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Applied Mathematics and Computation 246 (2014) 210–220
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier .com/ locate /amc
Generation of fractal curves and surfaces using ternary 4-pointinterpolatory subdivision scheme
http://dx.doi.org/10.1016/j.amc.2014.07.0780096-3003/� 2014 Elsevier Inc. All rights reserved.
⇑ Corresponding author.E-mail addresses: shahidsiddiqiprof@yahoo.co.uk (S.S. Siddiqi), usamaidrees.01@gmail.com (U. Idrees), kkashif_99@yahoo.com (K. Rehan).
Shahid S. Siddiqi a,⇑, Usama Idrees a, Kashif Rehan b
a Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistanb Department of Mathematics, University of Engineering and Technology, KSK Campus, Lahore, Pakistan
a r t i c l e i n f o a b s t r a c t
Keywords:Interpolatory subdivision schemeTernary 4-point schemeGeneration of fractal curves and surfacesFractal mountainsFractal properties
In this paper, the generation of fractal curves and surfaces along with their properties,using ternary 4-point interpolatory subdivision scheme with one parameter, are analyzed.The relationship between the tension parameter and the fractal behavior of the limitingcurve is demonstrated through different examples. The specific range of the tension param-eter has also been depicted, which provides a clear way to generate fractal curves. Since thefractal scheme introduces, in the paper, have more number of control points therefore itgives more degree of freedom to control the shape of the fractal curve.
� 2014 Elsevier Inc. All rights reserved.
1. Introduction
Generating smooth shapes of curves or surfaces, through subdivision techniques, are the easiest phenomena in the geo-metric modeling. These pleasing techniques give new direction to computer graphics, computer aided geometric design,reverse engineering and medical surgery simulations. Subdivision schemes have elegant mathematical ways to createsmooth curves or surfaces from discrete set of control points, by repeated refinements. Subdivision schemes can be relegatedas; approximating and interpolating subdivision schemes.
Fractals are apparently – random and irregular shapes (e.g. landscapes or cloud) or structures (e.g. plants and mountains)formed by recurring subdivisions of a basic form, and having a regular pattern in their apparent randomness. Every part of afractal is essentially a condensed-size copy of the whole shape, called self-similarity. Computer-generated fractals can createdetailed pictures of fractal landscapes, plants, waves, and planets. The astonishing fact about fractals is the assortment oftheir applications. Almost every part of the universe, from our body to bacteria cultures, comprises fractals.
Fractals are used in fractal antennas – small size antennas using fractal shapes, signal and image compression, computerand video game designs, classification of histopathology slides and coastline complexity, creation of digital photographicenlargements. Subdivision schemes generate self-similar curves. Therefore there is a close connection between curvesand surfaces generated by subdivision scheme and self similar fractals. Though, the fractal curves can be obtained throughdifferent manners but less efforts have been made for the generation of the fractal curves using subdivision schemes.
In 1956, commencing work in the field of subdivision was done by de Rham [1], a French mathematician, introduced thefirst piecewise linear corner cutting approximating subdivision scheme that generates C1 limiting curve. In 1974, Chaikin [2]proposed another piecewise linear binary corner cutting approximating subdivision scheme generating C1 curve. In 2002,Hassan and Dodgson [3] developed a ternary 3-point approximating subdivision scheme that generates C2 limiting curve.
S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220 211
In 2009, Siddiqi and Rehan [4] introduced a stationary binary subdivision scheme yielding C1 curve. In 2012, Siddiqi andRehan [5] proposed a new method of corner cutting subdivision scheme that generates the limiting curve of C1 continuity.
In 1987, Dyn et al. developed the first binary 4-point interpolatory subdivision scheme [6] that generates C1 limitingcurve. Later on, different interpolatory subdivision schemes were presented. In 1989, Deslauries and Dubuc [7] proposed bin-ary 4-point interpolating subdivision scheme generating C1 curve. In 2002, Hassan and Dodgson [3] introduced ternary threepoint interpolating subdivision scheme yielding C1 continuous curve. In 2005, Zheng et al. [8] developed ternary 3 pointinterpolatory subdivision scheme that generates the limiting curve of C1 continuity. In 2007, Amat et al. [9] introduced anew approach towards proving convexity preserving properties for interpolatory subdivision schemes. In 2007, RomildoMalaquias and Roberto Lopes [10] presented a computer algebra system that is both fast, and implemented in a stronglytyped language, and designed to accept compiled extensions i.e. programming software that needs both numerical compu-tation and computer algebra. In 2009, Zheng et al. [11] introduced 2n-1-point ternary interpolating subdivision schemes. In2012, Siddiqi and Rehan [12] proposed a 4-point interpolatory subdivision scheme yielding family of C1 limiting curves. In2013, Luo and Qi [13] deduced interpolatory subdivision scheme from approximating subdivision scheme.
In 2007, Bouboulis and Dalla presented the construction of fractal interpolation surfaces along with its properties [14]. In2007, Zheng et al. [15] proved that the limit curves generated by binary 4-point and ternary 3-point interpolatory subdivi-sion schemes are fractals, keeping the corresponding tension parameters within some particular ranges. Again in 2007,Zheng et al. [16] proposed that the limit curves generated by the ternary three point interpolating subdivision scheme withtwo parameters are fractal curves for some specific ranges of the parameters. In 2008, Feng [17] discussed the fractal inter-polation on the rectangular domain along with some special properties of fractal interpolation function. In 2011, Wang et al.[18] discussed the fractal properties of the generalized Chaikin corner-cutting subdivision scheme on the basis of its prop-erties of limit points. In 2014, Siddiqi et al. [19] explored the generation of fractal curves and surfaces using ternary 5-pointinterpolatory subdivision scheme.
In this paper, fractal scheme introduced by Zheng et al. [15] is followed to view the fractal behavior conforming to theternary 4-point interpolatory subdivision scheme proposed by Hassan et al. [20]. This scheme offers a faster rate of gener-ation of fractals as compared to the scheme proposed by Zheng et al. [15,16]. It may be noted that a subdivision scheme withmore number of control points give more control to obtain the desired curve, i.e. degree of freedom increases with increase innumber of control points.
The ternary 4-point interpolating subdivision scheme is given as follows.
Given the set of initial control points P0 ¼ fP0i 2 Rdg
nþ1
i¼�1. Let Pk ¼ fPki g
3knþ1
i¼�1 be the set of control points at level
kðk P 0; k 2 ZÞ and fPkþ1i g
3knþ1
i¼�1 satisfy the following rules, recursively
Pkþ13i ¼ Pk
i ; 0 6 j � 3k;
Pkþ13iþ1 ¼ � 1
18�l6
� �Pk
i�1 þ 1318þ
l2
� �Pk
i þ 718�
l2
� �Pk
iþ1 þ � 118þ
l6
� �Pk
iþ2; 0 6 j � 3k;
Pkþ13iþ2 ¼ � 1
18þl6
� �Pk
i�1 þ 718�
l2
� �Pk
i þ 1318þ
l2
� �Pk
iþ1 þ �118 �
l6
� �Pk
iþ2; 0 6 j � 3k;
8>><>>: ð1Þ
where l is the tension parameter.The scheme generates family of C0-continuous curves for �1 < l < 1 and family of C1-continuous for � 1
5 < l < 13 [17].
The rest of the article is organised as follows. The Section 2 presents generation of fractal curves corresponding to theternary 4-point interpolatory subdivision scheme. In the Section 3, several numerical examples are given. In this section,the comparison between the fractal scheme proposed in the Section 2 and the scheme proposed by Zheng et al. [15,16] isdiscussed. Finally Section 4 concludes our work.
2. Generation of fractal curves corresponding to the ternary 4-point interpolatory subdivision scheme
Consider two arbitrary fixed control points Pni and Pn
j after n subdivision steps, where 8n 2 Z;n P 0. The effect of theparameter l is needed to be analyzed on the sum of all the small edges between the two points after another k subdivisionsteps. For simplicity, the effect between the two initial control points, say, P0
0 and P01 is analyzed.
According to the subdivision scheme (1), it is known that Pk0 � P0
0, where k P 0, and
Pkþ11 ¼ � 1
18�l6
� �Pk�1 þ 13
18þl2
� �Pk
0 þ 718�
l2
� �Pk
1 þ � 118þ
l6
� �Pk
2;
Pkþ12 ¼ � 1
18þl6
� �Pk�1 þ 7
18�l2
� �Pk
0 þ 1318þ
l2
� �Pk
1 þ � 118�
l6
� �Pk
2:
(ð2Þ
Let the following three distinctive edge vectors be
Vk ¼ Pk1 � Pk
0;
Sk ¼ Pk2 � Pk
1;
Rk ¼ Pk3 � Pk
2;
then the difference equations for the edge vectors Vk, Sk and Rk can be obtained as follows.
212 S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220
Taking
Uk ¼ Pk1 � Pk
�1; Wk ¼ Pk0 � Pk
�1; Zk ¼ Pk2 � Pk
�2: So Uk ¼ Vk þWk;
Uk can be written as
Ukþ1 ¼ Pkþ11 � Pkþ1
�1 ¼�118þ 3l
18
� �Uk þ
818� 6l
18
� �Zk; ð3Þ
which can be rewritten as
Ukþ1 ��118þ 3l
18
� �Uk ¼
818� 6l
18
� �Zk: ð4Þ
Similarly,
Zkþ1 þ1
18þ 3l
18
� �Zk ¼
1418þ 6l
18
� �Uk: ð5Þ
Using Eqs. (4) and (5), Uk can be calculated as
Uk ¼ qk1c1 þ qk
2c2; ð6Þ
where q1 ¼ð1�9lÞ
18 and q2 ¼ 13,
c1 ¼ð�1þ 3lÞðP0
�2 � 2P0�1 þ 2P0
1 � P02Þ
5þ 9l; l – � 5
9
c2 ¼ð1� 3lÞP0
�2 � ð7þ 3lÞðP0�1 � P0
1Þ þ ð�1þ 3lÞP02
5þ 9l; l – � 5
9:
Similarly, Zk can be calculated as
Zk ¼ 18�kð�1� 3lÞkb1 þ qk1b2 þ qk
2b3; ð7Þ
where
b1 ¼�18c3
1þ 3lþ P0
�2 � P02;
b2 ¼ð7þ 3lÞð�P0
�2 þ 2P0�1 � 2P0
1 þ P02Þ
5þ 9l;
b3 ¼�2ðð�1þ 3lÞP0
�2 þ ð7þ 3lÞðP0�1 � P0
1Þ þ ð1� 3lÞP02Þ
5þ 9l:
Since
Vk ¼ Pk1 � Pk
0;
therefore,
Vkþ1 ¼ Pkþ11 � Pkþ1
0 ¼ 5� 9l18
� �Vk þ
�1þ 3l18
� l� �
Sk þ1þ 3l
18
� �Uk; ð8Þ
or
Vkþ1 �5� 9l
18
� �Vk ¼
�1þ 3l18
�� �
Sk þ1þ 3l
18
� �Uk: ð9Þ
Similarly,
Skþ1 þl3
Sk ¼1þ 3l
3Vk �
l3
Uk: ð10Þ
Using Eq. (6) in Eqs. (9) and (10), yields
Vkþ1 �5� 9l
18
� �Vk ¼
�1þ 3l18
�� �
Sk þ1þ 3l
18
� �ðqk
1c1 þ qk2c2Þ ð11Þ
and
Skþ1 þl3
Sk ¼1þ 3l
3Vk �
l3ðqk
1c1 þ qk2c2Þ: ð12Þ
−0.5 0 0.5−0.5
0
0.5
1
1.5
2
(a)−0.5 0 0.5
−0.5
0
0.5
1
1.5
2
(b)
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.5
0
0.5
1
1.5
2
2.5
3
(c)−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
1.5
2
2.5
3
(d)Fig. 1. Fractal curves generated using ternary 4-point subdivision scheme after five subdivision steps. (a) l ¼ � 13
25, (b) l ¼ � 920, (c) l ¼ 17
25, (d) l ¼ 79.
S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220 213
From Eqs. (11) and (12), it cab be written as
�54�1þ 3l
Vkþ2 þð15� 45lÞ�1þ 3l
Vkþ1 þð�1þ 5lÞ�1þ 3l
Vk ¼�18�kð1� 9lÞk
5þ 9lc1 þ
3kð1þ 5lÞ�1þ 3l
c2: ð13Þ
The corresponding characteristic equation is
�54�1þ 3l
k2 þ ð15� 45lÞ�1þ 3l
kþ ð�1þ 5lÞ�1þ 3l
¼ 0: ð14Þ
Case 1. When �1 < l < 1, where l – � 59 the roots of Eq. (14) are k1 ¼ 1�5l
6 ; k2 ¼ 19 then the solution of Eq. (13) is
Vk ¼ kk1c3 þ kk
2c4 þ qk1d1 þ qk
2d2; ð15Þ
where
c3 ¼ð�1þ 3lÞðP0
�2 � 4P0�1 þ 6P0
0 � 4P01 þ P0
2Þ2� 30l
;
c4 ¼ð�1þ 3lÞðP0
�2 þ P02Þ þ ð3þ 3lÞðP0
�1 þ P01Þ � 4ð1þ 4lÞP0
0
�2þ 30l;
d1 ¼ð�1þ 3lÞðP0
�2 � P02Þ þ 2ð�1þ 3lÞð�P0
�1 þ P01Þ
2ð5þ 9lÞ ;
d2 ¼ð1� 3lÞðP0
�2 � P02Þ � ð7þ 3lÞðP0
�1 � P01Þ
2ð5þ 9lÞ :
0.8 1 1.2 1.4 1.6 1.8 2 2.20.8
1
1.2
1.4
1.6
1.8
2
2.2
(a)0.8 1 1.2 1.4 1.6 1.8 2 2.2
0.8
1
1.2
1.4
1.6
1.8
2
2.2
(b)
0.8 1 1.2 1.4 1.6 1.8 2 2.21
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
(c)0.8 1 1.2 1.4 1.6 1.8 2 2.2
0.5
1
1.5
2
2.5
(d)Fig. 2. Fractal curves generated using ternary 4-point subdivision scheme after four subdivision steps. (a) l ¼ 3
4, (b) l ¼ � 59, (c) l ¼ 4
5, (d) l ¼ � 610.
214 S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220
Similarly, Sk can be calculated as
Sk ¼ rke1 þ qk1e2 þ qk
2e3 þ kk1e4 þ kk
2e5; ð16Þ
where r ¼ �ðl3Þ,
c5 ¼lðP0
1 � P02Þ
3;
e1 ¼�3c5 þ lP1 � lP2
l ;
e2 ¼ �3ð1þ lÞðP0
�2 � 2P0�1 þ 2P0
1 � P02Þ
5þ 9l;
e3 ¼ð1þ 3lÞðP0
�2 � 4P0�1 þ 6P0
0 � 4P01 þ P0
2Þ�1þ 15l
;
e4 ¼ð�1þ 3lÞðP0
2 � P0�2Þ þ ð7þ 3lÞðP0
1 � P0�1Þ
2ð5þ 9lÞ ;
e5 ¼ð�3þ 9lÞðP0
2 þ P0�2Þ � 12ð1þ 3lÞðP0
0Þ þ ð9þ 9lÞðP01 þ P0
�1Þ2ð�1þ 15lÞ :
From Eqs. (6), (15) and (16), the solution of equation
Rkþ1 ¼ð1þ 3lÞ
18Sk þ
ð�1þ 3lÞ18
Uk þð7� 9lÞ
18Vk; ð17Þ
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
(a)−5 0 5−4
−3
−2
−1
0
1
2
3
4
5
(b)
−5 0 5−8
−6
−4
−2
0
2
4
6
8
(c) (d)−6 −4 −2 0 2 4 6−8
−6
−4
−2
0
2
4
6
8
Fig. 3. Fractal curves generated using ternary 4-point subdivision scheme after five subdivision steps. (a) l ¼ � 1120, (b) l ¼ 41
50, (c) l ¼ � 610, (d) l ¼ 19
25.
S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220 215
can be determined as
Rk ¼ qk1h1 þ qk
2h2 þ kk1h3 þ kk
2h4 þ rkh5; ð18Þ
where
h1 ¼ð�1þ 3lÞc1 þ ð7� 9lÞd2 þ ð1þ 3lÞe2
1� 9l;
h2 ¼ð�1þ 3lÞc2 þ ð7� 9lÞd1 þ ð1þ 3lÞe3
3� 15l;
h3 ¼ð7� 9lÞc3 þ ð1þ 3lÞe4
6;
h4 ¼ð7� 9lÞc4 þ ð1þ 3lÞe5
2;
h5 ¼ �ð1þ 3lÞe1
6l:
Case 2. When l ¼ �59 , the solution of Eq. (4) is,
Uk ¼ 3�kðc01 þ kc02Þ; ð19Þ
where
c01 ¼ P01 � P0
�1;
c02 ¼49ðP0�2 � 2P0
�1 þ 2P02 � P0
2Þ:
−3 −2 −1 0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
(a)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
3.55
3.6
3.65
3.7
3.75
3.8
3.85
3.9
3.95
4
(b)
−3 −2 −1 0 1 2 3 40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(c)−2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3
2
2.1
2.2
2.3
2.4
2.5
(d)Fig. 4. Fractal curves generated using ternary 4-point subdivision scheme after four subdivision steps. (a) l ¼ � 9
20, (b) The amplified figure of selected partof (a), (c) l ¼ 7
10, (d) The amplified figure of selected part of (c).
216 S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220
From Eqs. (8) and (10), the roots of Vk and Sk can be determined as
Vk ¼ k0k1 c03 þ k0k2 c04 þ12
3�kðc01 þ kc02Þ; ð20Þ
Sk ¼ r0ke01 þ 3�kðe02 þ ke03Þ þ k0k2 e04 þ k0k2 e05; ð21Þ
where k01 ¼ 1727
� �, k02 ¼ 1
9 and r0 ¼ 527
� �,
c03 ¼1
40ð�4P0
�2 þ P0�1 � 10P0
0 þ P01 þ 4P0
2Þ;
c04 ¼1
40ð�4P0
�2 þ 19P0�1 � 30P0
0 þ 19P01 � 4P0
2Þ;
e01 ¼3
80ð4P0
�2 � 9P0�1 þ 10P0
0 � 9P01 þ 4P0
2Þ;
e02 ¼29ðP0�2 � 2P0
�1 þ 2P01 � P0
2Þ;
e03 ¼12ð�P0
�2 þ P0�1 � P0
1 þ P02Þ;
e04 ¼1
80ð4P0
�2 � 19P0�1 þ 30P0
0 � 19P01 þ 4P0
2Þ;
e05 ¼1
120ð4P0
�2 þ P0�1 � 10P0
0 þ P01 þ 4P0
2Þ:
Fig. 5. (a) Control Mesh [12], (b) Fractal mountains generated by Zheng et al. after six subdivision steps [12]. Fractal surfaces generated using proposedfractal subdivision scheme after four subdivision steps. (c) l ¼ � 1
4, (d) l ¼ � 13.
S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220 217
The solution of the Eq. (17) is,
Rk ¼ 3�kðh01 þ kh02Þ þ k0k1 h03 þ k0k2 h04 þ rkh05; ð22Þ
h01 ¼16ð�P0
�2 � P0�1 þ P0
1 þ P02Þ;
h02 ¼6
27ð�P0
�2 � 2P0�1 þ 2P0
1 � P02Þ;
h03 ¼999
23;120ð�4P0
�2 þ 19P0�1 � 30P0
0 þ 2P01 � P0
2Þ;
h04 ¼9
72ð4P0
�2 þ P0�1 � 10P0
0 þ P01 þ 4P0
2Þ;
h05 ¼3
400ð4P0
�2 � 9P0�1 þ 10P0
0 � 9P01 þ P0
2Þ:
Theorem. For �1 < l < � 15, and 3
5 < l < 1 the limit curve of the ternary 4-point subdivision scheme is a fractal curve.
Proof. For �1 < l < � 15, and 3
5 < l < 1 alongwith Eqs. (15), (16) and (18), it might be concluded by induction that 3k smalledge vectors between the two initial control points, say, P0
0 and P01, after k subdivision steps, can be expressed as
Case 1.
Eki ¼ Pk
i � Pki�1 ¼ a1iqk
1 þ a2iqk2 þ a3ik
k1 þ a4ik
k2 þ a5irk; i ¼ 1;2; . . . ;3k; ð23Þ
where aji – 0; j ¼ 1;2;3;4;5. in this case it can be proved that
13< k1 < 1; jk1j > jk2j; jk1j > jq1j and jk1j > jrj:
Fig. 6. (a) Control Mesh [13], (b) Fractal mountains generated by Zheng et al. [13]. Fractal mountains generated by Zheng et al. after six subdivision steps[13]. Fractal surfaces generated using proposed fractal subdivision scheme after four subdivision steps. (c) l ¼ 13
20, (d) l ¼ 4150.
218 S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220
Moreover, let a refer the length of a vector a and jEki0j ¼mini¼1;...;3k jEk
i j, then it gives
X3k
j¼1
jEki j P 3kjEk
i0j
¼ 3k a1iqk1 þ a2iqk
2 þ a3ikk1 þ a4ik
k2 þ a5irk
��� ���¼ ð3k1Þk a1i0
q1
k1
� �k
þ a2i0
q2
k1
� �k
þ a3i0 þ a4i0
k2
k1
� �k
þ a5i0
rk1
� �k�����
�����! þ1 ðk!1Þ:
Case 2. For l ¼ �59 alongwith Eqs. (20)–(22),
Eki ¼ Pk
i � Pki�1 ¼ f1j3
�k þ f2j3�kkþ f3jk
0k1 þ f4jk
0k2 þ f5jrk; ð24Þ
where f ji – 0; j ¼ 1;2;3;4;5.
Fig. 7. Fractal mountains generated using proposed fractal subdivision scheme after five subdivision steps. (a) Mesh (b) l ¼ � 920, (c) l ¼ � 11
20, (d) l ¼ � 34.
Fig. 8. Fractal mountains generated using proposed fractal subdivision scheme after five subdivision steps. (a) l ¼ 710 , (b) l ¼ 13
20.
S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220 219
X3k
j¼1
jEki j P 3kjEk
i0j
¼ 3k f1j3�k þ f2j3
�kkþ f3jk0k1 þ f4jk
0k2 þ f5jrk
��� ���¼ 3kk f1j
3�k
kþ f2j3
�k þ f3jk0k1kþ f4j
k0k2kþ f5j
rk
k
����������
! þ1 ðk!1Þ:
220 S.S. Siddiqi et al. / Applied Mathematics and Computation 246 (2014) 210–220
When k tends to infinity, the sum of the lengths of all the small edges between the initial points P00 and P0
1 has no bound afterk subdivision steps. The behavior of fractal curves for different values of tension parameter has been discussed in the follow-ing section.
3. Examples
In Figs. 1–3, fractal curves generated through ternary 4-point interpolatory subdivision scheme for closed polygon, afterfive subdivision steps, are shown. While in Fig. 4, fractal curves for open polygon are shown, after five subdivision steps. Theinitial control polygons are shown with the broken line segments while the solid line segments represent the correspondingfractal curves. Comparison between the fractal surfaces generated by Zheng et al. [15,16] and the proposed fractal scheme (1)and fractal range �1 < l < � 1
5
� � S 35 < l < 1� �
corresponding to different parametric values, is depicted in Figs. 5 and 6. Itmay be noted that the proposed fractal subdivision scheme gives faster rate of generating fractals as compared to that ofproposed by Zheng et al. [15,16]. Fractal mountains are shown in Figs. 7 and 8 by using proposed fractal subdivision scheme(1) and fractal range �1 < l < � 1
5
� � S 35 < l < 1� �
corresponding to different parametric values, after four subdivisionsteps.
4. Conclusion
The fractal properties of ternary 4-point interpolatory subdivision scheme with tension parameter, have been analyzedand discussed. It may be mentioned that the subdivision scheme used in this paper has also been discussed by Mustafaet al. [21] and the fractal range has been claimed to be l > 1
9, which does not satisfy the criteria of fractal formulation givenin [12,13].
The range of tension parameter l for the generation of fractal curves has also been obtained �1 < l < � 15
� �S 35 < l < 1� �
. The behavior of fractal curves conforming to different values of parameter has also been depicted throughdifferent examples. It may be noted that the fractal scheme proposed in this paper requires less number of subdivision stepsas compared to that proposed by Zheng et al. in [15,16].
Acknowledgement
The authors are thankful to Nadeem Ahmad for his valuable suggestions.
References
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