Applied Mathematics and Computation 246 (2014) 210–220

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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.

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