# Six-Point Subdivision Schemes with Cubic Precision

Authors: Jun Shi; Jieqing Tan; Zhi Liu; Li Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quinticB-spline refinement. By using the relation, we further propose the counterparts of cubic and quinticB-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 theB-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.


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## Body

## 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 cubicB-spline refinement algorithm [4], which actually generate uniform quadratic and cubicB-spline curves with C1 continuity and C2 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 cubicB-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 cubicB-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 C2 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 quinticB-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 quinticB-spline refinements and make the relations between the 6-point interpolatory subdivision and the counterparts of cubic and quinticB-spline refinements similar to those between the 4-point interpolatory subdivision and the cubic, quinticB-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)P2ik+1=1256αPi-2k+Pi+2k+βPi-1k+Pi+1k+256-2α-2βPik,P2i+1k+1=12563+α2Pi-2k+Pi+3k+α+β2-25Pi-1k+Pi+2k+150-α-β2Pik+Pi+1k.(1) is called the 6-point combined interpolatory and approximating binary subdivision scheme. If α=β=0, (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 C4 continuity and reproduce cubic polynomials, whereas theB-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 pointsP0={pi0∈R}i∈Z, the set of control points Pk+1={pik+1}i∈Z at level k+1 are recursively defined by the following binary subdivision rules: (2)pik+1=∑j∈Zai-2jpjk,i∈Z.The finite set a={ai}i∈Z is calledmask. The iterative algorithm based on the repeated application of (2) is termedsubdivision scheme and is denoted by Sa. The symbol of the scheme Sa is defined as az=∑i∈Zaizi.Theorem 1 (see [17]).
Let a binary subdivision schemeSa be convergent. Then the mask a={ai}i∈Z satisfies (3)∑i∈Za2i=∑i∈Za2i+1=1.Theorem 2 (see [17]).
Let subdivision schemeS with mask a={ai}i∈Z satisfy (3). Then there exists a subdivision scheme S1 (first-order divided difference scheme of S) with the property (4)dPk=S1dPk-1,where Pk=SkP0 and dPk=dPki=2kPi+1k-Pik∣i∈Z. The symbol of S1 is a1z=2z/1+zaz. Generally, if Sn (the nth-order divided difference scheme of S) exists with mask an={ain}i∈Z, then the symbol of Sn is anz=2z/1+znaz.Theorem 3 (see [17]).
(a) Let subdivision schemeS have mask a0={ai0}i∈Z, and its jth-order divided difference scheme Sj(j=1,2,…,n+1) exists with mask aj={aij}i∈Z satisfying (5)∑i∈Za2ij=∑i∈Za2i+1j=1,j=0,1,…,n.If there exists an integer L≥1, such that 1/2Sn+1L∞<1, then the subdivision scheme S is Cn continuous, where (6)12Sn+1L∞=max∑j∈Zbi-2LjL:0≤i<2L,bLz=bzbz2⋯bz2L-1,bz=12an+1z.In particular, when L=1,(7)12Sn+1∞=12max∑i∈Za2in+1,∑i∈Za2i+1n+1.
(b) Leta(z)=1+zn+1/2nbz with Sb being contractive (i.e., Sb maps any initial data to zero). Then, Sa is convergent and Cn continuous.Theorem 4 (see [18, 19]).
Letπd denote the space of all univariate polynomials with real coefficients up to degree d. Then a univariate subdivision scheme Sa(i)
generatesπd if and only if (8)a1=2,a-1=0,aj-1=0,j=1,…,d;(ii)
reproducesπd with respect to the parametrization tik=i+τ/2ki∈Z with τ=a(1)(1)/2 and k denoting the subdivision level, if and only if it generates πd and (9)aj1=2∏h=0j-1τ-h,j=1,…,d.


## 3. Construction of the New Family

This section first explains a new observation about the relation between 4-point interpolatory subdivision and quinticB-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{Pi0}, as shown in Figure 1, the rules of 4-point interpolatory subdivision for generating k+1 level vertices {Pik+1} are (10)P2ik+1=Pik,P2i+1k+1=-116Pi-1k+916Pik+916Pi+1k-116Pi+2k,and quinticB-spline refinement for generating k+1 level vertices {Qik+1} is (11)Q2ik+1=316Pi-1k+1016Pik+316Pi+1k,Q2i+1k+1=132Pi-1k+1532Pik+1532Pi+1k+132Pi+2k.Figure 1
The relation between 4-point interpolatory subdivision andB-spline refinement.
(a)
The relation between 4-point scheme and quinticB-spline scheme
(b)
The relation between 4-point scheme and cubicB-spline schemeDenote byΔ2i, Δ2i+1 the displacements of vertices from quinticB-spline refinement to 4-point interpolatory subdivision after one step of refinement, as shown in Figure 1(a), where the black lines represent the initial control polygon, the magenta lines represent the control polygon after one step of 4-point interpolatory subdivision, and the green lines represent the control polygon after one step of quinticB-spline refinement. Then, from (10) and (11), we can get(12)Δ2ik+1=P2ik+1-Q2ik+1=-316Pi-1k+616Pik-316Pi+1k,Δ2i+1k+1=P2i+1k+1-Q2i+1k+1=-332Pi-1k+332Pik+332Pi+1k-332Pi+2k.A new observation is(13)Δ2i+1k+1=12Δ2ik+1+Δ2i+2k+1,which shows that the relation between 4-point interpolatory subdivision and quinticB-spline refinement is similar to the one between 4-point interpolatory subdivision and cubicB-spline refinement discovered by Lin et al. in [7]; that is, Δ2i+1k+1=1/2Δ2ik+1+Δ2i+2k+1, as shown in Figure 1(b), where the blue lines represent the control polygon after one step of cubicB-spline refinement. So, from the point of view of displacements, 4-point interpolatory scheme has the same connections with cubicB-spline and quinticB-spline.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 quinticB-spline refinement.


### 3.2. Construction of the New 6-Point Combined Scheme

As is shown in [2], the rules of 6-point interpolatory subdivision for generating k+1 level vertices {Pik+1} are (14)P2ik+1=Pik,P2i+1k+1=3256Pi-2k+Pi+3k-25256Pi-1k+Pi+2k+150256Pik+Pi+1k.Suppose the new subdivision have the following rule:(15)P-2ik+1=1256αPi-2k+Pi+2k+βPi-1k+Pi+1k+256-2α-2βPik,where α,β are tension parameters, and then(16)P-2i+2k+1=1256αPi-1k+Pi+3k+βPik+Pi+2k+256-2α-2βPi+1k,Δ2ik+1=P2ik+1-P-2ik+1=1256-αPi-2k+Pi+2k-βPi-1k+Pi+1k+2α+2βPik,Δ2i+2k+1=P2i+2k+1-P-2i+2k+1=1256-αPi-1k+Pi+3k-βPik+Pi+2k+2α+2βPi+1k.Using relation (13), it can be deduced that (17)Δ2i+1k+1=1256-α2Pi-2k+Pi+3k-α+β2Pi-1k+Pi+2k+α+β2Pik+Pi+1k.So, we obtain (18)P-2i+1k+1=12563+α2Pi-2k+Pi+3k+α+β2-25Pi-1k+Pi+2k+150-α-β2Pik+Pi+1k,and then the new subdivision can be concluded from (15) and (18) as(19)P2ik+1=1256αPi-2k+Pi+2k+βPi-1k+Pi+1k+256-2α-2βPik,P2i+1k+1=12563+α2Pi-2k+Pi+3k+α+β2-25Pi-1k+Pi+2k+150-α-β2Pik+Pi+1k,which is the form of (1) in Section 1.The mask and symbol of subdivision (1) are (20)12563+α2,α,α+β2-25,β,150-α-β2,256-2α-2β,150-α-β2,β,α+β2-25,α,3+α2,(21)aα,βz=1+z48·1323+α2-12+αz+5+3α2+β2z2+40-2α-βz3+5+3α2+β2z4-12+αz5+3+α2z6,respectively. When β=-4α, symbol (21) can be written as(22)aαz=1+z632·183+α2-18+2αz+38+3αz2-18+2αz3+3+α2z4.In particular, when α=-10, (23)a-10z=1+z764·12-1+2z+2z2-z3.Denote the family of subdivision (1) by Saα,β and subfamily (22) by Saα. We call them the counterparts of cubic and quinticB-spline refinements based on the 6-point interpolatory subdivision. Figure 2 illustrates the limit curves of some members of Saα,β. In Section 4, we will prove that the family Saα,β generates curves with C3 continuity, and the subfamily Saα attains C4 continuity when α∈-14,-8 and reproduces cubic polynomials.Figure 2
Some limit curves generated bySaα,β all with β=-65,-50,-35,-20,-5,10,25,40,and55 from outside (red) to inside (yellow), respectively. The black is the initial control polygon.
(a)
α
=
-
30
(b)
α
=
-
10
(c)
α
=
10
(d)
α
=
30


## 3.1. A New Observation

Given an initial control polygon with vertices{Pi0}, as shown in Figure 1, the rules of 4-point interpolatory subdivision for generating k+1 level vertices {Pik+1} are (10)P2ik+1=Pik,P2i+1k+1=-116Pi-1k+916Pik+916Pi+1k-116Pi+2k,and quinticB-spline refinement for generating k+1 level vertices {Qik+1} is (11)Q2ik+1=316Pi-1k+1016Pik+316Pi+1k,Q2i+1k+1=132Pi-1k+1532Pik+1532Pi+1k+132Pi+2k.Figure 1
The relation between 4-point interpolatory subdivision andB-spline refinement.
(a)
The relation between 4-point scheme and quinticB-spline scheme
(b)
The relation between 4-point scheme and cubicB-spline schemeDenote byΔ2i, Δ2i+1 the displacements of vertices from quinticB-spline refinement to 4-point interpolatory subdivision after one step of refinement, as shown in Figure 1(a), where the black lines represent the initial control polygon, the magenta lines represent the control polygon after one step of 4-point interpolatory subdivision, and the green lines represent the control polygon after one step of quinticB-spline refinement. Then, from (10) and (11), we can get(12)Δ2ik+1=P2ik+1-Q2ik+1=-316Pi-1k+616Pik-316Pi+1k,Δ2i+1k+1=P2i+1k+1-Q2i+1k+1=-332Pi-1k+332Pik+332Pi+1k-332Pi+2k.A new observation is(13)Δ2i+1k+1=12Δ2ik+1+Δ2i+2k+1,which shows that the relation between 4-point interpolatory subdivision and quinticB-spline refinement is similar to the one between 4-point interpolatory subdivision and cubicB-spline refinement discovered by Lin et al. in [7]; that is, Δ2i+1k+1=1/2Δ2ik+1+Δ2i+2k+1, as shown in Figure 1(b), where the blue lines represent the control polygon after one step of cubicB-spline refinement. So, from the point of view of displacements, 4-point interpolatory scheme has the same connections with cubicB-spline and quinticB-spline.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 quinticB-spline refinement.


## 3.2. Construction of the New 6-Point Combined Scheme

As is shown in [2], the rules of 6-point interpolatory subdivision for generating k+1 level vertices {Pik+1} are (14)P2ik+1=Pik,P2i+1k+1=3256Pi-2k+Pi+3k-25256Pi-1k+Pi+2k+150256Pik+Pi+1k.Suppose the new subdivision have the following rule:(15)P-2ik+1=1256αPi-2k+Pi+2k+βPi-1k+Pi+1k+256-2α-2βPik,where α,β are tension parameters, and then(16)P-2i+2k+1=1256αPi-1k+Pi+3k+βPik+Pi+2k+256-2α-2βPi+1k,Δ2ik+1=P2ik+1-P-2ik+1=1256-αPi-2k+Pi+2k-βPi-1k+Pi+1k+2α+2βPik,Δ2i+2k+1=P2i+2k+1-P-2i+2k+1=1256-αPi-1k+Pi+3k-βPik+Pi+2k+2α+2βPi+1k.Using relation (13), it can be deduced that (17)Δ2i+1k+1=1256-α2Pi-2k+Pi+3k-α+β2Pi-1k+Pi+2k+α+β2Pik+Pi+1k.So, we obtain (18)P-2i+1k+1=12563+α2Pi-2k+Pi+3k+α+β2-25Pi-1k+Pi+2k+150-α-β2Pik+Pi+1k,and then the new subdivision can be concluded from (15) and (18) as(19)P2ik+1=1256αPi-2k+Pi+2k+βPi-1k+Pi+1k+256-2α-2βPik,P2i+1k+1=12563+α2Pi-2k+Pi+3k+α+β2-25Pi-1k+Pi+2k+150-α-β2Pik+Pi+1k,which is the form of (1) in Section 1.The mask and symbol of subdivision (1) are (20)12563+α2,α,α+β2-25,β,150-α-β2,256-2α-2β,150-α-β2,β,α+β2-25,α,3+α2,(21)aα,βz=1+z48·1323+α2-12+αz+5+3α2+β2z2+40-2α-βz3+5+3α2+β2z4-12+αz5+3+α2z6,respectively. When β=-4α, symbol (21) can be written as(22)aαz=1+z632·183+α2-18+2αz+38+3αz2-18+2αz3+3+α2z4.In particular, when α=-10, (23)a-10z=1+z764·12-1+2z+2z2-z3.Denote the family of subdivision (1) by Saα,β and subfamily (22) by Saα. We call them the counterparts of cubic and quinticB-spline refinements based on the 6-point interpolatory subdivision. Figure 2 illustrates the limit curves of some members of Saα,β. In Section 4, we will prove that the family Saα,β generates curves with C3 continuity, and the subfamily Saα attains C4 continuity when α∈-14,-8 and reproduces cubic polynomials.Figure 2
Some limit curves generated bySaα,β all with β=-65,-50,-35,-20,-5,10,25,40,and55 from outside (red) to inside (yellow), respectively. The black is the initial control polygon.
(a)
α
=
-
30
(b)
α
=
-
10
(c)
α
=
10
(d)
α
=
30


## 4. Analysis of the New Family

### 4.1. Smoothness Analysis

Proposition 5.
The schemeSaα,β defined by (1) converges and has smoothness C3 when β∈32,40 and α∈-4-β/4,4-β/4, or β∈40,72 and α∈-4-β/4,14-β/2; and when α∈-14,-8, the subfamily Saα generates C4 continuous limit curves.Proof.
The symbol ofSaα,β can be written as(24)aα,βz=∑iaizi=1+z48·bα,βz,where(25)bα,βz=1323+α2-12+αz+5+3α2+β2z2+40-2α-βz3+5+3α2+β2z4-12+αz5+3+α2z6.Let bi denote the coefficients of Laurent polynomial bα,β(z). By Theorem 3(b), if Sbα,β is contractive, then Saα,β is C3. When(26)β∈32,40,α∈-4-β4,4-β4orβ∈40,72,α∈-4-β4,14-β2,Sbα,β∞=max∑ib2i,∑ib2i+1=132max25+3α2+β2+3+α2,40-2α-β+212+α<1,which shows that Sbα,β is contractive; hence, when(27)β∈32,40,α∈-4-β4,4-β4orβ∈40,72,α∈-4-β4,14-β2,Saα,βisC3.
Whenβ=-4α, the symbol of the subfamily Saα is (28)aαz=1+z632·183+α2-18+2αz+38+3αz2-18+2αz3+3+α2z4,which can also be written as aαz=1+z5/16·bαz, where (29)bαz=1163+α2-15+3α2z+20+αz2+20+αz3-15+3α2z4+3+α2z5.When α∈-14,-8, Sbα∞=1/163+α/2+15+3α/2+20+α<1.
Hence, by Theorem3(b), Sbα is contractive and Saα is C4 when α∈-14,-8.


### 4.2. Generation Degree and Reproduction Degree

Polynomial generation and polynomial reproduction are desirable properties because any convergent subdivision scheme that reproduces polynomials of degreek has approximation order k+1 [18]. The polynomial generation of degree k is the capability of subdivision schemes to generate the full space of polynomials of degree k [20]. The polynomial reproduction is the capability of subdivision schemes to produce in the limit exactly the same polynomial from which the initial data is sampled. The generation degree is not less than the reproduction degree. For example, the generation degree of degree-n B-spline refinement is n, but the reproduction degree of degree-n B-spline refinement only attains 1. Hormann and Sabin [16] proposed a family of subdivision schemes Sk(k∈N) which is defined by the product of the symbol ofB-spline refinement with a degree-2 polynomial and increased the degree of polynomial reproduction ofB-spline schemes from 1 to 3.LetDα,β={α,β∈R∣Saα,βisconvergent} and suppose α,β∈Dα,β. Using Theorem 4, we get the following results.Proposition 6.
The subdivision schemeSaα,β generates(30)π3,ifβ≠-4α,π5,ifβ=-4α.In particular, when α=-10, Saα generates π7.Proof.
The symbol ofSaα,β can be written as (31)aα,βz=1256·AzBz,where Az=1+z4 and Bz=32·bα,βz.
Then,aα,β(n)z=1/256·∑i=0nCniA(i)(z)B(n-i)z, and aα,β(1)-1=aα,β(2)-1=aα,β(3)-1=0.
Moreover, whenβ=-4α, (32)aα4-1=aα5-1=0;and when α=-10, (33)aα6-1=aα7-1=0,aα8-1≠0.Hence, according to Theorem 4(i), we get that when β≠-4α, the subdivision scheme Saα,β generates π3; when β=-4α,Saα generates π5 and when α=-10, Saα generates π7.Proposition 7.
If applying the parameter shiftτ=5, the subdivision scheme Saα,β reproduces(34)π1,ifβ≠-4α,π3,ifβ=-4α,with respect to the parametrization tik=i+τ/2ki∈Z, where k denotes the subdivision level. In particular, when α=0, Saα reproduces π5.Proof.
To consider the reproduction degree of the subdivision schemeSaα,β, in view of Theorem 4(ii), we just need to consider aα,βj1,j=1,…,d. Using the notation in Proposition 6, we get that (35)aα,β11=10,aα,β21=40+4α+β16,aα,β31=120+384α+β,so τ=aα,β11/2=5, and when β=-4α,(36)aα21=40=2ττ-1,aα31=120=2ττ-1τ-2.Then aα41=240+3α, and when α=0, (37)aα41=240=2∏h=03τ-h,aα51=240=2∏h=04τ-h,aα61≠0=2∏h=05τ-h.
Hence, using Theorem4(ii), we conclude that when β≠-4α, the subdivision scheme Saα,β reproduces π1; when β=-4α,Saα reproduces π3 and when α=0, Saα reproduces π5.As the new family of subdivision schemesSaα,β is deduced from the 6-point interpolatory scheme using the relation between 4-point interpolatory scheme and cubic, quinticB-spline, the properties of all of them are summarized in Table 1. For the new subfamily Saα and Hormann-Sabin’s family Sk(k∈N) and both have cubic precision, we list corresponding properties for a comparison in Table 2.Table 1
Comparison between properties of cubicB-spline refinement, quintic B-spline refinement, 6-point interpolatory scheme, and the new family of schemes Saα,β, Saα.
Scheme
Support
Continuity
Generation degree
Reproduction degree
4-p interpolatory scheme
6
1
3
3
CubicB-spline
4
2
3
1
QuinticB-spline
6
4
5
1
6-p interpolatory scheme
10
2
5
5
S
a
α
,
β
10
3
3
1
S
a
α
10
4
5
3
S
a
α
(
α
=
-
10
)
10
4
7
3Table 2
Comparison between properties of Hormann-Sabin’s familySk,k∈N, and the new family of schemes Saα.
Scheme
Support
Continuity
Generation degree
Reproduction degree
S
4
6
1
3
3
S
5
7
2
4
3
S
6
8
3
5
3
S
7
9
4
6
3
S
8
10
5
7
3
S
9
11
6
8
3
S
k
k
≥
4
k
+
2
k
-
l
o
g
2
2
+
k
/
2
k
-
1
3
S
a
α
10
4
5
3
S
a
α
(
α
=
-
10
)
10
4
7
3


## 4.1. Smoothness Analysis

Proposition 5.
The schemeSaα,β defined by (1) converges and has smoothness C3 when β∈32,40 and α∈-4-β/4,4-β/4, or β∈40,72 and α∈-4-β/4,14-β/2; and when α∈-14,-8, the subfamily Saα generates C4 continuous limit curves.Proof.
The symbol ofSaα,β can be written as(24)aα,βz=∑iaizi=1+z48·bα,βz,where(25)bα,βz=1323+α2-12+αz+5+3α2+β2z2+40-2α-βz3+5+3α2+β2z4-12+αz5+3+α2z6.Let bi denote the coefficients of Laurent polynomial bα,β(z). By Theorem 3(b), if Sbα,β is contractive, then Saα,β is C3. When(26)β∈32,40,α∈-4-β4,4-β4orβ∈40,72,α∈-4-β4,14-β2,Sbα,β∞=max∑ib2i,∑ib2i+1=132max25+3α2+β2+3+α2,40-2α-β+212+α<1,which shows that Sbα,β is contractive; hence, when(27)β∈32,40,α∈-4-β4,4-β4orβ∈40,72,α∈-4-β4,14-β2,Saα,βisC3.
Whenβ=-4α, the symbol of the subfamily Saα is (28)aαz=1+z632·183+α2-18+2αz+38+3αz2-18+2αz3+3+α2z4,which can also be written as aαz=1+z5/16·bαz, where (29)bαz=1163+α2-15+3α2z+20+αz2+20+αz3-15+3α2z4+3+α2z5.When α∈-14,-8, Sbα∞=1/163+α/2+15+3α/2+20+α<1.
Hence, by Theorem3(b), Sbα is contractive and Saα is C4 when α∈-14,-8.


## 4.2. Generation Degree and Reproduction Degree

Polynomial generation and polynomial reproduction are desirable properties because any convergent subdivision scheme that reproduces polynomials of degreek has approximation order k+1 [18]. The polynomial generation of degree k is the capability of subdivision schemes to generate the full space of polynomials of degree k [20]. The polynomial reproduction is the capability of subdivision schemes to produce in the limit exactly the same polynomial from which the initial data is sampled. The generation degree is not less than the reproduction degree. For example, the generation degree of degree-n B-spline refinement is n, but the reproduction degree of degree-n B-spline refinement only attains 1. Hormann and Sabin [16] proposed a family of subdivision schemes Sk(k∈N) which is defined by the product of the symbol ofB-spline refinement with a degree-2 polynomial and increased the degree of polynomial reproduction ofB-spline schemes from 1 to 3.LetDα,β={α,β∈R∣Saα,βisconvergent} and suppose α,β∈Dα,β. Using Theorem 4, we get the following results.Proposition 6.
The subdivision schemeSaα,β generates(30)π3,ifβ≠-4α,π5,ifβ=-4α.In particular, when α=-10, Saα generates π7.Proof.
The symbol ofSaα,β can be written as (31)aα,βz=1256·AzBz,where Az=1+z4 and Bz=32·bα,βz.
Then,aα,β(n)z=1/256·∑i=0nCniA(i)(z)B(n-i)z, and aα,β(1)-1=aα,β(2)-1=aα,β(3)-1=0.
Moreover, whenβ=-4α, (32)aα4-1=aα5-1=0;and when α=-10, (33)aα6-1=aα7-1=0,aα8-1≠0.Hence, according to Theorem 4(i), we get that when β≠-4α, the subdivision scheme Saα,β generates π3; when β=-4α,Saα generates π5 and when α=-10, Saα generates π7.Proposition 7.
If applying the parameter shiftτ=5, the subdivision scheme Saα,β reproduces(34)π1,ifβ≠-4α,π3,ifβ=-4α,with respect to the parametrization tik=i+τ/2ki∈Z, where k denotes the subdivision level. In particular, when α=0, Saα reproduces π5.Proof.
To consider the reproduction degree of the subdivision schemeSaα,β, in view of Theorem 4(ii), we just need to consider aα,βj1,j=1,…,d. Using the notation in Proposition 6, we get that (35)aα,β11=10,aα,β21=40+4α+β16,aα,β31=120+384α+β,so τ=aα,β11/2=5, and when β=-4α,(36)aα21=40=2ττ-1,aα31=120=2ττ-1τ-2.Then aα41=240+3α, and when α=0, (37)aα41=240=2∏h=03τ-h,aα51=240=2∏h=04τ-h,aα61≠0=2∏h=05τ-h.
Hence, using Theorem4(ii), we conclude that when β≠-4α, the subdivision scheme Saα,β reproduces π1; when β=-4α,Saα reproduces π3 and when α=0, Saα reproduces π5.As the new family of subdivision schemesSaα,β is deduced from the 6-point interpolatory scheme using the relation between 4-point interpolatory scheme and cubic, quinticB-spline, the properties of all of them are summarized in Table 1. For the new subfamily Saα and Hormann-Sabin’s family Sk(k∈N) and both have cubic precision, we list corresponding properties for a comparison in Table 2.Table 1
Comparison between properties of cubicB-spline refinement, quintic B-spline refinement, 6-point interpolatory scheme, and the new family of schemes Saα,β, Saα.
Scheme
Support
Continuity
Generation degree
Reproduction degree
4-p interpolatory scheme
6
1
3
3
CubicB-spline
4
2
3
1
QuinticB-spline
6
4
5
1
6-p interpolatory scheme
10
2
5
5
S
a
α
,
β
10
3
3
1
S
a
α
10
4
5
3
S
a
α
(
α
=
-
10
)
10
4
7
3Table 2
Comparison between properties of Hormann-Sabin’s familySk,k∈N, and the new family of schemes Saα.
Scheme
Support
Continuity
Generation degree
Reproduction degree
S
4
6
1
3
3
S
5
7
2
4
3
S
6
8
3
5
3
S
7
9
4
6
3
S
8
10
5
7
3
S
9
11
6
8
3
S
k
k
≥
4
k
+
2
k
-
l
o
g
2
2
+
k
/
2
k
-
1
3
S
a
α
10
4
5
3
S
a
α
(
α
=
-
10
)
10
4
7
3


## 5. Conclusions

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 subfamilySaα attains cubic precision whereas theB-spline schemes have linear precision (see Figure 3). On the other hand, both having cubic precision, Hormann-Sabin’ family Skk∈N increases the degree of polynomial generation and smoothness in exchange of the increase of the support width, while Saα can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness. Moreover, the tension parameter α makes Saα able to provide more choices in applications (see Figures 4 and 5).Figure 3
The polynomial reproduction property ofSaαα=4 with (a) y=x2 and (b) y=x3. The blue is the initial control polygon.
(a)
(b)Figure 4
Comparison of limit curves (the blue curves) generated by (a) 4-p interpolatory scheme, cubicB-spline, and quinticB-spline refinement from outer to inner part and (b) 6-p interpolatory scheme, Saαα=-8 and Saα,βα=8,β=10, from outer to inner part. The red is the initial control polygon.
(a)
(b)Figure 5
Comparison of limit curves generated bySk (a) with k=4,5,6,8,and10 from outer to inner part and Saα (b) with α=8,4,0,-4,-8,-12,-20,and-32 from outer to inner part. The red is the initial control polygon.
(a)
(b)



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*Source: 2324893-2018-01-03.xml*