- Email: [email protected]
Modified form of binary and ternary 3-point subdivision schemes
Applied Mathematics and Computation 216 (2010) 970–982
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Modified form of binary and ternary 3-point subdivision schemes Shahid S. Siddiqi *, Kashif Rehan Department of Mathematics, Quaid-e-Azam Campus, University of the Punjab, Lahore 54590, Pakistan
a r t i c l e
i n f o
Keywords: Binary Ternary Approximating subdivision scheme Convergence and smoothness Mask Laurent polynomial
a b s t r a c t Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41–52], has been modified by introducing a tension parameter which generates a family of C 1 limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C 1 and C 2 limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Subdivision schemes are, mostly, used for the creation of smooth curves and surfaces from discrete set of data points in Computer Aided Geometric Designing (CAGD). They provide an efficient and flexible way to generate curves and surfaces. The vital schemes for applications are the schemes for surfaces design but schemes generating curves comprise an essential method for the understanding of schemes generating surfaces. A recursively corner cutting piecewise linear approximation scheme developed by de Rham [3] provided a basic foundation to subdivision which generates a limiting curve of C 0 continuity. Later on, Chaikin [2] presented another recursively corner cutting piecewise linear approximation method which generates a limiting curve of C 1 continuity. Both these techniques are very helpful for the creation of limiting curve in the early stages and became very famous. Hassan and Dodgson [9] presented a binary 3-point approximating scheme which generates C 3 limiting curve, also introduced ternary 3-point approximating scheme which generates C 2 limiting curve and ternary 3-point interpolatory scheme which gives C 1 limiting curve. Siddiqi and Ahmad [14] developed a binary 3-point approximating scheme that generates C 2 limiting curve with support on [3,2]. Zheng et al. [17] analyzed the fractal property of ternary 3-point interpolatory subdivision scheme, introduced by Hassan and Dodgson [9], with two parameters. Siddiqi and Rehan [15] recently introduced a ternary 3-point approximating subdivision scheme that gives C 2 limiting curve. Examples considered in the paper indicate that the scheme developed by Siddiqi and Rehan [15] behaves better than that proposed by Hassan and Dodgson [9]. Dyn et al. [5] presented a binary 4-point approximating subdivision scheme that generates C 2 limiting curve with support on [4,3] and is close to being interpolatory. Zhang et al. [16] developed another binary 4-point approximating subdivision scheme which generates limiting curve of C 3 continuity with the help of weights. Siddiqi and Ahmad [13] introduced a binary 4-point approximating subdivision scheme which gives C 4 limiting curve with support on [4,3]. Deslauriers and Dubuc * Corresponding author. E-mail addresses: [email protected] (S.S. Siddiqi), [email protected] (K. Rehan). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.115
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
971
[4] suggested a binary 4-point interpolatory subdivision scheme which generates C 1 limiting curve using polynomial reproducing property. Dyn et al. [6] introduced the general form of binary 4-point interpolatory subdivision scheme, independently, with the help of tension parameter which gives C 1 limiting curve. Kuijt et al. [8] examined the convexity preserving properties of the 4-point interpolatory subdivision scheme of Dyn et al. [6] which also generates C 1 limit functions. Yoon and co-workers [12] developed a new ternary 4-point approximating subdivision scheme derived from cubic polynomial interpolation which generates C 2 limiting curves. Hassan et al. [10] presented a ternary 4-point interpolatory subdivision scheme with tension parameter which gives C 2 continuous curves. Romani and co-workers [1] introduced a ternary non-stationary 4-point interpolatory subdivision scheme which generates C 2 limiting curve with a tension parameter. A subdivision scheme for generating curve or surface is defined as the limit of refined control polygon or mesh, according to some refining rules, recursively. In subdivision scheme, when the same mask is used in each refinement step then the scheme is called a stationary subdivision scheme. Each subdivision scheme is associated with a mask a ¼ fai g; i 2 Z. The binary and ternary subdivision schemes are the process which recursively define a sequence of control points f k ¼ fik ; i 2 Z by the rule of the form with mask a ¼ fai g; i 2 Z
X
fikþ1 ¼
ainj fjk ;
n ¼ f2; 3g;
j2Z k
which is formally denoted by f kþ1 ¼ Sf ¼ Sk f 0 . A subdivision scheme is said to be uniformly convergent if for every initial data f 0 ¼ ffi g; i 2 Z, there is a continuous function f such that for any closed interval ½a; b
lim
sup jfik f ðnk iÞj ¼ 0;
k!1 i2Z\nk ½a;b
n ¼ f2; 3g:
Obviously f ¼ S1 f 0 is considered to be a limit function of subdivision scheme S. Hormann and Sabin [11] proposed a binary 3-point approximating subdivision scheme. The subdivision rules to refine the control polygon are defined as
3 k 15 k 5 k f þ f þ f ; 32 i1 16 i 32 iþ1 5 k 15 k 3 k ¼ f þ f f : 32 i1 16 i 32 iþ1
f2ikþ1 ¼ kþ1 f2iþ1
It has been proved that the binary 3-point approximating subdivision scheme generates limiting curve of C 1 continuity. In this paper, the limitation of the scheme [11] has been removed by introducing a global tension parameter which generates a family of C 1 limiting curves for the certain range of tension parameter. The modified binary 3-point scheme is defined as
3 15 5 k k þ ; þ l fi1 2l fik þ þ l fiþ1 32 16 32 5 15 3 k k ¼ þ : þ l fi1 2l fik þ þ l fiþ1 32 16 32
f2ikþ1 ¼ kþ1 f2iþ1
ð1:1Þ
Siddiqi and Rehan [15] introduced a ternary 3-point approximating subdivision scheme. The subdivision rules to refine the control polygon are defined as
25 k 23 k 1 k f þ f þ f ; 72 i1 36 i 72 iþ1 1 k 3 1 k ¼ fi1 þ fik þ fiþ1 ; 8 4 8 1 k 23 k 25 k ¼ f þ f þ f : 72 i1 36 i 72 iþ1
f3ikþ1 ¼ kþ1 f3iþ1 kþ1 f3iþ2
It has been proved that the ternary 3-point approximating subdivision scheme generates limiting curve of C 2 continuity. In this paper, the limitation of the scheme [15] has been removed by introducing a global tension parameter which generates a family of C 1 and C 2 limiting curves for the certain range of tension parameter. The modified ternary 3-point scheme is defined as
25 23 1 k k þ ; þ l fi1 2l fik þ þ l fiþ1 72 36 72 1 3 1 k k ¼ þ ; þ l fi1 2l fik þ þ l fiþ1 8 4 8 1 23 25 k k ¼ þ : þ l fi1 2l fik þ þ l fiþ1 72 36 72
f3ikþ1 ¼ kþ1 f3iþ1 kþ1 f3iþ2
ð1:2Þ
972
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
Analysis of modified binary and ternary 3-point approximating subdivision scheme are presented in Sections 2 and 3 respectively. In Section 4, comparison of the modified ternary 3-point scheme is shown. Computational cost of the modified ternary 3-point scheme is compared in Section 5. In Section 6, examples are considered to demonstrate the usefulness of the modified schemes. The conclusion is drawn in Section 7. 2. Analysis of binary 3-point subdivision scheme For the convergent subdivision scheme S, the corresponding mask fai g; i 2 Z necessarily satisfies
X
a2j ¼
X
j2Z
a2jþ1 ¼ 1:
ð2:1Þ
j2Z
Introducing a symbol called the Laurent polynomial
aðzÞ ¼
X
ai zi ;
i2Z
of a mask fai g; i 2 Z with finite support. The corresponding symbols play an efficient role to analyze the convergence and smoothness of subdivision scheme. With the symbol, Dyn et al. [7] provided a sufficient and necessary condition for a uniform convergent subdivision scheme. A subdivision scheme S is uniform convergent if and only if there is an integer L P 1, such that
1 L S1 < 1; 2 1
2z subdivision S1 with symbol a1 ðzÞ is related to S with symbol aðzÞ, where a1 ðzÞ ¼ 1þz aðzÞ and satisfying
df k ¼ S1 df k1 ;
k ¼ 1; 2; . . . ;
k where f k ¼ Sk f 0 and df k ¼ fðdf k Þi ¼ 2k ðfiþ1 fik Þ : i 2 Zg and the norm kSk1 of a subdivision scheme S with a mask fai g; i 2 Z is defined by
( ) X X kSk1 ¼ max ja2i j; ja2iþ1 j : i2Z
i2Z
Theorem 1. Modified form of binary 3-point approximating subdivision scheme defined in Eq. (1.1) converges and have C 1 15 continuity for the range l 2 1 32 32 ½. Proof. Consider the refinement Eq. (1.1) and the Laurent polynomial aðzÞ for the mask of the scheme can be written as
aðzÞ ¼
3 5 15 15 5 3 þ l z3 þ þ l z2 þ 2l z1 þ 2l þ þ l z1 þ þ l z2 ; 32 32 16 16 32 32
Laurent polynomial method is used to prove the smoothness of the scheme to be C 1 . Taking ½m;L
b
ðzÞ ¼
where
am ðzÞ ¼
1 2L
a½L m ðzÞ;
m ¼ 1; 2; . . . ; L;
m 2z 2z aðzÞ; am1 ðzÞ ¼ 1þz 1þz
and
a½L m ðzÞ ¼
L1 Y
j
am ðz2 Þ:
j¼0
With a choice of m ¼ 1 and L ¼ 1, it can be written as ½1;1
b
ðzÞ ¼
1 2
a ðzÞ ¼ 1 1
3 1 11 1 3 þ l z2 þ z1 þ 2l þ z 1 þ þ l z2 : 32 4 16 4 32
Since the norm of subdivision 12 S1 is
( ) X ½1;1 1 3 11 1 S1 ¼ max þ ; jb j : c ¼ 0; 1 ¼ max 2 l l þ 2 cþ2b 2 32 16 2 < 1; 1 b
therefore the scheme S is convergent for the range of tension parameter
15 l 2 1 ½. 32 32
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
973
In order to prove that the scheme developed to be C 1 , consider m ¼ 2 and L ¼ 1; the Laurent polynomial gives ½2;1
b
1
ðzÞ ¼
2
a ðzÞ ¼ 1 2
3 11 11 3 þ 2l z1 þ 2l þ 2 l z1 þ þ 2l z 2 : 16 16 16 16
Since the norm of subdivision 12 S2 is
( ) X ½2;1 3 11 3 1 11 þ ; þ < 1; S2 ¼ max : b c ¼ 0; 1 ¼ max l l l l 2 þ 2 2 þ 2 c þ2b 16 16 16 2 16 1 b
15 therefore the scheme S 2 C 1 for the range of tension parameter l 2 1 ½. 32 32 3 It is to be mentioned that for global tension parameter l ¼ 32, the modified binary 3-point scheme coincides with the famous Chaikin’s [2] scheme and for l ¼ 0, the modified binary 3-point scheme coincides with the scheme introduced by Hormann and Sabin [11]. h
3. Analysis of ternary 3-point subdivision scheme For the convergent subdivision scheme S, the corresponding mask fai g; i 2 Z necessarily satisfies
X
a3j ¼
X
j2Z
a3jþ1 ¼
j2Z
X
a3jþ2 ¼ 1:
ð3:1Þ
j2Z
Introducing a symbol called the Laurent polynomial
aðzÞ ¼
X
ai zi ;
i2Z
of a mask fai g; i 2 Z with finite support. The corresponding symbols play an efficient role to analyze the convergence and smoothness of subdivision scheme. With the symbol, Hassan et al. [10] provided a sufficient and necessary condition for a uniform convergent subdivision scheme. A subdivision scheme S is uniform convergent if and only if there is an integer L P 1, such that
1 L S < 1; 3 1 1
2
3z subdivision S1 with symbol a1 ðzÞ is related to S with symbol aðzÞ, where a1 ðzÞ ¼ 1þzþz 2 aðzÞ and satisfying
df k ¼ S1 df k1 ;
k ¼ 1; 2; . . . ;
k 0
k where f k ¼ S f and df k ¼ fðdf k Þi ¼ 3k ðfiþ1 fik Þ : i 2 Zg and the norm kSk1 of a subdivision scheme S with a mask fai g; i 2 Z is defined by
( ) X X X kSk1 ¼ max ja3i j; ja3iþ1 j; ja3iþ2 j : i2Z
i2Z
i2Z
Theorem 2. Modified form of ternary 3-point approximating subdivision scheme defined in Eq. (1.2) converges and has 2 5 1 7 smoothness, C 1 for the range l 2 1 8 24 ½ and C for the range l 2 72 72 ½. Proof. Consider the refinement Eq. (1.2) and the Laurent polynomial aðzÞ for the mask of the scheme can be written as
aðzÞ ¼
1 1 25 23 3 23 25 þ l z4 þ þ l z3 þ þ l z2 þ 2l z1 þ 2l þ 2l z 1 þ þ l z2 72 8 72 36 4 36 72 1 1 3 4 þ þl z þ þl z 8 72
Laurent polynomial method is used to prove the smoothness of the scheme to be C 2 . Taking ½m;L
b
ðzÞ ¼
1 3L
a½L m ðzÞ;
m ¼ 1; 2; . . . ; L;
where
am ðzÞ ¼ and
m 3z2 3z2 a ðzÞ ¼ aðzÞ m1 1 þ z þ z2 1 þ z þ z2
974
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
a½L m ðzÞ ¼
L1 Y
j
am ðz3 Þ:
j¼0
With a choice of m ¼ 1 and L ¼ 1, it can be written as ½1;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 1
1 1 2 11 2 1 1 þ l z2 þ z1 þ þ 2l z 1 þ z 2 þ z 3 þ þ l z4 : 72 9 9 36 9 9 72
Since the norm of subdivision 13 S1 is
( ) X ½1;1 1 1 11 þ 2 1 þ l; 1 < 1; S1 ¼ max : b c ¼ 0; 1; 2 ¼ max l ; 2 cþ3b 3 3 3 36 72 1 b
23 therefore the scheme S is convergent for the range of tension parameter l 2 13 ½. 72 72 In order to prove that the scheme developed to be C 1 , consider m ¼ 2 and L ¼ 1; the Laurent polynomial gives
½2;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 2
1 7 1 7 1 þ 3l þ 3l z1 þ z2 þ 3l z3 þ þ 3l z4 : 24 24 3 24 24
Since the norm of subdivision 13 S2 is
( ) X ½2;1 1 1 1 7 þ þ 3l; 1 < 1; S2 ¼ max : b c ¼ 0; 1; 2 ¼ max l ; 3 cþ3b 3 3 3 24 24 1 b
5 therefore the scheme S 2 C 1 for the range of tension parameter l 2 1 ½. 8 24 In order to prove that the scheme developed to be C 2 , consider m ¼ 3 and L ¼ 1; the Laurent polynomial gives
½3;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 3
1 3 1 þ 9l z 2 þ 18l z3 þ þ 9l z 4 : 8 4 8
Since the norm of subdivision 13 S3 is
( ) X ½3;1 1 3 S3 ¼ max 18l; 1 þ 9l; 1 þ 9l < 1; : b c ¼ 0; 1; 2 ¼ max cþ3b 8 8 3 4 1 b
7 therefore the scheme S 2 C 2 for the range of tension parameter l 2 1 ½. 72 72 5 It is to be mentioned that for global tension parameter l ¼ 216, the modified ternary 3-point scheme coincides with the scheme developed by Hassan and Dodgson [9] and for l ¼ 0, modified ternary 3-point scheme coincides with the scheme introduced by Siddiqi and Rehan [15]. h
4. Comparison of modified ternary 3-point scheme Comparison Table 1 shows that support size of modified form of ternary 3-point approximating subdivision scheme is smaller than ternary 4-point approximating subdivision scheme [12] but gives the same order of derivative continuity. Table 1 also shows that support size and continuity of modified scheme are the same as ternary 3-point approximating subdivision scheme introduced by Hassan and Dodgson [9]. It is to be mentioned that the modified ternary 3-point scheme gives family 5 7 ½ and family of C 2 limiting curves for l 2 1 ½ which provides more freedom for curve of C 1 limiting curves for l 2 1 8 24 72 72 designing. In Table 1, the mask of the ternary 3-point interpolatory [9] scheme is defined as
½a; 0; b; 1 a b; 1; 1 a b; b; 0; a and the mask of the ternary 4-point interpolatory [10] scheme is
Table 1 Comparison of the Modified ternary 3-point subdivision scheme. Ternary scheme
Type
Support (size)
Continuity
Range
3-Point [9]
Interpolating
4
C1
3-Point [9]
Approximating
4
C2
a ¼ b 13 and b 2 29 39 ½ For some particular value
4-Point [10]
Interpolating
5
C2
1 l 2 19 15 ½
4-Point [12]
Approximating
5.5
C2
For some particular value
Modified 3-Point
Approximating
4
C1
Modified 3-Point
Approximating
4
C2
5 l 2 1 8 24 ½ 7 l 2 1 72 72 ½
975
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −1/128 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 3/64
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1 0
1
2
3
4
5
6
7
Siddiqi and Ahmad [14]
8
9
3
4
5
6
7
8
9
10
10
1
2
3
4
5
6
7
8
9
10
9
10
Chaikin [2] scheme and for µ = 3/32
10
0
2
Hormann and Sabin [11] and for µ = 0
10
0
1
0
0
1
2
3
4
5
6
7
8
Hassan and Dodgson [9]
Fig. 1. Results of modified binary 3-point scheme and comparison of subdivision results.
½a3 ; a0 ; 0; a2 ; a1 ; 1; a1 ; a2 ; 0; a0 ; a3 ; 7 where a0 ¼ 1 1 l; a1 ¼ 13 þ 12 l; a2 ¼ 18 12 l and a3 ¼ 1 þ 16 l. Hassan and Dodgson [9] introduced the ternary 3-point 18 6 18 18 approximating scheme whose mask is given by
976
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −1/128 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1 0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
µ = 3/64
0
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
7
Siddiqi and Ahmad [14]
8
9
2
3
4
5
6
7
8
9
10
Hormann and Sabin [11] and for µ = 0
10
0
1
10
0
0
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
Chaikin [2] scheme and for µ = 3/32
Hassan and Dodgson [9]
Fig. 2. Results of modified binary 3-point scheme and comparison of subdivision results.
977
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 2.5
2.5
2
2
1.5
1.5
1
1
0.5
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
0.5
2.5
2
2
1.5
1.5
1
1
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
0.5
2
2
1.5
1.5
1
1
3
4
5
6
7
Behavior of the scheme for
8
9
µ = 5/216
4
5
6
7
3
4
5
6
7
Behavior of the scheme for 2.5
2
2
µ = −5/344
2.5
0.5
3
Behavior of the scheme for
2.5
0.5
2
µ = −3/32
10
0.5
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
µ = −1/16
8
9
10
9
10
µ=0
8
µ = 13/216
Fig. 3. Results of modified ternary 3-point subdivision scheme for open curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter l ¼ 0, values of global tension parameter l ¼ 3 32 16 5 13 l ¼ 216 ; l ¼ 216 .
978
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5 1.5
2
2.5
3
Behavior of the scheme for
3.5
4
µ = −3/32
4.5
1.5 1.5
4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5 1.5
2
2.5
3
3.5
4
4.5
1.5 1.5
µ = −5/344
Behavior of the scheme for
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
2
2.5
3
Behavior of the scheme for
3.5
4
µ = 5/216
2.5
3
2
2.5
3
3.5
4.5
1.5 1.5
2
2.5
3
Behavior of the scheme for
4
4.5
µ = −1/16
3.5
4
4.5
4
4.5
µ=0
Behavior of the scheme for
4.5
1.5 1.5
2
Behavior of the scheme for
3.5
µ = 13/216
Fig. 4. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter values of global tension parameter l ¼ 3 32 16 5 13 13 l ¼ 0; l ¼ 216 ; l ¼ 216 . Moreover, for l ¼ 216 , it generates circle.
979
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = −3/32
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −5/344 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 5/216
3
4
5
6
9
1
2
3
4
5
6
Behavior of the scheme for
10
0
2
Behavior of the scheme for
10
0
1
10
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
9
10
9
10
µ = −1/16
7
8
µ=0
7
8
µ = 13/216
Fig. 5. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 5 13 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter l ¼ 0; l ¼ 216 ; l ¼ 216 . values of global tension parameter l ¼ 3 32 16
980
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = −3/32
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −5/344 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 5/216
3
4
5
6
9
1
2
3
4
5
6
Behavior of the scheme for
10
0
2
Behavior of the scheme for
10
0
1
10
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
9
10
9
10
µ = −1/16
7
8
µ=0
7
8
µ = 13/216
Fig. 6. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 values of global tension parameter l ¼ 3 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter 32 16 5 13 l ¼ 0; l ¼ 216 ; l ¼ 216 .
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
981
1 ½1; 4; 10; 16; 19; 16; 10; 4; 1: 27 Kwan et al. [12] developed the ternary 4-point approximating scheme whose mask is defined as
1 ½53; 81; 55; 231; 729; 1155; 729; 231; 55; 81; 53: 1296 5. Computational costs To produce an approximating curve with C 2 continuity using ternary approximating subdivision, it requires the 4-point approximating scheme introduced by Kwan et al. [12]. Each new point generated by 4-point scheme requires 4 multiplies and 3 adds: a total of 7 floating point operations. On the other hand, the modified form of ternary 3-point approximating subdivision scheme requires 3 multiplies and 2 adds: a total of 5 floating point operations to compute each new point which generates family of limiting curves of C 1 and C 2 continuity. From computational cost point of view, the modified scheme is better than the scheme developed by Kwan et al. [12] and coincides with the scheme presented by Hassan and Dodgson [9], while the modified scheme gives more flexibility using a global tension parameter.
6. Examples Two examples are depicted to show the usefulness of modified binary 3-point scheme for the different values of global tension parameter as shown in Figs. 1 and 2. Comparison of the proposed scheme reveals that the modified binary 3-point scheme behaves better than the existing one and it is closer to the original control polygon for some values of global tension parameter. Four examples are exposed to show the role of global tension parameter l when modified form of ternary 3-point approximating subdivision scheme applied on the different types of discrete data points as shown in Figs. 3–6. The behavior of the 5 ½. Moreover, for l ¼ 3 , limiting curve acts as a looseness if the choice of global tension parameter is in the range of l 2 1 8 344 32 the limiting curve is closed to original data points. In the same fashion, the limiting curve tends to shrink for the choice of 5 7 ½. global tension parameter in the range of l 2 ½344 72 7. Conclusion Modified form of binary 3-point approximating subdivision scheme is presented which generates a family of limiting 15 ½. Also modified ternary 3-point approximating curves of C 1 continuity for the range of global tension parameter l 2 1 32 32 1 subdivision scheme is presented which generates family of C limiting curves for the range of global tension parameter 5 7 l 2 1 ½ and family of C 2 limiting curves for the range of global tension parameter l 2 1 ½. The modified schemes are 8 24 72 72 analyzed using Laurent polynomial method and it is evident from examples that these modified schemes give flexibility to geometric designers for the creation of smooth curves according to their own requirements. Acknowledgement The authors are thankful to Nadeem Ahmad for his kind co-operation and valuable suggestions. References [1] C. Beccari, G. Casciola, L. Romani, An interpolatory 4-point C 2 ternary non-stationary subdivision scheme with tension control, Computer Aided Geometric Design 24 (2007) 210–219. [2] G.M. Chaikin, An algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (4) (1974) 346–349. [3] G. de Rham, Sur une courbe plane, Journal de Mathématiques Appliquées 35 (1956) 25–42. [4] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constructive Approximation 5 (1989) 49–68. [5] N. Dyn, M.S. Floater, K. Hormann, A C 2 four points subdivision scheme with fourth order accuracy and its extensions, Mathematical Methods for Curves and Surfaces: Troms/ 2004, Nashboro Press, Brentwood, 2005. pp. 145–156. [6] N. Dyn, J.A. Gregory, D. Levin, A 4-points interpolatory subdivision scheme for curve design, Computer Aided Geometric Design 4 (4) (1987) 257–268. [7] N. Dyn, A. Iske, E. Quak, M.S. Floater, Tutorials on multiresolution in geometric modelling, Summer school lecture notes series: Mathematics and Visualization, Springer, New York, 2002. ISBN: 3-540-43639-1. [8] N. Dyn, F. Kuijt, D. Levin, R.V. Damme, Convexity preservation of the four-point interpolatory subdivision scheme, Mathematical Methods for Curves and Surfaces, Memorandum No.1457, 1998, ISSN 0169-2690. [9] M.F. Hassan, N.A. Dodgson, Ternary and three point univariate subdivision schemes, in: Albert Cohen, Jean-Louis Merrien, L. Larry Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo 2002, Nashboro Press, Brentwood, 2003, pp. 199–208. [10] M.F. Hassan, I.P. Ivrissimitzis, N.A. Dodgson, M.A. Sabin, An interpolating 4-points C 2 ternary stationary subdivision scheme, Computer Aided Geometric Design 19 (2002) 1–18. [11] Kai Hormann, Malcolm A. Sabin, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design 25 (2008) 41–52.
982 [12] [13] [14] [15]
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
K.P. Ko, B.G. Lee, G.J. Yoon, A ternary 4-point approximating subdivision scheme, Applied Mathematics and Computation 190 (2007) 1563–1573. Shahid S. Siddiqi, Nadeem Ahmad, An approximating C 4 stationary subdivision scheme, European Journal of Scientific Research 15 (2006) 97–102. Shahid S. Siddiqi, Nadeem Ahmad, A new three-point approximating C 2 subdivision scheme, Applied Mathematics Letters 20 (2007) 707–711. Shahid S. Siddiqi and Kashif Rehan, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, 2009, in Press, doi: 10.1080/00207160802428220. [16] L. Zhang, Z. Luo, W. Liu, An approximating C 3 stationary subdivision scheme, Journal of Information and Computational Science 1 (1) (2004) 157–161. [17] H. Zheng, Z. Ye, Z. Chen, H. Zhao, Fractal range of a 3-point ternary interpolatory subdivision scheme with two parameters, Chaos, Solitons and Fractals 32 (2007) 1838–1845.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Modified form of binary and ternary 3-point subdivision schemes Shahid S. Siddiqi *, Kashif Rehan Department of Mathematics, Quaid-e-Azam Campus, University of the Punjab, Lahore 54590, Pakistan
a r t i c l e
i n f o
Keywords: Binary Ternary Approximating subdivision scheme Convergence and smoothness Mask Laurent polynomial
a b s t r a c t Binary 3-point scheme, developed by Hormann and Sabin [Hormann, K. and Sabin, Malcolm A., 2008, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design, 25, 41–52], has been modified by introducing a tension parameter which generates a family of C 1 limiting curves for certain range of tension parameter. Ternary 3-point scheme, introduced by Siddiqi and Rehan [Siddiqi, Shahid S. and Rehan, K., 2009, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, In Press, DOI: 10.1080/00207160802428220], has also been modified by introducing a tension parameter which generates family of C 1 and C 2 limiting curves for certain range of tension parameter. Laurent polynomial method is used to investigate the continuity of the subdivision schemes. The performance of modified schemes has been demonstrated by considering different examples along with its comparison with the established subdivision schemes. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Subdivision schemes are, mostly, used for the creation of smooth curves and surfaces from discrete set of data points in Computer Aided Geometric Designing (CAGD). They provide an efficient and flexible way to generate curves and surfaces. The vital schemes for applications are the schemes for surfaces design but schemes generating curves comprise an essential method for the understanding of schemes generating surfaces. A recursively corner cutting piecewise linear approximation scheme developed by de Rham [3] provided a basic foundation to subdivision which generates a limiting curve of C 0 continuity. Later on, Chaikin [2] presented another recursively corner cutting piecewise linear approximation method which generates a limiting curve of C 1 continuity. Both these techniques are very helpful for the creation of limiting curve in the early stages and became very famous. Hassan and Dodgson [9] presented a binary 3-point approximating scheme which generates C 3 limiting curve, also introduced ternary 3-point approximating scheme which generates C 2 limiting curve and ternary 3-point interpolatory scheme which gives C 1 limiting curve. Siddiqi and Ahmad [14] developed a binary 3-point approximating scheme that generates C 2 limiting curve with support on [3,2]. Zheng et al. [17] analyzed the fractal property of ternary 3-point interpolatory subdivision scheme, introduced by Hassan and Dodgson [9], with two parameters. Siddiqi and Rehan [15] recently introduced a ternary 3-point approximating subdivision scheme that gives C 2 limiting curve. Examples considered in the paper indicate that the scheme developed by Siddiqi and Rehan [15] behaves better than that proposed by Hassan and Dodgson [9]. Dyn et al. [5] presented a binary 4-point approximating subdivision scheme that generates C 2 limiting curve with support on [4,3] and is close to being interpolatory. Zhang et al. [16] developed another binary 4-point approximating subdivision scheme which generates limiting curve of C 3 continuity with the help of weights. Siddiqi and Ahmad [13] introduced a binary 4-point approximating subdivision scheme which gives C 4 limiting curve with support on [4,3]. Deslauriers and Dubuc * Corresponding author. E-mail addresses: [email protected] (S.S. Siddiqi), [email protected] (K. Rehan). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.115
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
971
[4] suggested a binary 4-point interpolatory subdivision scheme which generates C 1 limiting curve using polynomial reproducing property. Dyn et al. [6] introduced the general form of binary 4-point interpolatory subdivision scheme, independently, with the help of tension parameter which gives C 1 limiting curve. Kuijt et al. [8] examined the convexity preserving properties of the 4-point interpolatory subdivision scheme of Dyn et al. [6] which also generates C 1 limit functions. Yoon and co-workers [12] developed a new ternary 4-point approximating subdivision scheme derived from cubic polynomial interpolation which generates C 2 limiting curves. Hassan et al. [10] presented a ternary 4-point interpolatory subdivision scheme with tension parameter which gives C 2 continuous curves. Romani and co-workers [1] introduced a ternary non-stationary 4-point interpolatory subdivision scheme which generates C 2 limiting curve with a tension parameter. A subdivision scheme for generating curve or surface is defined as the limit of refined control polygon or mesh, according to some refining rules, recursively. In subdivision scheme, when the same mask is used in each refinement step then the scheme is called a stationary subdivision scheme. Each subdivision scheme is associated with a mask a ¼ fai g; i 2 Z. The binary and ternary subdivision schemes are the process which recursively define a sequence of control points f k ¼ fik ; i 2 Z by the rule of the form with mask a ¼ fai g; i 2 Z
X
fikþ1 ¼
ainj fjk ;
n ¼ f2; 3g;
j2Z k
which is formally denoted by f kþ1 ¼ Sf ¼ Sk f 0 . A subdivision scheme is said to be uniformly convergent if for every initial data f 0 ¼ ffi g; i 2 Z, there is a continuous function f such that for any closed interval ½a; b
lim
sup jfik f ðnk iÞj ¼ 0;
k!1 i2Z\nk ½a;b
n ¼ f2; 3g:
Obviously f ¼ S1 f 0 is considered to be a limit function of subdivision scheme S. Hormann and Sabin [11] proposed a binary 3-point approximating subdivision scheme. The subdivision rules to refine the control polygon are defined as
3 k 15 k 5 k f þ f þ f ; 32 i1 16 i 32 iþ1 5 k 15 k 3 k ¼ f þ f f : 32 i1 16 i 32 iþ1
f2ikþ1 ¼ kþ1 f2iþ1
It has been proved that the binary 3-point approximating subdivision scheme generates limiting curve of C 1 continuity. In this paper, the limitation of the scheme [11] has been removed by introducing a global tension parameter which generates a family of C 1 limiting curves for the certain range of tension parameter. The modified binary 3-point scheme is defined as
3 15 5 k k þ ; þ l fi1 2l fik þ þ l fiþ1 32 16 32 5 15 3 k k ¼ þ : þ l fi1 2l fik þ þ l fiþ1 32 16 32
f2ikþ1 ¼ kþ1 f2iþ1
ð1:1Þ
Siddiqi and Rehan [15] introduced a ternary 3-point approximating subdivision scheme. The subdivision rules to refine the control polygon are defined as
25 k 23 k 1 k f þ f þ f ; 72 i1 36 i 72 iþ1 1 k 3 1 k ¼ fi1 þ fik þ fiþ1 ; 8 4 8 1 k 23 k 25 k ¼ f þ f þ f : 72 i1 36 i 72 iþ1
f3ikþ1 ¼ kþ1 f3iþ1 kþ1 f3iþ2
It has been proved that the ternary 3-point approximating subdivision scheme generates limiting curve of C 2 continuity. In this paper, the limitation of the scheme [15] has been removed by introducing a global tension parameter which generates a family of C 1 and C 2 limiting curves for the certain range of tension parameter. The modified ternary 3-point scheme is defined as
25 23 1 k k þ ; þ l fi1 2l fik þ þ l fiþ1 72 36 72 1 3 1 k k ¼ þ ; þ l fi1 2l fik þ þ l fiþ1 8 4 8 1 23 25 k k ¼ þ : þ l fi1 2l fik þ þ l fiþ1 72 36 72
f3ikþ1 ¼ kþ1 f3iþ1 kþ1 f3iþ2
ð1:2Þ
972
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
Analysis of modified binary and ternary 3-point approximating subdivision scheme are presented in Sections 2 and 3 respectively. In Section 4, comparison of the modified ternary 3-point scheme is shown. Computational cost of the modified ternary 3-point scheme is compared in Section 5. In Section 6, examples are considered to demonstrate the usefulness of the modified schemes. The conclusion is drawn in Section 7. 2. Analysis of binary 3-point subdivision scheme For the convergent subdivision scheme S, the corresponding mask fai g; i 2 Z necessarily satisfies
X
a2j ¼
X
j2Z
a2jþ1 ¼ 1:
ð2:1Þ
j2Z
Introducing a symbol called the Laurent polynomial
aðzÞ ¼
X
ai zi ;
i2Z
of a mask fai g; i 2 Z with finite support. The corresponding symbols play an efficient role to analyze the convergence and smoothness of subdivision scheme. With the symbol, Dyn et al. [7] provided a sufficient and necessary condition for a uniform convergent subdivision scheme. A subdivision scheme S is uniform convergent if and only if there is an integer L P 1, such that
1 L S1 < 1; 2 1
2z subdivision S1 with symbol a1 ðzÞ is related to S with symbol aðzÞ, where a1 ðzÞ ¼ 1þz aðzÞ and satisfying
df k ¼ S1 df k1 ;
k ¼ 1; 2; . . . ;
k where f k ¼ Sk f 0 and df k ¼ fðdf k Þi ¼ 2k ðfiþ1 fik Þ : i 2 Zg and the norm kSk1 of a subdivision scheme S with a mask fai g; i 2 Z is defined by
( ) X X kSk1 ¼ max ja2i j; ja2iþ1 j : i2Z
i2Z
Theorem 1. Modified form of binary 3-point approximating subdivision scheme defined in Eq. (1.1) converges and have C 1 15 continuity for the range l 2 1 32 32 ½. Proof. Consider the refinement Eq. (1.1) and the Laurent polynomial aðzÞ for the mask of the scheme can be written as
aðzÞ ¼
3 5 15 15 5 3 þ l z3 þ þ l z2 þ 2l z1 þ 2l þ þ l z1 þ þ l z2 ; 32 32 16 16 32 32
Laurent polynomial method is used to prove the smoothness of the scheme to be C 1 . Taking ½m;L
b
ðzÞ ¼
where
am ðzÞ ¼
1 2L
a½L m ðzÞ;
m ¼ 1; 2; . . . ; L;
m 2z 2z aðzÞ; am1 ðzÞ ¼ 1þz 1þz
and
a½L m ðzÞ ¼
L1 Y
j
am ðz2 Þ:
j¼0
With a choice of m ¼ 1 and L ¼ 1, it can be written as ½1;1
b
ðzÞ ¼
1 2
a ðzÞ ¼ 1 1
3 1 11 1 3 þ l z2 þ z1 þ 2l þ z 1 þ þ l z2 : 32 4 16 4 32
Since the norm of subdivision 12 S1 is
( ) X ½1;1 1 3 11 1 S1 ¼ max þ ; jb j : c ¼ 0; 1 ¼ max 2 l l þ 2 cþ2b 2 32 16 2 < 1; 1 b
therefore the scheme S is convergent for the range of tension parameter
15 l 2 1 ½. 32 32
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
973
In order to prove that the scheme developed to be C 1 , consider m ¼ 2 and L ¼ 1; the Laurent polynomial gives ½2;1
b
1
ðzÞ ¼
2
a ðzÞ ¼ 1 2
3 11 11 3 þ 2l z1 þ 2l þ 2 l z1 þ þ 2l z 2 : 16 16 16 16
Since the norm of subdivision 12 S2 is
( ) X ½2;1 3 11 3 1 11 þ ; þ < 1; S2 ¼ max : b c ¼ 0; 1 ¼ max l l l l 2 þ 2 2 þ 2 c þ2b 16 16 16 2 16 1 b
15 therefore the scheme S 2 C 1 for the range of tension parameter l 2 1 ½. 32 32 3 It is to be mentioned that for global tension parameter l ¼ 32, the modified binary 3-point scheme coincides with the famous Chaikin’s [2] scheme and for l ¼ 0, the modified binary 3-point scheme coincides with the scheme introduced by Hormann and Sabin [11]. h
3. Analysis of ternary 3-point subdivision scheme For the convergent subdivision scheme S, the corresponding mask fai g; i 2 Z necessarily satisfies
X
a3j ¼
X
j2Z
a3jþ1 ¼
j2Z
X
a3jþ2 ¼ 1:
ð3:1Þ
j2Z
Introducing a symbol called the Laurent polynomial
aðzÞ ¼
X
ai zi ;
i2Z
of a mask fai g; i 2 Z with finite support. The corresponding symbols play an efficient role to analyze the convergence and smoothness of subdivision scheme. With the symbol, Hassan et al. [10] provided a sufficient and necessary condition for a uniform convergent subdivision scheme. A subdivision scheme S is uniform convergent if and only if there is an integer L P 1, such that
1 L S < 1; 3 1 1
2
3z subdivision S1 with symbol a1 ðzÞ is related to S with symbol aðzÞ, where a1 ðzÞ ¼ 1þzþz 2 aðzÞ and satisfying
df k ¼ S1 df k1 ;
k ¼ 1; 2; . . . ;
k 0
k where f k ¼ S f and df k ¼ fðdf k Þi ¼ 3k ðfiþ1 fik Þ : i 2 Zg and the norm kSk1 of a subdivision scheme S with a mask fai g; i 2 Z is defined by
( ) X X X kSk1 ¼ max ja3i j; ja3iþ1 j; ja3iþ2 j : i2Z
i2Z
i2Z
Theorem 2. Modified form of ternary 3-point approximating subdivision scheme defined in Eq. (1.2) converges and has 2 5 1 7 smoothness, C 1 for the range l 2 1 8 24 ½ and C for the range l 2 72 72 ½. Proof. Consider the refinement Eq. (1.2) and the Laurent polynomial aðzÞ for the mask of the scheme can be written as
aðzÞ ¼
1 1 25 23 3 23 25 þ l z4 þ þ l z3 þ þ l z2 þ 2l z1 þ 2l þ 2l z 1 þ þ l z2 72 8 72 36 4 36 72 1 1 3 4 þ þl z þ þl z 8 72
Laurent polynomial method is used to prove the smoothness of the scheme to be C 2 . Taking ½m;L
b
ðzÞ ¼
1 3L
a½L m ðzÞ;
m ¼ 1; 2; . . . ; L;
where
am ðzÞ ¼ and
m 3z2 3z2 a ðzÞ ¼ aðzÞ m1 1 þ z þ z2 1 þ z þ z2
974
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
a½L m ðzÞ ¼
L1 Y
j
am ðz3 Þ:
j¼0
With a choice of m ¼ 1 and L ¼ 1, it can be written as ½1;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 1
1 1 2 11 2 1 1 þ l z2 þ z1 þ þ 2l z 1 þ z 2 þ z 3 þ þ l z4 : 72 9 9 36 9 9 72
Since the norm of subdivision 13 S1 is
( ) X ½1;1 1 1 11 þ 2 1 þ l; 1 < 1; S1 ¼ max : b c ¼ 0; 1; 2 ¼ max l ; 2 cþ3b 3 3 3 36 72 1 b
23 therefore the scheme S is convergent for the range of tension parameter l 2 13 ½. 72 72 In order to prove that the scheme developed to be C 1 , consider m ¼ 2 and L ¼ 1; the Laurent polynomial gives
½2;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 2
1 7 1 7 1 þ 3l þ 3l z1 þ z2 þ 3l z3 þ þ 3l z4 : 24 24 3 24 24
Since the norm of subdivision 13 S2 is
( ) X ½2;1 1 1 1 7 þ þ 3l; 1 < 1; S2 ¼ max : b c ¼ 0; 1; 2 ¼ max l ; 3 cþ3b 3 3 3 24 24 1 b
5 therefore the scheme S 2 C 1 for the range of tension parameter l 2 1 ½. 8 24 In order to prove that the scheme developed to be C 2 , consider m ¼ 3 and L ¼ 1; the Laurent polynomial gives
½3;1
b
ðzÞ ¼
1 3
a ðzÞ ¼ 1 3
1 3 1 þ 9l z 2 þ 18l z3 þ þ 9l z 4 : 8 4 8
Since the norm of subdivision 13 S3 is
( ) X ½3;1 1 3 S3 ¼ max 18l; 1 þ 9l; 1 þ 9l < 1; : b c ¼ 0; 1; 2 ¼ max cþ3b 8 8 3 4 1 b
7 therefore the scheme S 2 C 2 for the range of tension parameter l 2 1 ½. 72 72 5 It is to be mentioned that for global tension parameter l ¼ 216, the modified ternary 3-point scheme coincides with the scheme developed by Hassan and Dodgson [9] and for l ¼ 0, modified ternary 3-point scheme coincides with the scheme introduced by Siddiqi and Rehan [15]. h
4. Comparison of modified ternary 3-point scheme Comparison Table 1 shows that support size of modified form of ternary 3-point approximating subdivision scheme is smaller than ternary 4-point approximating subdivision scheme [12] but gives the same order of derivative continuity. Table 1 also shows that support size and continuity of modified scheme are the same as ternary 3-point approximating subdivision scheme introduced by Hassan and Dodgson [9]. It is to be mentioned that the modified ternary 3-point scheme gives family 5 7 ½ and family of C 2 limiting curves for l 2 1 ½ which provides more freedom for curve of C 1 limiting curves for l 2 1 8 24 72 72 designing. In Table 1, the mask of the ternary 3-point interpolatory [9] scheme is defined as
½a; 0; b; 1 a b; 1; 1 a b; b; 0; a and the mask of the ternary 4-point interpolatory [10] scheme is
Table 1 Comparison of the Modified ternary 3-point subdivision scheme. Ternary scheme
Type
Support (size)
Continuity
Range
3-Point [9]
Interpolating
4
C1
3-Point [9]
Approximating
4
C2
a ¼ b 13 and b 2 29 39 ½ For some particular value
4-Point [10]
Interpolating
5
C2
1 l 2 19 15 ½
4-Point [12]
Approximating
5.5
C2
For some particular value
Modified 3-Point
Approximating
4
C1
Modified 3-Point
Approximating
4
C2
5 l 2 1 8 24 ½ 7 l 2 1 72 72 ½
975
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −1/128 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 3/64
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1 0
1
2
3
4
5
6
7
Siddiqi and Ahmad [14]
8
9
3
4
5
6
7
8
9
10
10
1
2
3
4
5
6
7
8
9
10
9
10
Chaikin [2] scheme and for µ = 3/32
10
0
2
Hormann and Sabin [11] and for µ = 0
10
0
1
0
0
1
2
3
4
5
6
7
8
Hassan and Dodgson [9]
Fig. 1. Results of modified binary 3-point scheme and comparison of subdivision results.
½a3 ; a0 ; 0; a2 ; a1 ; 1; a1 ; a2 ; 0; a0 ; a3 ; 7 where a0 ¼ 1 1 l; a1 ¼ 13 þ 12 l; a2 ¼ 18 12 l and a3 ¼ 1 þ 16 l. Hassan and Dodgson [9] introduced the ternary 3-point 18 6 18 18 approximating scheme whose mask is given by
976
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −1/128 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1 0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
µ = 3/64
0
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
7
Siddiqi and Ahmad [14]
8
9
2
3
4
5
6
7
8
9
10
Hormann and Sabin [11] and for µ = 0
10
0
1
10
0
0
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
Chaikin [2] scheme and for µ = 3/32
Hassan and Dodgson [9]
Fig. 2. Results of modified binary 3-point scheme and comparison of subdivision results.
977
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 2.5
2.5
2
2
1.5
1.5
1
1
0.5
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
0.5
2.5
2
2
1.5
1.5
1
1
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
0.5
2
2
1.5
1.5
1
1
3
4
5
6
7
Behavior of the scheme for
8
9
µ = 5/216
4
5
6
7
3
4
5
6
7
Behavior of the scheme for 2.5
2
2
µ = −5/344
2.5
0.5
3
Behavior of the scheme for
2.5
0.5
2
µ = −3/32
10
0.5
2
3
4
5
6
7
Behavior of the scheme for
8
9
10
µ = −1/16
8
9
10
9
10
µ=0
8
µ = 13/216
Fig. 3. Results of modified ternary 3-point subdivision scheme for open curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter l ¼ 0, values of global tension parameter l ¼ 3 32 16 5 13 l ¼ 216 ; l ¼ 216 .
978
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5 1.5
2
2.5
3
Behavior of the scheme for
3.5
4
µ = −3/32
4.5
1.5 1.5
4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5 1.5
2
2.5
3
3.5
4
4.5
1.5 1.5
µ = −5/344
Behavior of the scheme for
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
2
2.5
3
Behavior of the scheme for
3.5
4
µ = 5/216
2.5
3
2
2.5
3
3.5
4.5
1.5 1.5
2
2.5
3
Behavior of the scheme for
4
4.5
µ = −1/16
3.5
4
4.5
4
4.5
µ=0
Behavior of the scheme for
4.5
1.5 1.5
2
Behavior of the scheme for
3.5
µ = 13/216
Fig. 4. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter values of global tension parameter l ¼ 3 32 16 5 13 13 l ¼ 0; l ¼ 216 ; l ¼ 216 . Moreover, for l ¼ 216 , it generates circle.
979
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = −3/32
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −5/344 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 5/216
3
4
5
6
9
1
2
3
4
5
6
Behavior of the scheme for
10
0
2
Behavior of the scheme for
10
0
1
10
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
9
10
9
10
µ = −1/16
7
8
µ=0
7
8
µ = 13/216
Fig. 5. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 5 13 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter l ¼ 0; l ¼ 216 ; l ¼ 216 . values of global tension parameter l ¼ 3 32 16
980
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982 10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = −3/32
9
10
0
0
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
0
0
µ = −5/344 10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
µ = 5/216
3
4
5
6
9
1
2
3
4
5
6
Behavior of the scheme for
10
0
2
Behavior of the scheme for
10
0
1
10
0
0
1
2
3
4
5
6
Behavior of the scheme for
7
8
9
10
9
10
9
10
µ = −1/16
7
8
µ=0
7
8
µ = 13/216
Fig. 6. Results of modified ternary 3-point subdivision scheme for closed curve. Modified ternary 3-point scheme generates C 1 limiting curves for different 5 values of global tension parameter l ¼ 3 ; l ¼ 1 ; l ¼ 344 and C 2 limiting curves for the different values of global tension parameter 32 16 5 13 l ¼ 0; l ¼ 216 ; l ¼ 216 .
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
981
1 ½1; 4; 10; 16; 19; 16; 10; 4; 1: 27 Kwan et al. [12] developed the ternary 4-point approximating scheme whose mask is defined as
1 ½53; 81; 55; 231; 729; 1155; 729; 231; 55; 81; 53: 1296 5. Computational costs To produce an approximating curve with C 2 continuity using ternary approximating subdivision, it requires the 4-point approximating scheme introduced by Kwan et al. [12]. Each new point generated by 4-point scheme requires 4 multiplies and 3 adds: a total of 7 floating point operations. On the other hand, the modified form of ternary 3-point approximating subdivision scheme requires 3 multiplies and 2 adds: a total of 5 floating point operations to compute each new point which generates family of limiting curves of C 1 and C 2 continuity. From computational cost point of view, the modified scheme is better than the scheme developed by Kwan et al. [12] and coincides with the scheme presented by Hassan and Dodgson [9], while the modified scheme gives more flexibility using a global tension parameter.
6. Examples Two examples are depicted to show the usefulness of modified binary 3-point scheme for the different values of global tension parameter as shown in Figs. 1 and 2. Comparison of the proposed scheme reveals that the modified binary 3-point scheme behaves better than the existing one and it is closer to the original control polygon for some values of global tension parameter. Four examples are exposed to show the role of global tension parameter l when modified form of ternary 3-point approximating subdivision scheme applied on the different types of discrete data points as shown in Figs. 3–6. The behavior of the 5 ½. Moreover, for l ¼ 3 , limiting curve acts as a looseness if the choice of global tension parameter is in the range of l 2 1 8 344 32 the limiting curve is closed to original data points. In the same fashion, the limiting curve tends to shrink for the choice of 5 7 ½. global tension parameter in the range of l 2 ½344 72 7. Conclusion Modified form of binary 3-point approximating subdivision scheme is presented which generates a family of limiting 15 ½. Also modified ternary 3-point approximating curves of C 1 continuity for the range of global tension parameter l 2 1 32 32 1 subdivision scheme is presented which generates family of C limiting curves for the range of global tension parameter 5 7 l 2 1 ½ and family of C 2 limiting curves for the range of global tension parameter l 2 1 ½. The modified schemes are 8 24 72 72 analyzed using Laurent polynomial method and it is evident from examples that these modified schemes give flexibility to geometric designers for the creation of smooth curves according to their own requirements. Acknowledgement The authors are thankful to Nadeem Ahmad for his kind co-operation and valuable suggestions. References [1] C. Beccari, G. Casciola, L. Romani, An interpolatory 4-point C 2 ternary non-stationary subdivision scheme with tension control, Computer Aided Geometric Design 24 (2007) 210–219. [2] G.M. Chaikin, An algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (4) (1974) 346–349. [3] G. de Rham, Sur une courbe plane, Journal de Mathématiques Appliquées 35 (1956) 25–42. [4] G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constructive Approximation 5 (1989) 49–68. [5] N. Dyn, M.S. Floater, K. Hormann, A C 2 four points subdivision scheme with fourth order accuracy and its extensions, Mathematical Methods for Curves and Surfaces: Troms/ 2004, Nashboro Press, Brentwood, 2005. pp. 145–156. [6] N. Dyn, J.A. Gregory, D. Levin, A 4-points interpolatory subdivision scheme for curve design, Computer Aided Geometric Design 4 (4) (1987) 257–268. [7] N. Dyn, A. Iske, E. Quak, M.S. Floater, Tutorials on multiresolution in geometric modelling, Summer school lecture notes series: Mathematics and Visualization, Springer, New York, 2002. ISBN: 3-540-43639-1. [8] N. Dyn, F. Kuijt, D. Levin, R.V. Damme, Convexity preservation of the four-point interpolatory subdivision scheme, Mathematical Methods for Curves and Surfaces, Memorandum No.1457, 1998, ISSN 0169-2690. [9] M.F. Hassan, N.A. Dodgson, Ternary and three point univariate subdivision schemes, in: Albert Cohen, Jean-Louis Merrien, L. Larry Schumaker (Eds.), Curve and Surface Fitting: Sant-Malo 2002, Nashboro Press, Brentwood, 2003, pp. 199–208. [10] M.F. Hassan, I.P. Ivrissimitzis, N.A. Dodgson, M.A. Sabin, An interpolating 4-points C 2 ternary stationary subdivision scheme, Computer Aided Geometric Design 19 (2002) 1–18. [11] Kai Hormann, Malcolm A. Sabin, A family of subdivision schemes with cubic precision, Computer Aided Geometric Design 25 (2008) 41–52.
982 [12] [13] [14] [15]
S.S. Siddiqi, K. Rehan / Applied Mathematics and Computation 216 (2010) 970–982
K.P. Ko, B.G. Lee, G.J. Yoon, A ternary 4-point approximating subdivision scheme, Applied Mathematics and Computation 190 (2007) 1563–1573. Shahid S. Siddiqi, Nadeem Ahmad, An approximating C 4 stationary subdivision scheme, European Journal of Scientific Research 15 (2006) 97–102. Shahid S. Siddiqi, Nadeem Ahmad, A new three-point approximating C 2 subdivision scheme, Applied Mathematics Letters 20 (2007) 707–711. Shahid S. Siddiqi and Kashif Rehan, A ternary three point scheme for curve designing, International Journal of Computer Mathematics, 2009, in Press, doi: 10.1080/00207160802428220. [16] L. Zhang, Z. Luo, W. Liu, An approximating C 3 stationary subdivision scheme, Journal of Information and Computational Science 1 (1) (2004) 157–161. [17] H. Zheng, Z. Ye, Z. Chen, H. Zhao, Fractal range of a 3-point ternary interpolatory subdivision scheme with two parameters, Chaos, Solitons and Fractals 32 (2007) 1838–1845.