Koch Curves: Rewriting System, Geometry and Application

Transcription

1 Joural of Coputer Sciece 7 (9): , 0 ISSN Sciece Publicatios Koch Curves: Rewritig Syste, Geoetry ad Applicatio Mata Rai, Riaz Ul Haq ad Norrozila Sulaia Faculty of Coputer Systes ad Software Egieerig, Uiversity Malaysia Pahag, 6300, Gabag, Kuata, Malaysia Abstract: Proble stateet: Recetly, ew Koch curves have bee geerated by dividig the iitiator ito three uequal parts. There is o foral rewritig syste to geerate such kid of curves. Approach: It is required to easure the ew chaged geoetrical properties. Geeralized rewritig systes for the ew Koch curves have bee developed. Results: New forulas have bee give to easure their geoetrical properties. Coclusio/Recoedatios: The geoetrical properties of ew Koch curves ake the ore suitable as ateas i wireless couicatio tha the covetioal Koch curve. Key words: Koch curve, superior koch curve, koch loop, rewritig syste, fractal atea, fractal diesio, rewritig rules, fudaetal atheatical properties, atural uber, superior fractals, ulti-bad characteristics INTRODUCTION The classical Koch curve is, atheatically, cotiuous everywhere but differetiable owhere. It is a exaple of bouded curve of ifiite legth (Schroeder, 009; Falcoer, 003). Koch sowflake curve is used as a fractal atea. Natha (005) described the iportace of fractal ateas explicitly for wireless techologies with ephasis to ilitary services. For a detailed study o Koch fractal atea ad related various ew fidigs, oe ay refer to (Elkachouchi ad Nasr, 007; Ghatak et al., 009; Kordzadeh ad Kashai, 009; Krisha et al., 009; Mirzapour ad Hassai, 009; Sog et al., 008; Werer ad Sua, 003; Zhag ad Kishk, 006) ad several cross refereces thereof. McClure (008) has discussed the vibratio odes of a dru shaped like a Koch sowflake. Epstie ad Adeeb (008) derived stiffess of the Koch curve. Jibrael et al. (008) siulated quadratic Koch atea ad explored its atea properties. Further, Vioy et al. (00) preseted a ew way of geeratio of variats of a Koch curve by varyig idetatio agle ad gave a forula to calculate their fractal diesio ad studied the ipact of fractal diesio i the desig of ulti-resoat fractal ateas (Vioy et al., 004). Geerated ew Koch curves as ew exaples of superior fractals by dividig the iitiator ito three uequal parts. A coprehesive review of literature o superior fractals, which are costructed usig superior iterates, is give by Sigh et al. (0). The purpose of this study is to develop rewritig rules for superior Koch curves ad suggest forulas for calculatio of their fudaetal atheatical properties. Preliiaries: Costructio of the Koch curve (Fig. ) is well kow. It ca be expressed by followig rewrite syste (L-syste) ( Cos ta t : +, Pr oductio rule : F F + F F + F Correspodig Author: Mata Rai, Faculty of Coputer Systes ad Software Egieerig Uiversity Malaysia Pahag, 6300, Gabag, Kuata, Malaysia 330 () Here, F eas "draw forward", + eas "tur left 60 " ad - eas "tur right 60 ". To draw a Koch sowflake curve, the Prod. Rule is applied o axio F - - F - - F. Rewritig syste for koch curves: I this sectio, we develop geeral productio rules to draw superior Koch curves. We divide the productio rules for Koch curves at differet scalig factors ito two parts. All the sybols used i the followig rewritig systes carry siilar eaigs as i that of Koch

2 J. Coputer Sci., 7 (9): , 0 curve. We use the sae iitiator, as show i Fig. a, i all the geeratios. Case : Koch curves at scalig factor S =, where is a odd uber. Let S =, the oe of the possible Koch curves is 5 show i Fig.. The followig is the rewritig syste for such a Koch curve: Figure Koch iddle oe-fifth curve for (r, r, r3) = (/5, /5, /5) with its iitiator: Scalig factor : Cos ta t : +, Pr oductio rule : F FF + F F + FF At () S =, rewritig syste for Koch iddle oe- 7 seveth curve is as follows: The geeral rewritig syste for Koch curves at S=/, where = 3, 5, 7, 9,, which ca be derived fro Prod. Rule, ad 3 is as follows: Scalig factor S =, where is a odd atural uber ad 3: Cos ta t : +, Pr oductio rule : F (( ) / )F + F F + (( ) / )F (4) Case : Koch curves at scalig factor S=/, where is a eve uber. Let the scalig factor be ¼, the rewritig syste for the two possible Koch iddle oe-fourth curves (see the geerators i Fig. 3) is give below. Scalig factor : Cos ta t : +, Pr oductio rule : F FFF + F F + FFF (3) Fig. : Koch iddle oe-fifth curve for (r, r, r3) = (/5, /5, /5) with its iitiator (c) Fig. : Iterative costructio of koch curve Fig. 3: Geerators of two possible Koch iddle oe-fourth curves Geerator at (r, r, r3) = (/4, /4, /) Geerator at (r, r, r3) = (/, /4, /4) 33

3 J. Coputer Sci., 7 (9): , 0 Cos ta t : +, Productio rule : F ( / )F + F F + (( / )F Or F ( / )F + F F + (( / )F (7) Applicatios to fractal atea: Authors have geerated superior Koch curves at scalig factors s = /, for 4. Koch loop ca be geerated by applyig Prod. Rule 4 or 7 o the axio F - - F - - F. Followig are soe of the geoetric features of a Koch loop. Here, r deotes the radius of the circle which accoodates the Koch loop. The geeral forula to calculate the area of a Koch loop is give by: Fig. 4: Koch iddle oe-sixth curve At geerator (r, r, r3) = (/3, /6, /) At geerator (r, r, r3) = (/, /6, /3) 4 4 Area KochLoop = r ( + ), N ad 3 (8) Scalig factor: Cos ta t : +, S = Alphabet: F: 4 Pr oductio rule : F F + F F + FF or F FF + F F + F (5) At the scalig factor S =, rewritig syste for 6 two possible Koch iddle oe-sixth curve (Fig. 4) is as follows: Scalig factor : Alphabet :F Cos ta t : +, Pr oductio rule : F FF + F F + FFF (6) Therefore: Area KochLoop = li r ( + ) 3 3 = r 4 or F FFF + F F + FF li 3 3r r((4 / 3) ) The geeral rewritig syste for Koch curves at S =, where = 4, 6, 8, 0,, which ca be derived We see that Perieter KochLoop = fro Prod. Rule 5 ad 6 is as follows: Theoretically, a Koch loop ca accoodate wire Scalig factor: S =, where is a eve atural of ifiite legth. However, with the reducig scalig uber ad 4: factor, growth rate of the perieter decreases Thus saller the scalig factor, lesser the area of the Koch loops. The geeral forula to calculate the perieter of a Koch loop is give by: Perieter KochLoop = 3 3r + r ((4 / 3) ),, Nad 3 Where is the total uber of iteratios. Cosiderig: (9)

4 J. Coputer Sci., 7 (9): , 0 The geeral forula to calculate the diesio of a Koch curve is give by: log( + ) = (0) log Diesio Kochcurve, Nad 3 Fro (0), it ca be calculated that the diesio of Koch curves at = 4, 5 ad 6 is approxiately.6,.3 ad.086 respectively. Accordig to (Vioy et al., 004; Sog et al., 008), sice fractal ateas havig saller diesios show better ultibad characteristics, Koch ateas at S =, for 4, shall exhibit better ultibad characteristics. Theoretically, superior Koch loops, geerated at S =, for 4, are ore suitable as ateas tha the Koch atea at S =, as they are ore copact i 3 size (cf. (8)) ad ca accoodate log legth of wire (cf. (9)) ad therefore will have better ultibad characteristics due to saller fractal diesio (cf. (0)). CONCLUSION 4 4 Area KochLoop = r ( + ), Nad 3;) PeriterKochLoop = 3 3r + r((4 / 3) ),, Nad 3; log( + ) log Diesio KochLoop =, N, 3; Here, is the scalig factor ad r is the radius of the circle which accoodates the Koch loop. Fractal atea is preferred over circular atea because it is copact ad has ultibad characteristics. Theoretically, a superior Koch atea, geerated at S =, 4, are ore suitable as a atea tha the Koch atea at S =, as they are ore copact 3 (Result (i)), ca accoodate log legth of wire (Result (ii)) ad will have better ultibad characteristic due to saller fractal diesio (Result (iii)). at. Followig forulas have bee derived i this study: Rewritig syste for superior Koch curves ACKNOWLEDGMENT Researchers would like to thak Uiversity Malaysia Pahag for providig fud for this research. Scalig factor: s = /, where is a odd atural uber ad 3: Cos ta t : +, Productio rule : F (( ) / F + F F + (( ) / )F Scalig factor: s= /, where is a eve atural uber ad 4: Cos ta t : +, Productio rule : F ( / )F + F F + ( / )F or F ( / )F + F F + ( / )F Geoetrical properties of a superior Koch loop is give by: 333 REFERENCES Epstie, M. ad S.M. Adeeb, 008. The stiffess of self-siilar fractals. It. J. Solids Struct., 45: DOI: 0.06/J.IJSOLSTR Elkachouchi, H. ad M.A. Nasr, D-Fractal rectagular Koch dipole ad Hilbert dipole ateas. Proceedigs of the IEEE Iteratioal Coferece o Microwave ad Millieter Wave Techology, Apr. 8-, IEEE Xplore Press, Buili, -4. DOI: 0.09/ICMMT Falcoer, K.J., 003. Fractal Geoetry: Matheatical Foudatios ad Applicatios. d Ed., Wiley, Chichester, ISBN: , pp: 337. Ghatak, R., D.R. Poddar ad R.K. Mishra, 009. A oet-ethod characterizatio of V-Koch fractal dipole ateas. It. J. Electro. Cou., 63: DOI: 0.06/J.AEUE Jibrael, F.J., F.F. Shareef ad W.S. Muo, 008. Sall size ad dual bad of a quadratic Koch dipole fractal atea desig. A. J. Applied Sci. 5: DOI: /AJASSP

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