Engineering images designed by fractal subdivision scheme

Transcription

1 Mustafa et al SpringerPlus 06 5:493 DOI 086/s RESEARCH Open Access Engineering images designed by fractal subdivision scheme Ghulam Mustafa *, Mehwish Bari and Saba Jamil *Correspondence: ghulam Department of Mathematics, The Islamia University of Bahawalpur, Baghdad Campus, Bahawalpur 6300, Punjab, Pakistan Full list of author information is available at the end of the article Abstract This paper is concerned with the modeling of engineering images by the fractal properties of 6-point binary interpolating scheme Association between the fractal behavior of the limit curve/surface and the parameter is obtained The relationship between the subdivision parameter and the fractal dimension of the limit fractal curve of subdivision fractal is also presented Numerical examples and visual demonstrations show that 6-point scheme is good choice for the generation of fractals for the modeling of fractal antennas, bearings, garari s and rock etc Keywords: Interpolating subdivision scheme, Engineering images, Fractal antennas, Bearings Mathematics Subject Classification: 8A80, 65D05, 65D0 Background Fractals are infinitely complex patterns that are self-similar across different scales They are made by rehashing a basic procedure again and again in an ongoing feedback loop Its uses in various areas of the study of materials and of other areas of engineering are examples of practical prose Its uses in physical theory, especially in conjunction with the basic equations of mathematical physics Let us, instead, give a few typical examples: to rock base, an essential of engineering, furthermore a proceeding with objective of science, is to depict nature quantitatively Another illustration is viscus flow through porous media, ie the stream of water pushing oil, has demonstrated as of late to admit to a few administrations, one of which is a front, which is a compelling and attractive arrangement and another is fractal fingering which is undesirable A fractal antenna ie Fig is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter of material that can receive or transmit electromagnetic radiation within a given total surface area or volume Another mechanical parts in engineering such as gears, chain, grari etc can be constructed based on ideas of fractal geometry Owing to the rapid emergence and growth of techniques in the engineering application of fractals, it has become necessary to gather the most recent advances on a regular basis This study starts from the question, can we design mechanical parts in engineering by simple iterative techniques? Firstly, Mandelbrot 98 found fractal geometry that deals with geometric shape which is self-similar, irregular and has detailed structure at arbitrarily small scales 06 The Authors This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made

2 Mustafa et al SpringerPlus 06 5:493 Page of Fig Fractal antenna Nowadays, many techniques to generate fractals have been devised, such as IFS iterated function systems method Barnsley and Demko 985, L-system method Prusinkiewicz and Lindenmayer 990 and few others Recently it has been shown that subdivision technique is not only an important tool for the fast generation of smooth engineering objects, but also an efficient tool for the fast generation of fractals Zheng et al 007a, b analyzed fractal properties of 4-point binary and three point ternary interpolatory subdivision schemes Wang et al 0 discussed the fractal properties of the generalized chaikin corner-cutting subdivision scheme with two tension parameters They gave the fractal range of scheme on the basis of the discussion of limit points on the limit curve Li et al 03 designed the fractal curves by using the normal vector based subdivision scheme Sarfraz et al 05 designed some engineering images by using rational spline interpolation In this paper, we explore the properties of Weissman 989 fractal subdivision scheme in different areas including engineering images We conclude that 6-point scheme of Weissman can create engineering images for curves and surfaces with true fractal allotting and can provide some ways of shape control The paper is organized as follows In "Fractal properties of the scheme" section, fractal range of 6-point subdivision scheme is being discussed In "Numerical examples and demonstrations" section, some numerical examples are presented to confirm the correctness and effectiveness of the engineering images in the form of curve and surface Finally, we give some concluding remarks in "Conclusions" section Fractal properties of the scheme The well known 6-point binary interpolating scheme Weissman 989 is pi k = pi k p k i = µpk i 6 3µ 6 3µ p k i µpk i3 p k i 9 6 µ p k i 9 6 µ pi k The scheme produces C 0 - and C - continuous curves at 6 3 <µ< 3 6 and 63 <µ< 9 respectively By substituting µ = 0 and µ = 56 3, we get 4-point and 6-point schemes as given in Deslauriers and Dubuc 989 respectively

3 Mustafa et al SpringerPlus 06 5:493 Page 3 of According to interpolatory property p k 0 p0 0, k 0 Suppose pm i and pm j are two fixed control points after m subdivision steps, m Z, m 0 The role of parameter µ is required to be evaluated on the sum of all small edges among the two points after another k iterations First we discuss and analyze the effect of µ among the two initial control points p 0 0 and p0 For i = in the odd rule p k i p3 k = µ p4 k Substituting p4 k p3 k = µ pk = pk p k pk 6 3µ and pk of scheme, we have p k pk 0 = pk 6 3µ 3 in, we get p k 3 p µ p k pk Putting i =, 0 in odd rule pi k of scheme by using pk = pk and pk = p k, we have 9 p k = µ p3 k pk 6 3µ p k pk 6 µ p k p0 0, 9 6 µ p k pk and p k = µ p k pk µ p k pk 6 µ p k p0 0 Here, we define two edge vectors between the points p0 0 and p0 after k steps defined by v k = p k pk 0 and R k = p k pk Since pk i = i so we have p k k = = = p k k k and we can write as R k = p k p k 0 p k 0 pk = v k v k 3 Let U k = p k pk, W k = p0 k pk, we can write it as U k W k = p k pk pk 0 pk = p k pk 0 = v k Further U k = v k W k This implies U k =µ p k pk µ p3 k p3 k 6 3µ p k pk 9 6 3µ p k p k 6 µ p k p k Since by dyadic parametrization pi k = i then the term µ k written as µ Therefore k k k k p k pk 6 3µ p3 k p3 k = µ k k k k U k =µ p k pk µ p k p k pk p k p k 6 3µ p k 9 6 µ p k can be

4 Mustafa et al SpringerPlus 06 5:493 Page 4 of This implies U k = µu k 9 6 3µ U k 6 3µ U k 6 µ U k So we have U k = 0 6 6µ U k 6 3µ U k 4 Equation 4 is the second order linear difference equation, such that 0 U k 6 6µ U k 6 3µ U k = 0 The characteristic equation is 0 Q 6 6µ Q 6 3µ By solving 5, we get Q = 8 6µ and Q =, Q = Q when µ = Let U 0 = p 0 p0 and U = p p For k = 0 and i = 0 in, we get p = µp0 This implies Substitute k = 0 and i = in, we have This implies 6 3µ p = µ p 0 p0 3 p = µp µ p = µ p 0 3 p0 6 3µ p 0 µp0 3 = 0 p µ p µ 9 6 3µ p 0 p0 6 µ p0 0 p0 6 3µ p 0 µp0 p µ p p 0 6 µ 9 6 3µ p 0 p0 6 µ p 0 p0 0 p Subtracting 7 from 6, we have p p =µp0 3 p p 6 4µ 0 5µ 0 5µ 6 4µ p 0 µp0 3 p 0 8

5 Mustafa et al SpringerPlus 06 5:493 Page 5 of The solution of Eq 8 is k U k = 8 6µ c k c, where { c = µp3 p p µ 6 4µ 0 5µ 0 5µ } 6 4µ p 0 µp0 3 and { 8 c = µp3 p p µ 6 4µ 0 µ 0 µ } 6 4µ p 0 µp0 3 When µ = 6, the solution of Eq 4 will be U k = k ć k kć p 0 p 0 where ć = p 0 p0 and ć = 8 p3 0 5p0 7p0 7p0 5p0 p0 3 Since v k = µ p k pk µ p k p k Then by taking U k = p k p k and v k = p k pk 0 v k = 9 6 µ U k Since by dyadic parametrization µ v k = pk 9 6 µ 9 6 µ pk p k 0 U k 6 µ µ p k p k 0 v k µ p k p k 9 6 µ v k, we get p k p k pk µ p3 k p k µ p3 k p k = 0 then p k pk 0 p k pk 0 Since v k = p k pk 0 and W k = p0 k pk Then 9 9 v k = 6 µ U k 6 µ v k v k W k 9

6 Mustafa et al SpringerPlus 06 5:493 Page 6 of Substitute W k = U k v k in 9, we get 9 9 v k = 6 µ U k 6 µ v k v k Uk v k Simplifying, we have v k v k 6 µ v k = 6 µ U k 0 Now we will find the solution of Eq 0, first consider γ γ 6 µ = 0 Case When 6 3 <µ< 63, the solution of Eq is γ = 4 µ and γ = 4 µ, and γ = γ, γ, γ = 8 6µ, γ, γ = Equation 0 can be expressed as v k { k } k v k 6 µ v k = 6 µ c 8 6µ c The solution of above equation is v k = β γ k β γ k 65 k 44 c 8 6µ k c, where β and β are given as { β = wp 69 p 0 w 56 wp w 6 w 56 wp 3344 p w 896 wwp p ww 059 p 0 ww 5086 wp w 896 wwp wp w 3584 wwp 96 p p 09 p 3 p 850 p 3344 wp wp wp 846 p w 3568 p w 84 p 0 w 6067 w p w p p 0 w 739 p w 4990 w p p w 096 p w 876 p w }, { β = p 3344 p w 56 wp w 6 w 69 p 0 w 96 p0 878 p 3 p 739 p w 4990 w p w p 876 p w 846 p w 096 p w 6067 w p 3 84 p 0 w 896 wwp wp 56 wp 059 p 0 ww 5086 wp w p ww 896 wwp wp w 3584 wwp p 0 w 3344 wp 3568 wp wp p w 3568 p w 850 p } From Eqs 3 and, we have

7 Mustafa et al SpringerPlus 06 5:493 Page 7 of R k = β γ k γ γ k c β γ k 7 48µ 48µ 65 k 44 c 8 6µ 3 Case When 9 <µ< 3 6, the solution of Eq is ρ = 4 i µ and ρ = 4 i µ, ρ = ρ, ρ, ρ = 8 6µ, ρ, ρ = The solution of Eq 0 can be expressed as v k = χ ρ k χ ρ k 65 k 44 c 8 6µ k c, 4 where χ and χ are given as { χ = 384i 3568i wp 69ip 0 w 56i wp w 6 w 56i wp 3344ip w 896i wwp ip ww 059ip 0 ww 5086i wp w 896i wwp i wp w 3584i wwp 96 p p 09 p 3 p 850 p 3344 wp wp wp 846 p w 3568 p w 84 p 0 w 6067 w p w p p 0 w 739 p w 4990 w p p w 096 p w 876 p w }, { χ = 384i 09 p 3344ip w 56i wp w 6 w 69ip 0 w 96 p0 878 p 3 p 739 p w 4990 w p w p 876 p w 846 p w 096 p w 6067 w p 3 84 p 0 w 896i wwp i wp 56i wp 059ip 0 ww 5086i wp w 59776ip ww 896i wwp i wp w 3584i wwp p 0 w } 3344 wp 3568 wp wp p w 3568 p w 850 p From Eqs 3 and 4, we have R k = χ ρ k ρ ρ k c χ ρ k 7 48µ 48µ 65 k 44 c 8 6µ 5 From the Eqs 5, we have the following theorems Theorem For 9 <µ< 3 6, the limit curve of the 6-point scheme is a fractal curve

8 Mustafa et al SpringerPlus 06 5:493 Page 8 of Proof By following the Eqs 4 and 5, we may conclude that the k small edge vectors between p0 0 and p0 after k subdivision steps can be expressed by induction as: k k Ej k = pj k pj k = ς jρ k ς jρ k ς 3j 6 8 6µ ς 4j, for j =,,, k, where ρ ij = 0, i =,, 3, 4 The Eq 6 is written by the linear combination of 4 and 5 By taking ξ = 8 6µ and ξ > ρ, ρ Let v denote the length of a vector v and E k j 0 = min j=,, k E k j 0, then we get k j= E k j k E k j 0 = k ς j0 ρ k ς j 0 ρ k ς 3j 0 ξ k ς 4j0 k This implies k k Ej k ρ =ξk ς j0 ς j0 ξ j= ρ ξ k ς 3j0 ς 4j0 k as k ξ The sum of the length of all the small edges between p0 0 and p0 after k subdivision steps grows without bound when k approaches to infinity, so by Zheng et al 007a, 007b the limit curve of 6-point scheme is a fractal curve in the range 9 <µ< 3 6 Similarly, we get following theorem Theorem For 3 6 <µ< 63, the limit curve of the 6-point scheme is a fractal curve A straightforward generalization of 6-point scheme is its tensor-product version which generates fractal surfaces over the parametric ranges 6 3 <µ< 63 and 9 <µ< 3 6 Complexity of subdivision fractal We measure the complexity of the 6-point binary interpolating subdivision fractal by estimating its box dimension By using Theorem and adopting the procedures of Zheng et al 00, Wang and Qian 003, we get following theorem Theorem 3 If pi k k are the control points at level k and fi = k α pi k pk i then the sequence fi k is bounded above, where α = log µ l, l = 9 and 9 <µ< 3 6 By using Theorem 3 and taking up the methods of Zheng et al 00, Wang and Qian 003, we get following theorem Theorem 4 The fractal dimension of the 6-point binary interpolatory subdivision fractal is d = α = 5430

9 Mustafa et al SpringerPlus 06 5:493 Page 9 of a b c d Fig Fractals: Dotted lines with initial control points show the initial control polygon a, b, c, d whereas solid lines show the fractal antennas at third, seventh, tenth and thirteenth level with µ = 078 respectively a b Fig 3 Smooth curves: a shows the initial structure of bearing and b, c shows the bearing at third level with µ = 0 and µ = 3 56 respectively From Theorem, we know that subdivision fractal can be gotten by keeping the corresponding subdivision parameter µ within the interval 9 <µ< 3 6 Therefore by Theorem 4 the fractal dimension of the 6-point subdivision fractal as a limit will be no more than d = α = 5430 Similarly, one can compute the fractal dimension of subdivision fractal for the interval 6 3 <µ< 63 Numerical examples and demonstrations The proposed work is used to construct the engineering structures such as fractal antennas, bearings and garari s etc Figure a d present the fractal antennas generated after third, seventh, tenth and thirteenth subdivision levels at µ = 078 The fractal dimension of these fractals is 4066 The initial sample of another fractal antenna is shown in Fig 3a Figure 3b, c show the fractal antennas generated by the scheme at µ =0099 Figure 4a shows the initial sample for a bearing and Fig 4b, c show the actual bearing generated at parametric values µ = 0 and µ = 56 3 respectively The initial mesh for rock surface is shown in Fig 5a Figure 5b, c show the rock surfaces at third level with µ =0 and µ =005 respectively Figure 6 presents the structure of garari type shapes In the case of given initial control points, shapes and dimensions of the fractals can be adjusted and controlled by adjusting the parameter µ Hence the obtained results in "Fractal properties of the scheme" section can be used to generate fractal in a fast and efficient way Conclusions In this article, we have reorganized the engineering images by fractal subdivision scheme We have identified two different parametric intervals to generate different types of engineering models The relationship between the subdivision parameter and the c

10 Mustafa et al SpringerPlus 06 5:493 a Page 0 of c b d Fig 4 Fractals: a shows the initial structure of fractal antenna and b d show the fractal antennas at first, second and fourth iteration of the scheme at parametric value µ = 0099 respectively a b c Fig 5 Fractal surfaces: a shows the initial control polygon whereas b, c show the rock surfaces at third level with µ = 0and µ = 005 respectively a b c Fig 6 Fractal surfaces: a shows the initial structure of garari and b, c show the garari at third level with µ = 087 and 008 respectively fractal dimension of the limit fractal curve of the 6-point binary interpolatory subdivision fractal is also presented It is concluded that 6-point subdivision scheme is an efficient tool for the fast generation of self similar fractals useful in fractal antennas It is also an appropriate technique for the designing of bearings and garari s etc Authors contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Author details Department of Mathematics, The Islamia University of Bahawalpur, Baghdad Campus, Bahawalpur 6300, Punjab, Pakistan National College of Business Administration and Economics, Bahawalpur Campus, Bahawalpur, Punjab, Pakistan Acknowledgements This work is supported by NRPU Project No 383 and the Indigenous PhD Scholarship Scheme of Higher Education Commission HEC Pakistan Competing interests The authors declare that there is no competing interest regarding the publication of this article and regarding the funding that they have received Received: 4 May 06 Accepted: 8 August 06

11 Mustafa et al SpringerPlus 06 5:493 Page of References Barnsley MF, Demko S 985 Iterated function systems and the global construction of fractals Proc R Soc A 399:43 75 Deslauriers G, Dubuc S 989 Symmetric iterative interpolation processes Constr Approx 5:49 68 Li Y, Zheng H, Peng G, Zhou M 03 Normal vector based subdivision scheme to generate fractal curves TELKOMNIKA 8: Mandelbrot BB 98 The fractal geometry of nature Freeman, New York Prusinkiewicz P, Lindenmayer A 990 The algorithmic beauty of plants Springer, New York Sarfraz M, Ishaq M, Hussain MZ 05 Shape designing of engineering images using rational spline interpolation Adv Mater Sci Eng Wang J, Zheng H, Xu F, Liu D 0 Fractal properties of the generalized Chaikin corner-cutting subdivision scheme Comput Math Appl 6:97 00 Wang J, Qian X 003 Dimensionality estimation of the fractal interpolatory curve generated by 4-point interpolatory subdivision scheme J Gansu Univ Technol 9:0 Weissman A 989 A 6-point interpolatory subdivision scheme for curve design, MSc thesis, Tel-Aviv University Zheng H, Ye Z, Lei Y, Liu X 007 Fractal properties of interpolatory subdivision schemes and their application in fractal generation Chaos Solitons Fractals :3 3 Zheng H, Ye Z, Lei Y, Chen Z, Zhao H 007 Fractal range of a 3-point ternary interpolatory subdivision scheme with two parameters Chaos Solitons Fractals : Zheng H, Wang J, Liu D, Peng G 00 Dimension estimation of subdivision fractal and its application In: The 5th international conference on computer science & education, Hefei, China August 4 7