points are computed as (12)f2ik-1,1=fik-1+2ωd2ik-1,1,f2i+1k-1,1=fik-1+2ωd2i+1k-1,1.Now by substituting (9) in (11), we get the following subdivision scheme which will refine the course polygon fk-1,0 into refined polygon fk-1,1:(13)f2ik-1,1=α1fi-1k-1+α2fik-1+α3fi+1k-1,f2i+1k-1,1=α3fi-1k-1+α2fik-1+α1fi+1k-1,where (14)α1=141+2ω2-l,α2=341+2ωl,α3=-ωl+1.Now the second step in our frame work is to smooth the refined polygonfk-1,1=fik-1,1i∈Z by applying l-times smoothing operator. That is, smoothing operator maps fik-1,1 into fik=fik,0=fik-1,l+1 by using the following rule:(15)fik-1,n+1=112fi-1k-1,n+512fik-1,n+512fi+1k-1,n+112fi+2k-1,n,where n=1,2,…,l.By applying one time refining operator andl-times smoothing operator, we get the new family of subdivision schemes with symbol(16)alz=z-l-l/2-2112z+1z2+4z+1l-ωl+1+141-2ωl-2z+341+2ωlz2+341+2ωlz3+141-2ωl-2z4-ωl+1z5,where  is a ceiling function.The first member of our proposed family is a primal 5-point relaxed subdivision scheme with two refinement rules. The brief description of these refinement rules is given below.(i)
Edge splitting rule: for every edge in the coarse polygon, a new edge point is calculated by using the affine combination of 4 neighboring control points as shown in Figure2(a).(ii)
Vertex updating rule: for every vertex in the coarse polygon, a new vertex point is calculated by using the affine combination of 5 neighboring control points as shown in Figure2(b).Figure 2
Schematic overview of the refinement rules when we apply one time refining operator forl=1 and one time smoothing operator. Solid circles show the points at (k-1)th subdivision level while circles show the points at kth subdivision level after applying one refining and one smoothing stage.
(a)
Edge rule (b)
Vertex ruleRemark 1.
Forω=0, symbol (16) takes the form (17)alz=z-l-l/2-1z2+4z+13lz+1l+34l+1.Now we shall construct family of bivariate nontensor product subdivision schemes for quadrilateral meshes by choosing Doo-Sabin points as origin.


### 2.2. Bivariate Family

Let us denote the Doo-Sabin points for one face of quadrilateral mesh byC2i,2jk-1,1, C2i+1,2jk-1,1, C2i,2j+1k-1,1, and C2i+1,2j+1k-1,1 that are computed from the initial points fi,jk-1, fi+1,jk-1, fi,j+1k-1, and fi+1,j+1k-1 as shown in Figure 3. In this figure, black solid circles show initial points at (k-1)th subdivision level, blue cross symbols show Doo-Sabin points, and blue solid circles show refining points at (k-1)th level; red, gray, green, and orange vectors are involved in the insertion of the refining points f2i,2jk-1,1, f2i+1,2jk-1,1, f2i,2j+1k-1,1, and f2i+1,2j+1k-1,1, respectively.Figure 3
The geometrical construction of the refining stage for quadrilateral meshes.We compute vectorsD2i,2jk-1,1, D2i+1,2jk-1,1, D2i,2j+1k-1,1, and D2i+1,2j+1k-1,1 by taking the linear combination of two vectors that are drawn on the Doo-Sabin points C2i,2jk-1,1, C2i+1,2jk-1,1, C2i,2j+1k-1,1, and C2i+1,2j+1k-1,1, respectively, as shown in Figure 3.(18)D2i,2jk-1,1=l+1C2i,2jk-1,1-fi+1,j+1k-1+l-1C2i,2jk-1,1-fi,jk-1,D2i+1,2jk-1,1=l+1C2i+1,2jk-1,1-fi,j+1k-1+l-1C2i+1,2jk-1,1-fi+1,jk-1,D2i,2j+1k-1,1=l+1C2i,2j+1k-1,1-fi+1,jk-1+l-1C2i,2j+1k-1,1-fi,j+1k-1,D2i+1,2j+1k-1,1=l+1C2i+1,2j+1k-1,1-fi,jk-1+l-1C2i+1,2j+1k-1,1-fi+1,j+1k-1,where l∈N.Similarly to the curve case, we compute refining points as(19)f2i,2jk-1,1=C2i,2jk-1,1+ωD2i,2jk-1,1,f2i+1,2jk-1,1=C2i+1,2jk-1,1+ωD2i+1,2jk-1,1,f2i,2j+1k-1,1=C2i,2j+1k-1,1+ωD2i,2j+1k-1,1,f2i+1,2j+1k-1,1=C2i+1,2j+1k-1,1+ωD2i+1,2j+1k-1,1,where ω∈R.Now by substituting (18) in (19), we get the subdivision scheme which will refine the course mesh fk-1,0 into refined mesh fk-1,1.(20)f2i,2jk-1,1=β1fi,jk-1+β2fi+1,jk-1+β2fi,j+1k-1+β3fi+1,j+1k-1,f2i+1,2jk-1,1=β2fi,jk-1+β1fi+1,jk-1+β3fi,j+1k-1+β2fi+1,j+1k-1,f2i,2j+1k-1,1=β2fi,jk-1+β3fi+1,jk-1+β1fi,j+1k-1+β2fi+1,j+1k-1,f2i+1,2j+1k-1,1=β3fi,jk-1+β2fi+1,jk-1+β2fi,j+1k-1+β1fi+1,j+1k-1,where β1=1/169+2l+8ω, β2=1/1632ωl+1, and β3=1/161-27l+8ω.Now we smooth the refined meshfk-1,1=fi,jk-1,1i,j∈Z by applying l-time smoothing operator. That is, we map fi,jk-1,1 into fi,jk=fi,jk,0=fi,jk-1,l+1 by using the following smoothing operator:(21)fi,jk-1,n+1=112112fi-1,j-1k-1,n+512fi,j-1k-1,n+512fi+1,j-1k-1,n+112fi+2,j-1k-1,n+512112fi-1,jk-1,n+512fi,jk-1,n+512fi+1,jk-1,n+112fi+2,jk-1,n+512112fi-1,j+1k-1,n+512fi,j+1k-1,n+512fi+1,j+1k-1,n+112fi+2,j+1k-1,n+112112fi-1,j+2k-1,n+512fi,j+2k-1,n+512fi+1,j+2k-1,n+112fi+2,j+2k-1,n,where n=1,2,…,l. By applying one time refining operator R and l-times smoothing operator S, we get the family of bivariate primal/dual subdivision schemes for quadrilateral meshes with symbol(22)blz1,z2=z1z2-1-l-l/21144z2+1z22+4z2+1z1+1z12+4z1+1l1169+2l+8ωz1z2+11631+2ωlz13z2+11631+2ωlz1z23+1161-27l+8ωz13z23+11631+2ωlz2+1169+2l+8ωz12z2+1161-27l+8ωz23+11631+2ωlz12z23+11631+2ωlz1+1161-27l+8ωz13+1169+2l+8ωz1z22+11631+2ωlz13z22+1161-27l+8ω+11631+2ωlz12+11631+2ωlz22+1169+2l+8ωz12z22.Remark 2.
The bivariate family forω=0 corresponding to the symbol (22) reduces to the tensor product univariate family. This tensor product family of schemes can also be obtained from the symbol defined by (16). If we apply one time refining operator (20) for l=1 and one time smoothing operator (21), we get the primal approximating subdivision scheme with four refinement rules. The schematic overview of these refinement rules is given in Figure 4.Figure 4
Schematic overview of the refinement rules forl=1. Solid circles show points at k-1th subdivision level where solid squares show the new points at level k.
(a)
Vertex rule (b)
Edge rule (c)
Edge rule (d)
Face ruleRemark 3.
Forz1=1 and z2=z or z1=z and z2=1, our family of bivariate schemes reduce to the family of univariate schemes with the symbol (23)blz=12bl1,z=12blz,1=z-l+1-l/21+zl+12l+2z2+4z+16l1-2l+2ωz2+21+2l+2ωz+1-2l+2ω. This family of schemes is different from our proposed univariate family of schemes.


## 2.1. Univariate Family

Here we denote the Chaikin points byC2ik-1,1 and C2i+1k-1,1 that are computed around the point fik-1,0=fik-1 as shown in Figure 1. In this figure, black solid circles show control points at (k-1)th subdivision level, blue crosses show the Chaikin points, and blue solid circles show refining points. Red vectors are involved in the insertion of even point f2ik-1,1 and green vectors are involved in the insertion of odd point f2i+1k-1,1. In this figure, we have shown the refinement of only two Chaikin points which have been inserted around point fik-1.In our framework, we compute vectorsd2ik-1,1 and d2i+1k-1,1 by taking the linear combinations of two vectors, as shown in Figure 1(a):(9)d2ik-1,1=l+1C2ik-1,1-fi+1k-1+l-1C2ik-1,1-fi-1k-1,d2i+1k-1,1=l+1C2i+1k-1,1-fi-1k-1+l-1C2i+1k-1,1-fi+1k-1,where in Figure 1(b) vectors are computed as (10)d2ik-1,1=l+3C2ik-1,1-C2i+1k-1,1+l-12fik-1-fi-1k-1,d2i+1k-1,1=l+3C2i+1k-1,1-C2ik-1,1+l-12fik-1-fi+1k-1, with l∈N.In our framework, we compute refining point by adding each Chaikin point to the corresponding vector; that is,(11)f2ik-1,1=C2ik-1,1+ωd2ik-1,1,f2i+1k-1,1=C2i+1k-1,1+ωd2i+1k-1,1,where ω∈R. While in Romani’s framework refining points are computed as (12)f2ik-1,1=fik-1+2ωd2ik-1,1,f2i+1k-1,1=fik-1+2ωd2i+1k-1,1.Now by substituting (9) in (11), we get the following subdivision scheme which will refine the course polygon fk-1,0 into refined polygon fk-1,1:(13)f2ik-1,1=α1fi-1k-1+α2fik-1+α3fi+1k-1,f2i+1k-1,1=α3fi-1k-1+α2fik-1+α1fi+1k-1,where (14)α1=141+2ω2-l,α2=341+2ωl,α3=-ωl+1.Now the second step in our frame work is to smooth the refined polygonfk-1,1=fik-1,1i∈Z by applying l-times smoothing operator. That is, smoothing operator maps fik-1,1 into fik=fik,0=fik-1,l+1 by using the following rule:(15)fik-1,n+1=112fi-1k-1,n+512fik-1,n+512fi+1k-1,n+112fi+2k-1,n,where n=1,2,…,l.By applying one time refining operator andl-times smoothing operator, we get the new family of subdivision schemes with symbol(16)alz=z-l-l/2-2112z+1z2+4z+1l-ωl+1+141-2ωl-2z+341+2ωlz2+341+2ωlz3+141-2ωl-2z4-ωl+1z5,where  is a ceiling function.The first member of our proposed family is a primal 5-point relaxed subdivision scheme with two refinement rules. The brief description of these refinement rules is given below.(i)
Edge splitting rule: for every edge in the coarse polygon, a new edge point is calculated by using the affine combination of 4 neighboring control points as shown in Figure2(a).(ii)
Vertex updating rule: for every vertex in the coarse polygon, a new vertex point is calculated by using the affine combination of 5 neighboring control points as shown in Figure2(b).Figure 2
Schematic overview of the refinement rules when we apply one time refining operator forl=1 and one time smoothing operator. Solid circles show the points at (k-1)th subdivision level while circles show the points at kth subdivision level after applying one refining and one smoothing stage.
(a)
Edge rule (b)
Vertex ruleRemark 1.
Forω=0, symbol (16) takes the form (17)alz=z-l-l/2-1z2+4z+13lz+1l+34l+1.Now we shall construct family of bivariate nontensor product subdivision schemes for quadrilateral meshes by choosing Doo-Sabin points as origin.


## 2.2. Bivariate Family

Let us denote the Doo-Sabin points for one face of quadrilateral mesh byC2i,2jk-1,1, C2i+1,2jk-1,1, C2i,2j+1k-1,1, and C2i+1,2j+1k-1,1 that are computed from the initial points fi,jk-1, fi+1,jk-1, fi,j+1k-1, and fi+1,j+1k-1 as shown in Figure 3. In this figure, black solid circles show initial points at (k-1)th subdivision level, blue cross symbols show Doo-Sabin points, and blue solid circles show refining points at (k-1)th level; red, gray, green, and orange vectors are involved in the insertion of the refining points f2i,2jk-1,1, f2i+1,2jk-1,1, f2i,2j+1k-1,1, and f2i+1,2j+1k-1,1, respectively.Figure 3
The geometrical construction of the refining stage for quadrilateral meshes.We compute vectorsD2i,2jk-1,1, D2i+1,2jk-1,1, D2i,2j+1k-1,1, and D2i+1,2j+1k-1,1 by taking the linear combination of two vectors that are drawn on the Doo-Sabin points C2i,2jk-1,1, C2i+1,2jk-1,1, C2i,2j+1k-1,1, and C2i+1,2j+1k-1,1, respectively, as shown in Figure 3.(18)D2i,2jk-1,1=l+1C2i,2jk-1,1-fi+1,j+1k-1+l-1C2i,2jk-1,1-fi,jk-1,D2i+1,2jk-1,1=l+1C2i+1,2jk-1,1-fi,j+1k-1+l-1C2i+1,2jk-1,1-fi+1,jk-1,D2i,2j+1k-1,1=l+1C2i,2j+1k-1,1-fi+1,jk-1+l-1C2i,2j+1k-1,1-fi,j+1k-1,D2i+1,2j+1k-1,1=l+1C2i+1,2j+1k-1,1-fi,jk-1+l-1C2i+1,2j+1k-1,1-fi+1,j+1k-1,where l∈N.Similarly to the curve case, we compute refining points as(19)f2i,2jk-1,1=C2i,2jk-1,1+ωD2i,2jk-1,1,f2i+1,2jk-1,1=C2i+1,2jk-1,1+ωD2i+1,2jk-1,1,f2i,2j+1k-1,1=C2i,2j+1k-1,1+ωD2i,2j+1k-1,1,f2i+1,2j+1k-1,1=C2i+1,2j+1k-1,1+ωD2i+1,2j+1k-1,1,where ω∈R.Now by substituting (18) in (19), we get the subdivision scheme which will refine the course mesh fk-1,0 into refined mesh fk-1,1.(20)f2i,2jk-1,1=β1fi,jk-1+β2fi+1,jk-1+β2fi,j+1k-1+β3fi+1,j+1k-1,f2i+1,2jk-1,1=β2fi,jk-1+β1fi+1,jk-1+β3fi,j+1k-1+β2fi+1,j+1k-1,f2i,2j+1k-1,1=β2fi,jk-1+β3fi+1,jk-1+β1fi,j+1k-1+β2fi+1,j+1k-1,f2i+1,2j+1k-1,1=β3fi,jk-1+β2fi+1,jk-1+β2fi,j+1k-1+β1fi+1,j+1k-1,where β1=1/169+2l+8ω, β2=1/1632ωl+1, and β3=1/161-27l+8ω.Now we smooth the refined meshfk-1,1=fi,jk-1,1i,j∈Z by applying l-time smoothing operator. That is, we map fi,jk-1,1 into fi,jk=fi,jk,0=fi,jk-1,l+1 by using the following smoothing operator:(21)fi,jk-1,n+1=112112fi-1,j-1k-1,n+512fi,j-1k-1,n+512fi+1,j-1k-1,n+112fi+2,j-1k-1,n+512112fi-1,jk-1,n+512fi,jk-1,n+512fi+1,jk-1,n+112fi+2,jk-1,n+512112fi-1,j+1k-1,n+512fi,j+1k-1,n+512fi+1,j+1k-1,n+112fi+2,j+1k-1,n+112112fi-1,j+2k-1,n+512fi,j+2k-1,n+512fi+1,j+2k-1,n+112fi+2,j+2k-1,n,where n=1,2,…,l. By applying one time refining operator R and l-times smoothing operator S, we get the family of bivariate primal/dual subdivision schemes for quadrilateral meshes with symbol(22)blz1,z2=z1z2-1-l-l/21144z2+1z22+4z2+1z1+1z12+4z1+1l1169+2l+8ωz1z2+11631+2ωlz13z2+11631+2ωlz1z23+1161-27l+8ωz13z23+11631+2ωlz2+1169+2l+8ωz12z2+1161-27l+8ωz23+11631+2ωlz12z23+11631+2ωlz1+1161-27l+8ωz13+1169+2l+8ωz1z22+11631+2ωlz13z22+1161-27l+8ω+11631+2ωlz12+11631+2ωlz22+1169+2l+8ωz12z22.Remark 2.
The bivariate family forω=0 corresponding to the symbol (22) reduces to the tensor product univariate family. This tensor product family of schemes can also be obtained from the symbol defined by (16). If we apply one time refining operator (20) for l=1 and one time smoothing operator (21), we get the primal approximating subdivision scheme with four refinement rules. The schematic overview of these refinement rules is given in Figure 4.Figure 4
Schematic overview of the refinement rules forl=1. Solid circles show points at k-1th subdivision level where solid squares show the new points at level k.
(a)
Vertex rule (b)
Edge rule (c)
Edge rule (d)
Face ruleRemark 3.
Forz1=1 and z2=z or z1=z and z2=1, our family of bivariate schemes reduce to the family of univariate schemes with the symbol (23)blz=12bl1,z=12blz,1=z-l+1-l/21+zl+12l+2z2+4z+16l1-2l+2ωz2+21+2l+2ωz+1-2l+2ω. This family of schemes is different from our proposed univariate family of schemes.


## 3. Analysis of Univariate and Bivariate Schemes

This section deals with the analysis of proposed families of schemes. The analysis includes support width and smoothness/continuity via Laurent polynomial method. The capacity of generating and reproducing polynomials has also been discussed mathematically.


### 3.1. Support of Basic Limit Function

Support of basic limit function of a subdivision scheme is equal to the number of spans of the curve affected when one control point is moved from its initial place.A convergent subdivision schemeSal defines a basic limit function ϕl=Sal∞δi with δi being the initial data such that (24)δi=1fori=0,0otherwise.Support of basic limit function of our univariate family of schemes depends on parameterω.(i)
Forω≠0∈Ωl, the numbers of nonzero coefficients in the symbol of refining operator (13) and smoothing operator (15) are 6 and 4, respectively. So by [18] support size of basic limit function of the schemes Sal for the general symbol al(z) is 3l+5 and its support width is -3l+5/2,3l+5/2.(ii)
Forω=0, the number of nonzero coefficients in the symbol of both refining and smoothing stages is 4; therefore support size of basic limit function of the schemes Sal for the general symbol al(z) is 3(l+1) and its support width is -3/2l+1,3/2l+1.A convergent bivariate subdivision schemeSbl defines a basic limit function ϕl=Sbl∞δi,j, where δi,j is the initial data such that (25)δi,j=1fori=j=0,0otherwise.Support of basic limit of our bivariate family of schemes is 3(l+1)×3(l+1) and its support width is -3/2(l+1),3/2(l+1)×-3/2(l+1),3/2(l+1).Basic limit functions for the first three family members of univariate family for three different values of shape parameters are presented in Figure5, whereas basic limit functions for the first family member of bivariate family for three different values of shape parameter are presented in Figure 6. From these figures, we see that the support width for positive value is less than the support width for negative value of the parameter involved in the bivariate family.Figure 5
Basic limit functions of subdivision schemesSal for three values of shape parameter ω. Cross symbols show control points, while solid lines, dashed dotted lines, and dash lines represent curves obtained by subdivision schemes Sa1, Sa2, and Sa3, respectively.
(a)
ω = - 1 / 8 (b)
ω = 0 (c)
ω = 1 / 8Figure 6
Basic limit functions of subdivision schemeSb1 for three values of shape parameter ω.
(a)
ω = - 1 / 3 (b)
ω = 0 (c)
ω = 1 / 3


### 3.2. Continuity Analysis

Here we present the smoothness of proposed families of schemes by using the Laurent polynomial method. For this purpose, we have used the methods that are given ([3], Theorems  4.11, 4.13, 4.29, and 4.30).Theorem 4.
The subdivision schemeSa1 is convergent for -7/8<ω<13/38, generates C1 limiting curve for -3/10<ω<1/6, and generates C3 limiting curve for ω=0.Proof.
Since the scheme with the symbolz/1+za1(z) is contractive for -7/8<ω<13/38, therefore the scheme Sa1 is convergent for -7/8<ω<13/38. Similarly, the scheme with the symbol 2z/1+z2a1(z) is contractive for -3/10<ω<1/6; this implies that the scheme Sa1 is C1 continuous for -3/10<ω<1/6. Moreover, the scheme with symbol 4z/1+z3a1(z) is contractive for ω=0; accordingly the scheme Sa1 is C2 continuous for ω=0. Furthermore, the scheme with symbol 6z/1+z4a1(z) is contractive for ω=0; hence the scheme Sa1 is C3 continuous for ω=0.Theorem 5.
(i) The subdivision schemeSa2 is convergent for -53/100<ω<7/26, generates C1 and C2 limiting curves for -53/100<ω<7/26 and -11/27<ω<19/108, respectively, and generates C4 limiting curve for ω=0.
(ii) The subdivision schemeSa3 is convergent for -1723/3322<ω<585/1954, generates C1, C2, and C3 limiting curves for -114/271<ω<269/1154, -114/271<ω<269/1154, and -19/58<ω<61/358, respectively, and generates C5 limiting curve for ω=0.
(iii) The subdivision schemeSa4 is convergent for -5021/11121<ω<2099/7808, generates C1, C2, C3, and C4 limiting curves for -5021/11121<ω<2099/7808, -2635/7128<ω<854/3993, -2635/7128<ω<854/3993, and -703/2404<ω<391/2384, respectively, and generates C6 limiting curve for ω=0.Proof.
Proof of this theorem is same as Theorem4.Theorem 6.
The subdivision schemeSb1 is convergent for -41/36<ω<11/21, generates C1 limiting surface for -1/2<ω<19/45, and generates C3 limiting surface for ω=0.Proof.
Since the schemes with symbolsz1/1+z1b1(z1,z2) and z2/1+z2b1(z1,z2) are contractive for -41/36<ω<11/21, therefore the scheme Sb1 is convergent for -41/36<ω<11/21. Similarly, the schemes with symbols 2z1/1+z12b1(z1,z2) and 2z2/1+z22b1(z1,z2) are contractive for -1/2<ω<5/18 and the scheme with symbol 2z1/1+z1z2/1+z2b1(z1,z2) is contractive for -17/15<ω<19/45; this implies that the scheme Sb1 is C1 for -1/2<ω<19/45. Moreover, the schemes with symbols 4z1/1+z13b1(z1,z2), 4z2/1+z23b1(z1,z2), 4z1/1+z12z2/1+z2b1(z1,z2), and 4z1/1+z1z2/1+z22b1(z1,z2) are contractive for ω=0; accordingly the scheme Sb1 is C2 for ω=0. Furthermore, the schemes with symbols 6z1/1+z14b1(z1,z2), 6z2/1+z24b1(z1,z2), 6z1/1+z13z2/1+z2×b1(z1,z2), 6z1/1+z1z2/1+z23b1(z1,z2), and 6z1/1+z12z2/1+z22b1(z1,z2) are contractive for ω=0; thus the scheme Sb1 is C3 for ω=0.Theorem 7.
The subdivision schemeSb2 is convergent for -14075/12444<ω<1191/1600, generates C1 and C2 limiting surfaces for -5463/4928<ω<801/1388 and -11/16<ω<1089/2716, respectively, and generates C4 limiting surface for ω=0.Proof.
Proof is same as Theorem6.When the initial control points are taken at the same intervals from some polynomial and if the new data points after applying subdivision scheme are located on the polynomial of same degree, then we say that the subdivision scheme generates polynomials up to that certain degree (degree of polynomial). But if the new data points after applying subdivision scheme are located on the same polynomial, then we say that the subdivision scheme reproduces polynomials up to that certain degree. See [16] for details. We will discuss the polynomial generation and reproduction of our families in the upcoming subsections.


### 3.3. Generation Degree

LetΩl and Ωl′ be the range of ω for which the subdivision schemes Sal and Sbl are convergent, respectively.Here we compute the polynomial generation of our proposed family of schemes by using the conditions given in [1] (Conditions Z1 and Zk).Theorem 8.
The subdivision schemeSal generates polynomials (i)
up to degreel for ω∈Ωl,(ii)
up to degreel+2 for ω=0.Proof.
Sinceal(1)=2, al(-1)=0, and for ω∈Ωl, Dnalzz=-1=0, n=1,2,3,…,l. Then by [1] (Condition Zk), Sal generates polynomial up to degree l. For ω=0, Dn+2alzz=-1=0, n=1,2,3,…,l. Therefore again by [1] (Condition Zk), Sal generates polynomial up to degree l+2.Theorem 9.
The subdivision schemeSb1 generates (i)
linear polynomials forω∈Ω1′,(ii)
cubic polynomials forω=0.Proof.
It is clear that(26)b11,1=4,b11,-1=0,b1-1,1=0,b1-1,-1=0.Let n1,n2∈N0=N∪0 and ω∈Ω1′, so (27)Dn1,n2b1z1,z2z1=1,z2=-1=0,Dn1,n2b1z1,z2z1=-1,z2=1=0,Dn1,n2b1z1,z2z1=-1,z2=-1=0,where n1+n2≤1. Therefore Sb1 generates linear polynomials for ω∈Ω1′.
Again letω=0; then (28)Dn1,n2b1z1,z2z1=1,z2=-1=0,Dn1,n2b1z1,z2z1=-1,z2=1=0,Dn1,n2b1z1,z2z1=-1,z2=-1=0,where n1+n2≤3. Therefore Sb1 generates cubic polynomials for ω=0.Theorem 10.
The subdivision schemeSb2 generates (i)
quadratic polynomials forω∈Ω2′,(ii)
quartic polynomials forω=0.Proof.
Proof is same as Theorem9.


### 3.4. Reproduction Degree

Now we evaluate the polynomial reproduction of proposed family of schemes by using the methods given in [1] (Corollaries  2.7 and 2.3).Theorem 11.
The subdivision schemesSal reproduce polynomials as follows: (i)
Ifω∈Ωl, Sal reproduces linear polynomials.(ii)
Ifω=23/216 and l=2, Sal reproduces quadratic polynomials.(iii)
Ifω=9+7l/48+84l and l>2, Sal reproduces cubic polynomials.Proof.
Forp∈N, the parametric value τ is given below: (29)τ=0ifl=2p-1,12ifl=2p.
Forω∈Ωl, we have D1alzz=1=2τ. Combining this condition with Theorem 8, we get the required result.
Similarly forω=23/216 and l=2, Sal reduces to the subdivision scheme with mask (30)a2=-2310368,-532592,-1372592,28110368,18075184,9071296,9071296,18075184,28110368,-1372592,-532592,-2310368.Since D(2)a2(z)z=1=-36ω+10/3 then it is equal to 2τ(τ-1) for ω=23/216∈Ω2. Now by combining this condition with Theorem 8, we get the required result.
Again forω=9+7l/48+84l∈Ωl and l>2, we have D2alzz=1=2ττ-1 and D3alzz=1=2ττ-1τ-2. Once more by combining this condition with Theorem 8, we get the result.Theorem 12.
The subdivision schemeSb1 reproduces linear polynomials for ω∈Ω1′.Proof.
Since(31)τ1=14D1,0b1z1,z2z1=1,z2=1=0,τ2=14D0,1b1z1,z2z1=1,z2=1=0then (32)4τ1=D1,0b1z1,z2z1=1,z2=1=0,4τ2=D0,1b1z1,z2z1=1,z2=1=0.Now by combing these results with Theorem 9, the required result has been proved.Theorem 13.
The subdivision schemeSb2 reproduces linear polynomials for ω∈Ω2′ and quadratic polynomials for ω=23/48.Proof.
Since(33)τ1=14D1,0b2z1,z2z1=1,z2=1=12,τ2=14D0,1b2z1,z2z1=1,z2=1=12then(34)4τ1=D1,0b2z1,z2z1=1,z2=1=2,4τ2=D0,1b2z1,z2z1=1,z2=1=2.
Also, forω=23/48, we have (35)4τ1τ2=D1,1b2z1,z2z1=1,z2=1=1,4τ1τ1-1=D2,0b2z1,z2z1=1,z2=1=-1,4τ2τ2-1=D0,2b2z1,z2z1=1,z2=1=-1.Now by combing these results with Theorem 10, the required result has been proved.Remark 14.
If we takeω=0, the symbols (16) and (22) reduce to the symbol of “univariate family of subdivision schemes” and “bivariate family of subdivision schemes.” These families produce Cl+2 limiting curves and surfaces, respectively. Also these families generate polynomials up to degree l+2, while polynomial reproduction of these families is linear.


## 3.1. Support of Basic Limit Function

Support of basic limit function of a subdivision scheme is equal to the number of spans of the curve affected when one control point is moved from its initial place.A convergent subdivision schemeSal defines a basic limit function ϕl=Sal∞δi with δi being the initial data such that (24)δi=1fori=0,0otherwise.Support of basic limit function of our univariate family of schemes depends on parameterω.(i)
Forω≠0∈Ωl, the numbers of nonzero coefficients in the symbol of refining operator (13) and smoothing operator (15) are 6 and 4, respectively. So by [18] support size of basic limit function of the schemes Sal for the general symbol al(z) is 3l+5 and its support width is -3l+5/2,3l+5/2.(ii)
Forω=0, the number of nonzero coefficients in the symbol of both refining and smoothing stages is 4; therefore support size of basic limit function of the schemes Sal for the general symbol al(z) is 3(l+1) and its support width is -3/2l+1,3/2l+1.A convergent bivariate subdivision schemeSbl defines a basic limit function ϕl=Sbl∞δi,j, where δi,j is the initial data such that (25)δi,j=1fori=j=0,0otherwise.Support of basic limit of our bivariate family of schemes is 3(l+1)×3(l+1) and its support width is -3/2(l+1),3/2(l+1)×-3/2(l+1),3/2(l+1).Basic limit functions for the first three family members of univariate family for three different values of shape parameters are presented in Figure5, whereas basic limit functions for the first family member of bivariate family for three different values of shape parameter are presented in Figure 6. From these figures, we see that the support width for positive value is less than the support width for negative value of the parameter involved in the bivariate family.Figure 5
Basic limit functions of subdivision schemesSal for three values of shape parameter ω. Cross symbols show control points, while solid lines, dashed dotted lines, and dash lines represent curves obtained by subdivision schemes Sa1, Sa2, and Sa3, respectively.
(a)
ω = - 1 / 8 (b)
ω = 0 (c)
ω = 1 / 8Figure 6
Basic limit functions of subdivision schemeSb1 for three values of shape parameter ω.
(a)
ω = - 1 / 3 (b)
ω = 0 (c)
ω = 1 / 3


## 3.2. Continuity Analysis

Here we present the smoothness of proposed families of schemes by using the Laurent polynomial method. For this purpose, we have used the methods that are given ([3], Theorems  4.11, 4.13, 4.29, and 4.30).Theorem 4.
The subdivision schemeSa1 is convergent for -7/8<ω<13/38, generates C1 limiting curve for -3/10<ω<1/6, and generates C3 limiting curve for ω=0.Proof.
Since the scheme with the symbolz/1+za1(z) is contractive for -7/8<ω<13/38, therefore the scheme Sa1 is convergent for -7/8<ω<13/38. Similarly, the scheme with the symbol 2z/1+z2a1(z) is contractive for -3/10<ω<1/6; this implies that the scheme Sa1 is C1 continuous for -3/10<ω<1/6. Moreover, the scheme with symbol 4z/1+z3a1(z) is contractive for ω=0; accordingly the scheme Sa1 is C2 continuous for ω=0. Furthermore, the scheme with symbol 6z/1+z4a1(z) is contractive for ω=0; hence the scheme Sa1 is C3 continuous for ω=0.Theorem 5.
(i) The subdivision schemeSa2 is convergent for -53/100<ω<7/26, generates C1 and C2 limiting curves for -53/100<ω<7/26 and -11/27<ω<19/108, respectively, and generates C4 limiting curve for ω=0.
(ii) The subdivision schemeSa3 is convergent for -1723/3322<ω<585/1954, generates C1, C2, and C3 limiting curves for -114/271<ω<269/1154, -114/271<ω<269/1154, and -19/58<ω<61/358, respectively, and generates C5 limiting curve for ω=0.
(iii) The subdivision schemeSa4 is convergent for -5021/11121<ω<2099/7808, generates C1, C2, C3, and C4 limiting curves for -5021/11121<ω<2099/7808, -2635/7128<ω<854/3993, -2635/7128<ω<854/3993, and -703/2404<ω<391/2384, respectively, and generates C6 limiting curve for ω=0.Proof.
Proof of this theorem is same as Theorem4.Theorem 6.
The subdivision schemeSb1 is convergent for -41/36<ω<11/21, generates C1 limiting surface for -1/2<ω<19/45, and generates C3 limiting surface for ω=0.Proof.
Since the schemes with symbolsz1/1+z1b1(z1,z2) and z2/1+z2b1(z1,z2) are contractive for -41/36<ω<11/21, therefore the scheme Sb1 is convergent for -41/36<ω<11/21. Similarly, the schemes with symbols 2z1/1+z12b1(z1,z2) and 2z2/1+z22b1(z1,z2) are contractive for -1/2<ω<5/18 and the scheme with symbol 2z1/1+z1z2/1+z2b1(z1,z2) is contractive for -17/15<ω<19/45; this implies that the scheme Sb1 is C1 for -1/2<ω<19/45. Moreover, the schemes with symbols 4z1/1+z13b1(z1,z2), 4z2/1+z23b1(z1,z2), 4z1/1+z12z2/1+z2b1(z1,z2), and 4z1/1+z1z2/1+z22b1(z1,z2) are contractive for ω=0; accordingly the scheme Sb1 is C2 for ω=0. Furthermore, the schemes with symbols 6z1/1+z14b1(z1,z2), 6z2/1+z24b1(z1,z2), 6z1/1+z13z2/1+z2×b1(z1,z2), 6z1/1+z1z2/1+z23b1(z1,z2), and 6z1/1+z12z2/1+z22b1(z1,z2) are contractive for ω=0; thus the scheme Sb1 is C3 for ω=0.Theorem 7.
The subdivision schemeSb2 is convergent for -14075/12444<ω<1191/1600, generates C1 and C2 limiting surfaces for -5463/4928<ω<801/1388 and -11/16<ω<1089/2716, respectively, and generates C4 limiting surface for ω=0.Proof.
Proof is same as Theorem6.When the initial control points are taken at the same intervals from some polynomial and if the new data points after applying subdivision scheme are located on the polynomial of same degree, then we say that the subdivision scheme generates polynomials up to that certain degree (degree of polynomial). But if the new data points after applying subdivision scheme are located on the same polynomial, then we say that the subdivision scheme reproduces polynomials up to that certain degree. See [16] for details. We will discuss the polynomial generation and reproduction of our families in the upcoming subsections.


## 3.3. Generation Degree

LetΩl and Ωl′ be the range of ω for which the subdivision schemes Sal and Sbl are convergent, respectively.Here we compute the polynomial generation of our proposed family of schemes by using the conditions given in [1] (Conditions Z1 and Zk).Theorem 8.
The subdivision schemeSal generates polynomials (i)
up to degreel for ω∈Ωl,(ii)
up to degreel+2 for ω=0.Proof.
Sinceal(1)=2, al(-1)=0, and for ω∈Ωl, Dnalzz=-1=0, n=1,2,3,…,l. Then by [1] (Condition Zk), Sal generates polynomial up to degree l. For ω=0, Dn+2alzz=-1=0, n=1,2,3,…,l. Therefore again by [1] (Condition Zk), Sal generates polynomial up to degree l+2.Theorem 9.
The subdivision schemeSb1 generates (i)
linear polynomials forω∈Ω1′,(ii)
cubic polynomials forω=0.Proof.
It is clear that(26)b11,1=4,b11,-1=0,b1-1,1=0,b1-1,-1=0.Let n1,n2∈N0=N∪0 and ω∈Ω1′, so (27)Dn1,n2b1z1,z2z1=1,z2=-1=0,Dn1,n2b1z1,z2z1=-1,z2=1=0,Dn1,n2b1z1,z2z1=-1,z2=-1=0,where n1+n2≤1. Therefore Sb1 generates linear polynomials for ω∈Ω1′.
Again letω=0; then (28)Dn1,n2b1z1,z2z1=1,z2=-1=0,Dn1,n2b1z1,z2z1=-1,z2=1=0,Dn1,n2b1z1,z2z1=-1,z2=-1=0,where n1+n2≤3. Therefore Sb1 generates cubic polynomials for ω=0.Theorem 10.
The subdivision schemeSb2 generates (i)
quadratic polynomials forω∈Ω2′,(ii)
quartic polynomials forω=0.Proof.
Proof is same as Theorem9.


## 3.4. Reproduction Degree

Now we evaluate the polynomial reproduction of proposed family of schemes by using the methods given in [1] (Corollaries  2.7 and 2.3).Theorem 11.
The subdivision schemesSal reproduce polynomials as follows: (i)
Ifω∈Ωl, Sal reproduces linear polynomials.(ii)
Ifω=23/216 and l=2, Sal reproduces quadratic polynomials.(iii)
Ifω=9+7l/48+84l and l>2, Sal reproduces cubic polynomials.Proof.
Forp∈N, the parametric value τ is given below: (29)τ=0ifl=2p-1,12ifl=2p.
Forω∈Ωl, we have D1alzz=1=2τ. Combining this condition with Theorem 8, we get the required result.
Similarly forω=23/216 and l=2, Sal reduces to the subdivision scheme with mask (30)a2=-2310368,-532592,-1372592,28110368,18075184,9071296,9071296,18075184,28110368,-1372592,-532592,-2310368.Since D(2)a2(z)z=1=-36ω+10/3 then it is equal to 2τ(τ-1) for ω=23/216∈Ω2. Now by combining this condition with Theorem 8, we get the required result.
Again forω=9+7l/48+84l∈Ωl and l>2, we have D2alzz=1=2ττ-1 and D3alzz=1=2ττ-1τ-2. Once more by combining this condition with Theorem 8, we get the result.Theorem 12.
The subdivision schemeSb1 reproduces linear polynomials for ω∈Ω1′.Proof.
Since(31)τ1=14D1,0b1z1,z2z1=1,z2=1=0,τ2=14D0,1b1z1,z2z1=1,z2=1=0then (32)4τ1=D1,0b1z1,z2z1=1,z2=1=0,4τ2=D0,1b1z1,z2z1=1,z2=1=0.Now by combing these results with Theorem 9, the required result has been proved.Theorem 13.
The subdivision schemeSb2 reproduces linear polynomials for ω∈Ω2′ and quadratic polynomials for ω=23/48.Proof.
Since(33)τ1=14D1,0b2z1,z2z1=1,z2=1=12,τ2=14D0,1b2z1,z2z1=1,z2=1=12then(34)4τ1=D1,0b2z1,z2z1=1,z2=1=2,4τ2=D0,1b2z1,z2z1=1,z2=1=2.
Also, forω=23/48, we have (35)4τ1τ2=D1,1b2z1,z2z1=1,z2=1=1,4τ1τ1-1=D2,0b2z1,z2z1=1,z2=1=-1,4τ2τ2-1=D0,2b2z1,z2z1=1,z2=1=-1.Now by combing these results with Theorem 10, the required result has been proved.Remark 14.
If we takeω=0, the symbols (16) and (22) reduce to the symbol of “univariate family of subdivision schemes” and “bivariate family of subdivision schemes.” These families produce Cl+2 limiting curves and surfaces, respectively. Also these families generate polynomials up to degree l+2, while polynomial reproduction of these families is linear.


## 4. Applications and Comparisons

In this section, we have shown the capacity of parameterω to control shapes in different ways. The optical presentation of our proposed families of univariate and bivariate (nontensor product for quadrilateral meshes) subdivision schemes also has been shown. In Figure 7, we have shown limit curves that are obtained by our proposed family of univariate subdivision schemes at five values of shape parameter. In Figures 8 and 9, we have also shown limit surfaces. These surfaces are obtained by Sb1 (i.e., scheme with one time smoothing operator) and Sb2 (i.e., scheme with two times smoothing operator) on quadrilateral meshes, respectively. From these figures, we observe that shapes of limit curves/surfaces are affected by shape parameter and number of applied smoothing stages both.Figure 7
In (b)–(f), solid lines, dashed dotted lines, and dash lines show the curves obtained by subdivision schemesSa1, Sa2, and Sa3, respectively.
(a)
Initial polygon (b)
ω = - 1 / 10 (c)
ω = - 1 / 20 (d)
ω = 0 (e)
ω = 1 / 20 (f)
ω = 1 / 10Figure 8
(b)–(f) are limit surfaces obtained by applying one time our refining and one time smoothing operators for different values of shape parameterω after 3 iterations.
(a)
Initial mesh (b)
ω = - 1 / 3 (c)
ω = - 1 / 10 (d)
ω = 0 (e)
ω = 1 / 10 (f)
ω = 1 / 3Figure 9
(b)–(f) are limit surfaces obtained by applying one time our refining and two times our smoothing operators for different values of shape parameterω after 3 iterations.
(a)
Initial mesh (b)
ω = - 1 / 2 (c)
ω = - 1 / 3 (d)
ω = 0 (e)
ω = 1 / 3 (f)
ω = 1 / 2Tables1 and 2 show comparison between our univariate and bivariate families with Romani’s [8] univariate and bivariate families, respectively. From these tables it is clear that our schemes give better smoothness than Romani’s schemes for ω=0; however in this case the support size is larger than Romani’s schemes and the reproduction degree is only one.Table 1
Comparison between our univariate familySal and Romani’s univariate family Sal,ω via different properties. Let S, SS, RD, and GD be the integer smoothness, support size, reproduction degree, and generation degree, respectively.
RS-schemes S SS RD GD S a l for ω∈Ωl C l 3 l + 5 1 l S a l for ω=0∈Ωl C l + 2 3 l + 3 1 l + 2 S a 2 for ω=23/216∈Ω2 C 2 11 2 2 S a l for ω=9+7l/48+84l∈Ωl and l>2 C l 3 l + 5 3 l S a l , ω for ω∈Ωl C l l + 5 1 l S a l , ω for ω=1/16 C l l + 5 3 l + 2Table 2
Comparison between our bivariate familySbl and Romani’s bivariate family Sbl,ω via different properties. Let S, SS, RD, and GD be the integer smoothness, support size, reproduction degree, and generation degree, respectively.
RS-schemes withl≤2 S SS RD GD S b l for ω∈Ωl′ C l ( 3 l + 3 ) × ( 3 l + 3 ) 1 l S b l for ω=0∈Ωl′ C l + 2 ( 3 l + 3 ) × ( 3 l + 3 ) 1 l + 2 S b 2 for ω=23/48∈Ω2′ C 1 9 × 9 2 2 S b l , ω for ω∈Ωl′ C l ( l + 5 ) × ( l + 5 ) 1 l S b l , ω for ω=1/16 C l ( l + 5 ) × ( l + 5 ) 3 l + 2The geometrical construction of our refining operator allows us to extend our subdivision schemes to the nonuniform setting according to the geometry of the initial polygon/mesh by defining local parameterωi. Sometimes choosing a variable shape parameter ωi for each edge of the initial polygon/mesh gives better result than choosing a constant shape parameter ω for an entire initial polygon/mesh. Figures 10(a) and 10(b) demonstrate the visual effects of the proposed scheme Sa2 and the scheme proposed by Romani Sa2,ω, respectively, for ω=1/8. Similarly Figures 11(a) and 11(b) display the visual performance of the proposed scheme Sb2 and Romani’s scheme Sb2,ω, respectively, for ω=1/8, while Figures 10(c) and 11(c) show the performance of nonuniform setting of the schemes Sa2 and Sb2, respectively. Also Figures 10(d) and 11(d) represent the performance of nonuniform setting of Romani’s schemes Sa2,ωi and Sb2,ωi, respectively. These figures show that the limit curves/surfaces generated by nonuniform schemes are better than the limit curves/surfaces obtained from uniform schemes.Figure 10
Solid lines represent initial polygons while dash lines represent limit curves obtained by subdivision schemes proposed ((a) and (c)) by us and ((b) and (d)) by Romani.
(a)
Proposed uniform schemeSa2, ω=1/8 (b)
Uniform Romani’s schemeSa2,ω, ω=1/8 (c)
Nonuniform proposed schemeSa2, ωi=1/32,1/8,1/8,1/32,1/32,1/8,1/8,1/32 (d)
Nonuniform Romani’s schemeSa2,ωi, ωi=1/32,1/8,1/8,1/32,1/32,1/8,1/8,1/32Figure 11
Solid red lines represent initial mesh, while red and blue bullets show initial points.
(a)
Proposed uniform schemeSb2, ω=1/8 (b)
Romani’s uniform schemeSb2,ω, ω=1/8 (c)
Nonuniform proposed schemeSb2, ωi=1/8 at blue initial points and ωi=0 at red initial points (d)
Nonuniform Romani’s schemeSb2,ωi, ωi=1/8 at blue initial points and ωi=0 at red initial points


## 5. Conclusion

In this paper, we have presented univariate and bivariate families of approximating subdivision schemes depending on two parameters. These families are called the Refining-Smooth families where refining operator is the operator in which each refining pointfik-1,1 moves around corresponding Chaikin point Cik-1,1 in curve case and each refining point fi,jk-1,1 moves around corresponding Doo-Sabin point Ci,jk-1,1 in surface case for different values of shape parameter ω∈R and smoothing parameter l∈N, while each smoothing operator is the weighted average of the four adjacent points fik-1,n, i∈Z, in curve case and each smoothing operator is the weighted average of sixteen adjacent neighboring points fi,jk-1,n,i,j∈Z, in surface case, where n=1,2,…,l.We also have analyzed some basic properties of our proposed families. Our families of schemes give cubic polynomial reproduction for specific values of shape parameterω. Polynomial generation and continuity of our schemes are l for specific range of parameter ω, while for ω=0 they are l+2. However for ω=0, our schemes reproduce only linear polynomials. Nonuniform setting of the proposed family members gives better performance than that of uniform setting of the schemes.



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*Source: 1092476-2016-12-06.xml*