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82 

J. mt. area res., Vol. 8, 2023 

   

        Journal of Mountain Area Research 
   

          
 

 

A UNIQUE COMBINATION OF MASK IN BINARY FOUR-POINT SUBDIVISION 

SCHEME 
Uzma Mukhtar1and Kashif Rehan2, * 

 

1. Department of Mathematics, University of Engineering and Technology, Lahore, Pakistan. 

2. Department of Mathematics, University of Engineering and Technology, New Campus, Lahore, Pakistan. 

 

ABSTRACT 

A unique binary four-point approximating subdivision scheme has been developed in which 

one part of binary formula have stationary mask and other part have the non-stationary mask. 

The resulting curves have the smoothness of C3 continuous for the wider range of shape control 

parameter. The role of the parameter has been depicted using the square form of discrete 

control points. 

 

Keywords: Binary, Stationary and Non-stationary, Approximating, S ubdivision Scheme. 

2010 AMS Subject Classifications: 41A05, 41A25, 41A30, 65D17, 65D10. 

 

*Corresponding authors: (Email: Kashif Rehan: kkashif_99@yahoo.com) 

 

1. INTRODUCTION 

Geometric modelling is the heart of Computer 

Graphics and Computer Aided Geometric 

Design and covers a wide range of 

applications. Computer Aided Geometric 

Design is a branch of applied mathematics that 

designs smooth curves/ surfaces with algorithms. 

The Computer Aided Geometric Design field 

compiles the visual display. Computer Aided 

Geometric Design studies the design and 

handling of curves / surfaces provided by a 

collection of points. The development of new 

geometric objects and shapes is an important 

task in the field of Computer Aided Geometric 

Design, Computer Graphic, Computer 

Animation Industry, and Image Processing etc. 

Subdivision is an important tool to create 

different geometric objects and shapes in 

Computer Aided Geometric Design. Algorithms 

of subdivisions are suitable for computer 

applications. Subdivision develops different 

types of smooth curves and surfaces using 

refining rules by subdividing them from a 

collection of discrete data points. 

For the development of subdivision field, G. de 

Rham firstly developed a subdivision scheme in 

1947. Subdivision is a method that generates 

smooth curves and surfaces just as successive 

arrangement of refined control polygons. The 

current polygon is included to new points at 

each refining level and the basic points 

continue to remain or are thrown away across 

all resultant control polygons. The number of 

points placed from step 𝑘  to level 𝑘 + 1 

between two successive points is termed also 

the scheme’s arity. If there are 2,3,4 ,…, n 

inserted then the subdivision schemes are 

known as binary, ternary,…, n-ary respectively. 

Subdivision is also a technique for construction 

of smooth curves / surfaces, which first applied 

as an extension of splines to arbitrary topology 

control nets. Simplicity and flexibility of the 

subdivision algorithms make them suitable for 

Vol. 8, 2023 

https://doi.org/10.53874/jmar.v8i0.168 

Full length article 

mailto:kkashif_99@yahoo.com


 

 

 

Uzma et al., J. mt. area res. 08 (2023) 82-89 

83 

J. mt. area res., Vol. 8, 2023 

many interactive computer graphics 

applications. Purity of the subdivision lies in the 

construction of smooth curves / surfaces. 

However, the uses such as special aspects of an 

objects, animations, and construction of 

composite geometric shapes which mostly like 

real world geometry. 

An important characteristic of these schemes is 

that these schemes have local effect and not 

global effect on curve by changing the control 

points. Although the mathematical surface 

analysis resulting from subdivision algorithms is 

not always easy. Subdivision scheme takes the 

attention of scholars and researchers in this field 

because of its simplicity and understanding. In 

the last two decades, many papers have been 

published. There will be a lot of new Computer 

Aided Geometric Design applications in the 

future. In Computer Aided Geometric Design, 

the most frequently used and attractive 

schemes are subdivision schemes that draw 

different types of objects in the form of curves 

and surfaces. Iterative refinements use the 

subdivision schemes to build continuous 

curves/surfaces from the collection of specific 

control points. Subdivision schemes have been 

valued in various areas, just as Image 

Processing, Computer Graphic and Computer 

Animation, due to their clarity and easily 

handling. 

Dyn et al. [1] conditions are algebraic and easy 

to check by considering the subdivision scheme 

symbol, but also relate to the scheme’s 

parameterization. The convexity of four-point 

ternary interpolating subdivision scheme by Cai 

[2] is analysed with a tension parameter. It is 

shown that the resulting curve is C2 for a certain 

range of the tension parameter. 

Victoria et al. [3] shows that the subdivision 

curve converges and is continuous. In addition, 

starting with the initial polygon’s chord-length 

parameterization. We get a subdivision curve 

parameterized by an arc-length multiple. The 

weights of the masks of the scheme are defined 

by Daniel and Shummugaraj [4] in terms of some 

values of trigonometric B-spline functions. 

Mustafa et al. [5] proposed and analysed the m-

point approximating subdivision scheme with 

single parameter where 𝑚 > 1.  Compared to 

the existing binary and ternary subdivision 

scheme, smoothness of schemes is higher.  

Zheng and Peng [6] presented an explicit 

formula that unifies the mask of ternary 

interpolation and approximation of subdivision 

scheme (2𝑛 − 1)  points. Augsdorfer et al. [7] 

proposed a different form of binary four-point 

scheme that is helpful in computer designing. 

Deng and Guozhao, [8] developed an in-centre 

subdivision scheme for curve interpolation. 

Siddiqi and Rehan [9] improved the binary 4-

point approximating subdivision scheme by 

introducing a global tension parameter. A new 

Subdivision Scheme for corner cutting was also 

proposed, which generates a limiting curve of 

C1 continuity. Siddiqi and Rehan [10] presented 

ternary three-point non-stationary 

approximating subdivision scheme that 

generates a limiting curve C2. Tan et al. [11] 

proposed a new four-point shape preserving C3 

subdivision technique. Tan et al. [12] developed 

a non-stationary three-point binary 

approximating subdivision scheme that 

generates C3 smooth curves using the discrete 

control points. Kim et al. [13] proposed a binary 

subdivision scheme with four points that 

generates a smooth C3-continuous limiting 

curve. With the help of local cubic polynomial 

fitting, Floater [14] derived an approximation 

property of four-point interpolatory curve 

subdivision. 



 

 

 

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84 

J. mt. area res., Vol. 8, 2023 

2. STATIONARY BINARY FOUR-POINT 

SUBDIVISION TECHNIQUE 

Kim et al. [13] introduced a binary subdivision 

technique with four points which creates a 

smooth C3-continuous limiting curve. Given the 

set of control points 𝑞0 = {𝑞𝑖
0}𝑖∈𝑍 at step 0, the 

binary four-point subdivision scheme for the 

design of curves develops a new collection of 

control points {𝑞𝑖
𝑘+1}𝑖∈𝑍 at step 𝑘 + 1 using the 

following subdivision rules 

𝑞2𝑖
𝑘+1 = −𝛽𝑞𝑖−2

𝑘 + 4𝛽𝑞𝑖−1
𝑘 + (1 − 6𝛽)𝑞𝑖

𝑘 + 4𝛽𝑞𝑖+1
𝑘

− 𝛽𝑞𝑖+2
𝑘  

                    

 𝑞2𝑖+1
𝑘+1 = −

1

16
𝑞𝑖−1

𝑘 +
9

16
𝑞𝑖

𝑘 +
9

16
𝑞𝑖+1

𝑘 −
1

16
𝑞𝑖+2

𝑘       (1) 

Where 𝑞0 = {𝑞𝑖
0}𝑖∈𝑍  is the collection of initial 

control points at step 0 and the sum of the mask 

of the scheme should be one as described in 

equation (1). 

They found that the scheme is C1-continuous 

when −0.12 < 𝛽 < 0.21 and the scheme is C2-

continuous and C3-continuous when −0.88 <

𝛽 < 0.13. 

 

3. NON-STATIONARY BINARY FOUR-

POINT SUBDIVISION TECHNIQUE  

The refining rules of the binary non-stationary 

four points subdivision scheme is defined as 

𝑞2𝑖
𝑘+1 = −𝛽0

𝑘 𝑞𝑖−2
𝑘 + 𝛽1

𝑘 𝑞𝑖−1
𝑘 + 𝛽2

𝑘 𝑞𝑖
𝑘 + 𝛽3

𝑘 𝑞𝑖+1
𝑘 − 𝛽4

𝑘 𝑞𝑖+2
𝑘  

 𝑞2𝑖+1
𝑘+1 = −

1

16
𝑞𝑖−1

𝑘 +
9

16
𝑞𝑖

𝑘 +
9

16
𝑞𝑖+1

𝑘 −
1

16
𝑞𝑖+2

𝑘       (2) 

Where 𝑞𝑘 = {𝑞𝑖
𝑘 }

𝑖∈𝑍
 is the collection of initial 

control point at any level ‘k’ and the mask of the 

scheme must satisfy the relationship 𝛽0 + 𝛽1 +

𝛽2 + 𝛽3 + 𝛽4 = 1. 

The binary non-stationary four-point 

subdivision scheme (2) is counter part of 

stationary scheme [13]. The mask of the 

proposed scheme is given by  

𝛽0
𝑘 = 𝑔(𝛽𝑘+1) = 𝛽4

𝑘 , 𝛽1
𝑘 = 4𝑔(𝛽𝑘+1) = 𝛽3

𝑘 ,  

𝛽2 = 1 − 6𝑔(𝛽
𝑘+1); 

𝑔(𝛽𝑘+1) =
1

2
[

 (𝛽𝑘+1)2−1

(𝛽𝑘+1)2+60
]                (3) 

With  

 𝛽𝑘+1 = √𝛽𝑘 + 2, 𝑎𝑛𝑑 𝛽0 ∈ [−2, +∞).          

(4) 

 

In this way, the coefficients 𝛽𝑖
𝑘 at each different 

steps k can be calculated using the given 

formula and an initial parameter 𝛽0 ∈ [−2, +∞). 

Starting with any 𝛽0 ≥ −2,  we have 𝛽𝑘 + 2 ≥

0, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘 ∈ 𝑍+, so 𝛽
𝑘+1 is constantly 

characterized. A wide range of definitions 

enables to achieve significant modification in 

the form of the smooth curves. 

Remark 1. As the initial values of 𝛽0 extend in its 

definition range, the behaviour of the smooth 

curve develops significantly from the figure (1) 

and tends toward approximation the initial 

control polygon just as 𝛽0 → +∞. 

Remark 2. Equation (4), the sequence {𝛽𝑘 }𝑘∈𝑁 

will be stationary, if we have 𝛽0 = 2 then 𝛽𝑘 =

2 , ∀ 𝑘 ∈ 𝑍+ . So that the non-stationary 

subdivision scheme is then retrograde to the 

stationary subdivision scheme. 

If 𝛽0 > 2 , then 𝛽𝑘 > 2 , the sequence {𝛽𝑘 }𝑘∈𝑁 

will be decreases strictly that 𝛽𝑘 converges to 2 

as the 𝑘 → ∞. 

If 𝛽0 < 2 , then 𝛽𝑘 < 2 , the sequence {𝛽𝑘 }𝑘∈𝑁 

will be increases strictly and 𝛽𝑘 converges to 2 

as the 𝑘 → ∞. 

Therefore, in the definition domain of any 𝛽0 

we always have  

                lim
𝑘→+∞

𝛽𝑘 = 2. 

4. SMOOTHNESS  

In this section, we will demonstrate that for the 

given initial discrete polygon, the subdivision 

scheme (3) gives a smooth C3 continuous curves 

for any selection of the parameter values 𝛽0 in 

its definition range. Considering, Dyn and Levin’s 

well-known results [14], smoothness about non-



 

 

 

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85 

J. mt. area res., Vol. 8, 2023 

stationary scheme to its counterpart stationary 

scheme, asymptotically equivalent method has 

been used. 

Theorem.  A non-stationary subdivision scheme 

defined in equation (3), the masks are 

asymptotically equivalent to the stationary 

scheme (1). So the proposed scheme has C3-

continuous limit curves for the shape control 

parameter 𝛽0 ∈ [−2, +∞). 

Proof. To check the smoothness of the proposed 

scheme C3-continuous, it is necessary to obtain 

the second divided difference mask. The mask 

of the scheme is 

𝑚𝑘

= [−𝑔( 𝛽𝑘+1), −  
1

16
, 4𝑔( 𝛽𝑘+1),

9

16
, 1

− 6𝑔( 𝛽𝑘+1),
9

16
 , 4𝑔( 𝛽𝑘+1), −  

1

16
, −𝑔( 𝛽𝑘+1)] 

Then it turns out its first divided difference masks 

are 

𝑒(1)
𝑘 = 2 [−𝛽, (𝛽 −

1

16
) , (3𝛽 +

1

16
) , (

1

2
− 3𝛽) , (

1

2

− 3𝛽) , (3𝛽 +
1

16
)  , (𝛽 −

1

16
) , −𝛽] 

 

Then it turns out its 2nd divided difference masks 

are 

𝑒(2)
𝑘 = 4 [−𝛽, (2𝛽 −

1

16
) , (𝛽 +

1

8
) , (

3

8
− 4𝛽) , (𝛽

+
1

8
)  , (2𝛽 −

1

16
) , −𝛽] 

Then it turns out its 3rd divided difference masks 

are 

 𝑒(3)
𝑘 = 8 [−𝛽, (3𝛽 −

1

16
) , (−2𝛽 +

3

16
) , (−2𝛽 +

3

16
)  , (3𝛽 −

1

16
) , −𝛽] 

The application of Remark 2 now provides  

 𝑒(3)
∞ = lim

𝑘→+∞
𝑒3

𝑘 = 8 [−
3

128
,

1

128
 ,

18

128
,

18

128
,

1

128
, −

3

128
] 

This is only the coefficients of the stationary 

technique’s third divided differences with 

equation coefficients in (1). In this case, the 

stationary technique is C3-continuous, the 

technique associated with 𝑒3
∞  will be C3 

smooth. If it is as  

                                           

∑ ‖𝑒(3)
𝑘 − 𝑒3

∞‖
∞

< +∞+∞𝑘=0 .                      (5) 

The two techniques are then equivalent 

asymptotically. And one can conclude this that 

the 𝑒3
∞ of the technique is C3, since then 

𝑒(3)
𝑘 − 𝑒3

∞ = 8[−2𝑔( 𝛽𝑘+1)

+
6

128
 , −3 (−2𝑔( 𝛽𝑘+1)

+
6

128
, 2(−2𝑔( 𝛽𝑘+1) +

6

128
)]  

‖𝑒(3)
𝑘 − 𝑒3

∞‖
∞

= 8𝑚𝑎𝑥 {3 |−2𝑔( 𝛽𝑘+1)

+
6

128
 | , |−3| |−2𝑔( 𝛽𝑘+1)

+
6

128
 | , 2 |−2𝑔( 𝛽𝑘+1) +

6

128
 |} 

                 = 24|−2𝑔( 𝛽𝑘+1) +
6

128
 | 

                 = 48|
3

128
−  𝑔( 𝛽𝑘+1)| 

Now we are proving the series smoothness  

                                             

∑ |
3

128
− 𝑔( 𝛽𝑘+1)|+∞𝑘=0                           

(6) 

Which depends on the 𝑔( 𝛽𝑘+1) function. Now, 

since 𝑔( 𝛽𝑘+1) is expressed in terms of the 𝛽𝑘+1, 

the behaviour of  𝛽𝑘+1 varies in the interval [0, 

+∞). From now on  

3

128
− 𝑔( 𝛽𝑘+1) = 0   𝛽𝑘+1 = 2(i.e., 𝛽𝑘 = 2)). 

3

128
−  𝑔( 𝛽𝑘+1) > 0   𝛽𝑘+1 ∈ (−1, 2)(𝑖. 𝑒. , 𝛽0 ∈

[−2,2)). 

3

128
− 𝑔( 𝛽𝑘+1) < 0   𝛽𝑘+1 ∈ (2, +∞). 



 

 

 

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86 

J. mt. area res., Vol. 8, 2023 

Now, we are talking about smoothness of (6) in 

the three cases. 

Case 1: 𝜷𝟎 = 𝟐 (𝒊. 𝒆. , 𝜷𝒌+𝟏 = 𝟐) 

Then        ‖𝑒(3)
𝑘 − 𝑒3

∞‖
∞

= 0. 

Smoothness of (6) follows. 

 

Case 2: 𝜷𝟎 ∈ [−𝟐, 𝟐)(𝒊. 𝒆. , 𝜷𝒌+𝟏 ∈ [𝟎, 𝟐)). 

In this case 

‖𝑒(3)
𝑘 − 𝑒3

∞‖
∞

= 48 [
3

128
−

1

2
[

 (𝛽𝑘+1)2−1

(𝛽𝑘+1)2+60
]]  

Thus  

∑ (
3

128
− 𝑔( 𝛽𝑘+1))+∞𝑘=0 =  ∑ (

3

128
−+∞𝑘=0

1

2
[

 (𝛽𝑘+1)2−1

(𝛽𝑘+1)2+60
]) < +∞.                                                           

We use the ratio test to this end. Since  
3

128
−

𝑔( 𝛽𝑘+1) > 0  and the sequence {𝛽𝑘 }𝑘∈𝑁  is 

increases surely in this case, we got it  

3
128

−
1
2

[
 (𝛽𝑘+2)2 − 1
(𝛽𝑘+2)2 + 60

]

3
128

−
1
2

[
 (𝛽𝑘+1)2 − 1
(𝛽𝑘+1)2 + 60

]
 < 1. 

The smoothness of (6) has therefore been 

proven.  

Case 3:     𝜷𝟎 ∈ (𝟐, +∞)(𝒊. 𝒆. , 𝜷𝒌+𝟏 ∈ (𝟐, +∞)). 

In this case               

‖𝑒(3)
𝑘 − 𝑒3

∞‖
∞

= 48 [
1

2
{

 (𝛽𝑘+1)2 − 1

(𝛽𝑘+1)2 + 60
} −

3

128
] 

Thus  

∑ (𝑔( 𝛽𝑘+1) −
3

128
)

+∞

𝑘=0

=  ∑ (
1

2
{

 (𝛽𝑘+1)2 − 1

(𝛽𝑘+1)2 + 60
} −

3

128
)

+∞

𝑘=0

< +∞. 

Case 3.1 

In this case, the sequence {𝛽𝑘 }𝑘∈𝑁  is reducing 

strictly. 

Use ratio test to this end. The sequence {𝛽𝑘 }𝑘∈𝑁 

reducing strictly as follows. 

1
2

[
 (𝛽𝑘+2)2 − 1
(𝛽𝑘+2)2 + 60

] −
3

128

1
2

[
 (𝛽𝑘+1)2 − 1
(𝛽𝑘+1)2 + 60

] −
3

128

 < 1 

The smoothness of (6) has therefore been 

proven. 

5. GRAPHICAL REPRESENTATION  

We would like to give an example in this section 

to depict the benefit of the developed scheme 

(3). As discussed in section 2, generated curves 

appear to approximate the basic discrete 

control polygon while 𝛽0 → +∞. 

In Figure 3.1, Generating wide range of C3-

continuous limiting curves for different values of 

parameters. (a) 𝛽0 = −2 (b) 𝛽0 = −1, (c) 𝛽0 =

0, (d) 𝛽0 = 1, (e) 𝛽0 = 5, (f) 𝛽0 = 10, (g) 𝛽0 =

25, (h) 𝛽0 = 50, (i) 𝛽0 = 100. 

 

 



 

 

 

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J. mt. area res., Vol. 8, 2023 

 

 

 

 

 

 



 

 

 

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88 

J. mt. area res., Vol. 8, 2023 

 

Figure 1. Generating wide range of C3-continuous 

limiting curves using the proposed scheme (3) for 

different values of parameter. (a) 𝛽0 = −2, (b) 𝛽0 =

−1 , (c)  𝛽0 = 0 , (d)  𝛽0 = 1 , (e)  𝛽0 = 5 , (f)  𝛽0 = 10 , 

(g) 𝛽0 = 25, (h) 𝛽0 = 50, (i) 𝛽0 = 100. 

 

CONCLUSION 

A nonstationary binary four-point approximating 

subdivision scheme with the shape control 

parameter has been developed that generates 

the smooth resulting curve of C3 continuous for 

the wider range of shape control parameter 

𝛽0 ∈ [−2, +∞). 

 

DECLARATIONS 

Funding: No funding was received to conduct this 

study. 

Conflicts of interest/Competing interests: The 

authors declare no conflict of interest or 

competing interest.  

Authors’ contributions:  

The authors Uzma Mukhtar and Kashif Rehan have 

equally contributed to this study in all stages from 

conceptualization to the write up of the final draft.  

 

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Received: 04 October 2022. Revised/Accepted: 20 February 2023. 

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