ON GEOMETRIC CHAIKIN'S APPROACH TO FRACTAL MODELING OF CONTOURS Wieslaw Kotarski (kotarski@gate.math.us.edu.pl) & Agnieszka Lisowska (alisow@ux2.math.us.edu.pl) Institute of Computer Science Silesian University Bedzinska 39 41-200 Sosnowiec, Poland

This worksheet can be treated as the third part of our previous Maple applications (http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651 and http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657) in which we used analytical representation of Bezier curves in 2D contour modeling. Here we demonstrate how one can model fractally any contour basing on purely geometric Chaikin's approach.

Bezier curves that are widely used in modeling of 2D shapes are based on analytical representation using Berstein polynomials [3]. Chaikin's approach [1] to modeling is purely geometric. Namely, according to his paradigm a curve is generated using "corner cutting scheme". It works in the following way. For a given control polygon , {P0,P1,...Pn} we create a new one by generating a sequence of control points {Q0,R0,Q1,R1,...,Qn-1,Rn-1} , where Qi and Ri are calculated according to the formulae [4]: 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 and 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After few iterations as the result one obtains a smooth curve. Chaikin's method (see Fig. 1) provides a very simple and elegant curve drawing mechanism. MFNWtKUb<ob<R=MDLCdNFZX]h@[=Zih\\V`JHZJ_cNgB>J<J:?j:D:;BZ>j:BZ;^:Ar:yYvYyyyjywy:ryyyy<J:>:::::::::::::a[`kFCm<jy=jSrFakST`mFMaCmHmyoFTpSnf`cSmNmlf`wYm^SX@mfS`sFGmFU`yQ`CmaK`U`FIMSHmlvywy:>ZyY@yiSxyF=jymfyyymB:<jSH]`W@mvFlvSvymV@akSH]`W`mFmmV`FY`mvymnFTpSnf`W``GmmnfFY``wymFM`sSFm`W`SHmmFmFY`Sxym^SlvS^s`W`FImm^sFY`FyymvYasyasFyiSasymvFyImasyyiS<Z:V@;bSGmFU`FqFyiSaC:GmSV``oFmfSlVmvy`oFGmST``oFmfSGEmGeSyiSm^SX@mfS`sFUpFmVm`syUP`CmaK`U`FIm`_sFqfFyy`wYGuyacSyiSGuymfSyImGuyyY`<Z:V@;bS<jFacFIMyY`aC:mnSV``sFmV`lvFwySpf:FMaS`T``GmSDmmnyasFGMaS`TpFGmSDmmV`yY`S`FY@ShSlvF_sFImFyySxYmvyaS`yiSmvymV`yImmvyyIm<Z:V@;bS<jF?ZFaSyImaC:SpSV``sFmnSlV`wyFqf:^SaCmT``GMGEmSpymvF=J`W`FeSSpFGmFaSyImSP`W`FeSSpFmfFaSyImyQ`yySlvyUP`yyFmvymV`yyyyI<jS>Z`C:mN:lf:vyymykST`mFMaCmHmyyIyay]`GMGEmTpyyIyaymykFmfSxyymysyvYxIlvyyIyaymysyvYLNk<UbB:]BW_dMatyyyyyA:B:ESemfR>\\fIc>GbN?iTiwWwifW]Oi]ywl`vvDW^Wad>_^^h`vIlmG^xof>qopNenninWfTWjn@uOFaUOg\\_`@GZOhhZHx=O^CajgAckO`G>tpyb?A\\Fpc>IZsa]bhuc_nU``GfcTTUUvnUxu_ctQFgGw`kFR?tbAepWiDgufqdpEy[uuwwXy[YIqwuqykEXHyXaywwuIiiYYyxQYYIWt;SfpifHkF<oX`oSfWg^CR;?hborbSVXCN>ALDTKNHx_DXt\\j_PSfhTtuopakitNuYXnhpGQSXLnf\\lNMurQtyAsaETqaoLQOjqj@PNVEQZYnG]kqeK:AU;<R?pPxPjuTnvLLveqgPnVHRDXRC<nc]QWMVvTPq<QlTZTahmR`WC_?BAyhF[iC?Ug=F[_CW=tukDXmfJgGZgF]IxdOchEEEIIymgp]dKcV\\]IX]gWcdq=W=wX[Ew<ACV]WOouESidurKMuv]VX]gWSFNIihqyxOukuTYwxV?Xf[g;QrT[cR[iFOURkUGCTCyiH_HdCy\\_RV=ujqEUkgJAG@idVIWdCvW=sKEtfQIkuGVmfjKYJeF[kw]ODXIR;wWcoueOXsMG_ACc_Wo=bAgG__eEUgMIS`EgnggcefqiRR?hTyDXYDW;BD]SVeWDsgv?G<_R@GcegS[iwLIFB?gF]UgMk_tYgAU^MxduSrUX[Qw[dQEavOqmjQnTmPYLsFmUgEKiirKuP:YN`<pp]jqdX[pnMXkjttbhO[iO^xOvIkd\\SZeWBtw_\\TQeT`hmUALXIuIUwgYN<TjqhK[iOWMmHEpOhT`epY\\qlQw_`wP=VCHSfhwZiOIdUdapK\\rnXXUIrU=uIiYtIsQ<VpXXuHreXWexlytmW]V\\TWm<YEmy:ivIyREaS^eW@tW_<vy`ySxTRhwVqVjQsautvYpmds^mtP=x\\\\xRQmyQlslNmpxdiq<QuDMvZTLSUY;YofxoMtK_EmJQuCluY@keqyGpLR@tTI[OhxtAa@QvPgw>vw\\xi=I[BaeYq`<x`mXi=IkZomCgsV@_SpldOa=?\\GygrF^LG`QA_S?ub>brvcI>q_YvigodW`eOnw_p<YeiOdiNxxoxUgbtfni`hoqmgieOgjmhqa_Zs>r?^^dot>_l`WrMIbw_gpnvsvbUOd@yuyA_rxh=Ime_f_qdIiZr@pwYwWxtVgushiEqxXPw[Qghxj=vpYFn?nuNYyIywgIZVimjijI_qhA[`WfioiM?\\LXjN^keV[UHnFNt\\Otafjnir>N_V`_NHw@xpXNlvPwsQ`FNqtohmIs:f]CQsF>]w`xK?i?_nUnucHtGyv\\h\\Iyfe^vwWbUxc@IoRHrROasFp`yq_qidHv?Wjooc@^aYP_bvw]Qwjhb?wvwx`wpgKgk>A`?Ax\\>pgpsS?elQuc`orQwxP[?Vnio`jpaBy`HIcNVrUwpA_wdNiVXti@goa_LAb>Yy:Xkxf`qHvFAma@xTVa@WkQquPwvfqxY_i`qZAI];whv_x:yvp`ux_ndQdRPtUwyt>diaxuxqwXu?HfxI__Ire?qUykVyko`bN_v[ikSVteglqaujqkJI^sIuMiaqHmCXsjxxdgbhpwsIZQ^gN?mT^eXPjEi`MW\\ShcMvefxq`_\\ygmEn^w@i]_hsfZQX^ehd^hcWa]NGkiOaq^^sH``?rWOvIQh;hfKGZ:AsuQ`>xh;xoy`]HajUN[AfmKWZ:fvO>^Y>_SvlUQ_bneqaiJwh]PsxxoPfn=gbS>w:X`eVZXFoMOoN^xsPvAoxCQgGhoW>bKAaSQ]vhqxQyEoqsPpJ^mYosvpdgglE@^snmlneD`oIGoipw>vd>FoZFgX>wJOZV^vGolBWefYkPIau^vHGZ>PlSxZZwvm>bBF]>qpGvbNPf]hjmQbSIdaYiiAcjPgDibAnr<GhmHvJyuBWfgNayXu=pk\\G`:PadWmcAmtoel?qXysAYph_Zcyl<G`ZVj[PgIWxU_]ugl?qbEglG?aY?daXyHydIXt[Hgw_ufQtcpbk@nBI[aictYx[P]mw^?^k:ph_@dYoemvZV?wuhpwGgnQhj^gvGmKwwOydUf\\HIqjxpFvdmFv[^sdX``qgiW^KWy>X\\HxoWxrQQjJGbfP[<i^Rwylv`<`x<@h;_ep_yXgpkAdM^hcp]PO_RXyhpZGW_fivTiaDIs@OtHWiPNmnNjQ_cnnyDim:ilXOcjonQfasFuRiwnQ`hgno^uOosu`c_Hn_auRWt?vcrPrGAsh_[_QkTvn[_aGwoX`uSAeJP\\w?x\\VZe^lQ^ubqq=IdF^hGfmY@kAhl<GmuQ]xg`THnInu^Y\\@V^uxu?PhaI^KvbEhfG`k?ajqgdRi\\=hvdyhpQgnixXW^Yqt?`ao`tOvrs`oVP`rQt=QcLFhLAbSiuVXnGne]udYRHuRBOCSyw]eRykThWHtQEtmHAmRoKWkIR?oHrUx@ei>[BvmCIkibsIIgheKBT[HF[S?iiLqs_UrBeg:OcNGrUMHnevssfq=gROEAYcakD=mbYUY`QE_oUl[TdkEO=in[bQEI]]E\\eb_uRAGii]RkaeTcSokVEwbXcXYerr;h?CDW=xHqRGWBFCh]qSHki=CCFuty?rgeHx?F`ispugLqI]evMkG[KituWBavHui=ot<mS?iCfAG;SdKErqmrM;YEUYF]fOigMaBieRmSEVubWWhoifvQH`idTIImcEx;wCqwwMU?aSVMrVcSjqdtaSN?FRywREW>Kd`Gh]uBa[Sl;VOIrQYCUchZYxUGeXeBT?v?kImovBYwHCE=OW`cwAqw_KbcgrHcijGToUtgSBs;T[?CXQhsErG=u[OiMIbAIxmCBi=emgFNMeYCiL]VmQEtAvu[x:YcHwIhmwciSsSvLcS>?xiksXcHieXTWFmqhnWeBKvpKWcudciHSGGDOYikrxafx=v>GHR_u@]TpgbKSy:oc^SGyGy>=c>gG[GS\\QX@ku\\ks>gejWBkiuVeb>QfN?XNAIvsv:_EMoYLoTQwr[YGnsRG[isgyRsx@awUSbLQte[FQwSGWVV]FPsTfUxOgB>Ucekdk]upAykKWZitj]HM]RpWESAgBEClGBk[ckEDayyagwPuT`SvemCawxHKstcFjqdkQD@kYUoIBOybmebmt`kV:QV:qgPEUNoFCQH`;Uoeyv=YTODrer=?UqsrfmtR_cfIvBogImfjoGuqeGQs`cRTcX_UvuIhLcFjQBvOX[IbWkhxUYrKIHmX=Krdsv:kXkGYnuSNiDSUXtOS?[iLkfDawJ]w>lVZDvFmlLlWjUSVDp_XT@eUTyRXywiLt;twx]qUXJpMZmP_cQsYyc]Qi\\HhuV\\Nwb@`udVc`YqKf\\jqdjWqOxst^q_Qb<`xPf[QwfTI_qochAkup^fI^^Pd:vgKVa<y]S?pTq]EWfm^n_aa_fyd^gx`_IobCgd`_uMnhJXoJsu?cWkc=wEU]X_axuUt>irTeDBgufQrIAUNsBdsSrGGgqfAkgdIrEow<qiwWu\\QDQMvr=erqe\\kiI_Ek?g>ch^abeWxRmrYATvsbQCDrmDKSWh_uR]bVaYKatg?uQQXPSXtyXIobViDqYvmoe=YUH;vq?vRYEPMdeGs;_BsiSE;VBsu@ecSMhWwB`MfhScuCFsmCiQxHMbuygm?HrQSlgDQYR=_B[kg:_WCufrwI=AhCYgRgW@wTwcSlQdPKFBCYcSrKWHdwvBMrMaeHmId]FuOejgBQmevMC]ovb[ilsRlkyfIFpacNydkYSeotx_f<Mt_IF^kX;Mxbqe`[Xa;w][SLWGskuK=Y;uECiy<Cw<mctAswkdN=FQYSNivwafjqdHgf?oF:YCl_VTIUyMXa;XNkuqKIBomnENAAkbeWhAwIlJBpOaiT<emp=VvHQMUy[uPKlqK@XBITrepfDUTtV@`oThUvHu\\]siTSpuUFmNeqqwYxGMuxtkE`OL<U]XV\\AQj]LXiT\\aYTDSqDSDll]PR>hmEySNuOSymhMul`MkdJYuuGHxC<vf\\qSMTWIKYpNjEkdtxS]vMMNW]ku\\Rl<W^TmsxVCMJuHvC=rhQvyUmZiQ@MovTf=xgnvfHPeove`w^CvgBP\\roq^IqCWaNFlkvbYGcTfxu^[ipnKPgc@[<itHQyGIiEYkGP_^`gkiyRqmiGi[Hq??dm>ixxgsVaaWi>AuBvw<FxMndIYvXY\\rNgsH`T?sunZkxZGHp]`eMGvOnvkOrePlr_uxQsuX][iiL@iep`;v`xXaNPphWvI@lewsnngRIqyojAfs_>ekvsr?cshfAnicXr`HjDrDih[Et@EE>ovXqb:eyEesYUshEy_UHmQGMwYG;IIsWKQbtacZ[EOIGsaWZWcf?U^EHf?XrMHdYBGyfCahk_sLwvVGSw[W:wHaEf@AtAWTkeTLOTp[TZmRn?Htkc]AWaaGycw;Ki:Kc@GfaEHj=dmUBXycg_rh]BFSudArFgEUsYvuc?WevErU_Su?bnMY<;Y<EHXAf]cC>OvHcrywti]XM]RuqhMwI]auiKr`uXoyGiuh=iVIgfuWVqgECmic=Ei?GxOyV;xBoxxoydUBPaf[ae]UEpagAwxNsV_Irq[S?ecqUtKQvI[Y;YCKkx>qCesCvQeMKEgCIYEIr]fAYGrErWifYsteiDbYiy=YCIiWMfPYrUwCTMh\\sYOYVvUUtwvKsg>?YeIvHCwoeYcoYeQylUhlACoEwHeVPuurYy?yuyHrCiw:iMrETx=uLUWPykUlXtawAHWIdnBTVjiy]evqTYptqSAx[iq:YSthLx@YdyoeUMXEySyPq@QUQvPIwgdlqpxfAYDAtvEOIMYddb<Aq^x^evaWA\\OGaJfxtWqh?mWOitAejOv\\Xc;PlT>uwfwqoxQnt;Yq\\GuXW`TyabVd\\AjQa\\=F`s?]F_oG?r^pZAgaaWkEfscHmUpawy^knp\\UiKrgGsgSGikbgkhaEGm_VZ]H=KUb=E?kws_hX=hNMe`obiDqoXK>QUyxpQEpTLig_]RA[^IwKPvvQxhHuif^i^fLPsS`\\w^i[_gB_v:PhhtdoD_IR=gYNaxkGhZcR\\]B]AB_mf<SIqUgcGIY;i<;Wg[iLgxp[DW]gyuWdWI;EuESrscg:OCbmE<WGHkUeYf^yhXSgESV;USMyxDaYFUTnYG=cUJIwFYV>gTwoc\\uG_uvc_YcmQVENnprnPmoHlMevpitSHqdpMRHJF@W;xP]uVR`uPunf@YZmUZMJ`pJX=VaLVLDTpdLNiKI=R`tUd=vgTuDhOCLwciKaMl\\o^XwocyycfjAp]oXlAPw>_]>NtRIoAxqW_uyIbgNdYXyZA`gvwbhwtVxWxkhW]HoajhjhFuEW^yoifAdcY_Y^wgOZvPsIy^fgqwyeXiolGy[atynypQsioeranD`hxIatOmjImcYrIyyhYgAogv^cSwutafdQoqwmuisEvypw_SniSGaYXxSY_Af[QQ[dapdA[vQeg_[Y_l`A_LVf_@eXIhJ_aIgeVPc=OcG_eWicjYiYHpSh]=hdeFuax_tYfZ@]CVjKXdvfb\\_xePxXFbShnGI[ailDgdVg^RFg<^qdhuphqT_jdvdgW^DPipghpOl`^h;IwTOhbGhnigWycQwvgQk<>^Q_y?XqVWlPwr_Wqs?s`YlX@iU>_tgd;wsPpcHh^>Oo^AqoV[:N_^P[nGnE^lTqkTGq<a\\aqeY?qNxwGqmpheLGhsg`THoXOulA[Q_qYYofgpwA_[YtlHx^prJinFne>pxSqiNqwEhb]iZNPf\\gngxxdYufwad_y<apqPvfOm]Of_`[XguKhunXgMQnX`maGl`OwZistFfdqc^@jpwxe_oiQtAvgk^w:Yp@wyEn[>YZFonNIrUOxsNlShlMH^AW_=IlmVoYVaxHfmNlWFiFP^qF[XyGcvWiXaufMMSiUIlAe?_c=YUxOSH_wkKC`UshWruEH]]huqSAmVDiwGsUgAGmueygrKOTjgGq_UeagJQYVGvewVAqXqGvw?saWRe=ducWmSycOYFWviIu>sD]WBjKWyUsSGD=wdKUC<YoOLMSTtneS[XT^MoXQQjqWCeqqAtOExWttAQKq\\xJerRymHLxy\\xYUOPloaXkgf]da\\iAyuI^CnxnpZvxtf`uyAsMfm;pyYww_PkkxslYqHfwuPaJ>y_ydcQkloiNiy?ynhwjJNd`no@PsSWdeFsg?_VVe[XcQNhlPaOfb;GtqPbQ@rrX`dOpDydfgbeg[rwsli^koceiluQicPffg\\NvwDW_<PnhAnE?funsQfrcgeQajhpd:xpM>o_@_>FnKqoZfdOyg:gbc?[\\V^;pgDQsQgllgli@fDw\\spt:xlehgNv_tydAF_]aoppwX_kM>gEq]Ef`x`bhxieW_ZXqhGwNnbf^o;Ak;YsBHrE=ZZif>Av__R::<YSDPrEMk::BXmRFkU@eG=uy=krROgfadamGf;B:;j^PNaLNQENjD5B Fig. 1. Chaikin's corner cutting method (from left to right 0,1 and 4 iteration)It should be pointed out that Chaikin's curve is equivalent to a quadratic uniform B-spline [7] that in turn is equivalent to a piecewise quadratic Bezier curve. Similarly as to de Casteljau subdivision algorithm the matrices that are used for obtaining control points of the left and the right parts of the divided curve have the following form [4]: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 and 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

By analogy to the nice result of Goldman [2] who showed how Bezier curves can be generated fractally we find Iterated Function System (IFS) for fractal generation of Chaikin's curve. The IFS has the following form: 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 where P is the matrix of points P0=[x0,y0], P1=[x1,y1], P2[x2,y2] in homogeneous coordinates: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It is interesting to mention that Goldman's IFS ( for u=1/2 ) for fractal generating of Bezier curve is the same as the one obtained for Chaikin's curve. It is also worth to point out that control points P0, P2 of Chaikin's curve do not interpolate its ends. The ends of the curve i.e. points Q0 and Q1 lie in the middle between P0 , P1 and P1 , P2, respectively. Points Q0,P1 and Q1 should not be colinear. In Fig. 2 Chaikin's curve and all its characteristic points are presented. MFNWtKUb<ob<R=MDLCdNFZTj:hK>H:DmRhuPr:<O`Lo\\jyyyyk;Hj;LJ<Lj=Xj=`j>`j?hj?pj@pjAxjA@kB@KCDKCLKDLkEXkE\\KF\\KGdKGpkHpkIxkI@lJ@lKHlKPlLPlMXlM`lN`lOhlOplPplQxlQ<MR<MSDMSLMTLmUXmU`mV`mWhmWpmXpMYtMY@nZ@n[Hn[LN\\Ln]Xn]`n^`n_hn_pn`pnaxna@ob@ocHocPodPoeXoe\\Of\\OgdOgpohpoixoi<Pj<pkHpkLPlLpmXpm`pn`pohpolPplPqtPq<Qr<qsHqsPqtPquXqu`qv`qwhqwlQxlqyxqy@r:Ar;Ir;Qr<QR=UR=]R>]r?ir?qr@qrAyrA=SB=sCIsCQsDQsEYsEasFaSGeSGmSHmsIysIAtJAtKItKQtLQtMYtMatNaTOeTOqtPqtQytQAuRAUSEUSQuTQuUYuU]UV]UWeUWquXqUYuUY=VZ=v[Iv[Qv\\Qv]Yv]av^aV_eV_mV`mvayvaAwbAwcIwcMWdMWeUWe]Wf]WgeWgqwhqwiywi=Xj=xkIxkQxlQXmUXm]Xn]XoeXoqxpqxqyxqAyrAysIysMYtMYuUYu]Yv]ywiywqyxqyyyyyK:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::RIoSYME:tJ?`N\\@Nd\\QgqxHaJdLkeTK=yxSaPG<s\\Pnb<n[LNKDT>dlKdrW=LkEKlHO\\IND`JJLn<LNl@qZ<LMtpRhV\\HUFqWV`NT@Y@`VFqPC]U\\HM`lR=lRAerbLk@erbDT]@j\\<SGuXkhP[Ijw`Tqutluym]whpohqodtQxYYYiYipUOUL<<OtPQ[LJ>Ujb=LGLn;]PBpnWesa=MmuJ@HS@<MEAP]ALGPmXYmAtxihVFLYsEOFHsl=vEHlGAyk=MpPsyUmltSNIrt<X`=toLwdxTw\\u]uLsQR@xQc=QQLtD<R;HWrhkB\\qZIx<ut_uskIKGtSleK=tXHEl]urjHx`Hq`hVK@NTdXOupfltPUNTdP?qqUaujDQwqYBymalS@Inf\\w=ENT\\R]DQnDSvMQRDMiPtjIxBXLbQMd`MiiwFxRQAtnLxNmkMYyKxNsQXr\\QdXWP]QeQXKLJHdT\\LmpPKgIx`<RVYuHhvR<TghpFmvphTsaTVXxs`qXTtjUOWdQjemwmU=]s_EQbQMoelTmPGttNQstdmAaUMQN@eWFqNqQWGXSxLn[XsyukcykExOspruXSXPyEiqp\\s>\\R=QO_AoAUOn<TcepieP@\\tYUPGUn\\huDuy;@WZlnZMwHqpeeovpY>]PneswuYTyTLusoTRS`TTQwgYkx`MeMUUAmVUo:YpWxX\\eL^Du^uj?yJQaw>@ppmusuwc<uVhO_aUMdXwuSuyWpQOIIY@MpspV@eSA\\VZQoUQm_PT?UP]YpHhkUMwtYjNUJDaulysA`Y\\EX_huYinbQmciPMpMpIlgtpXMw]DSQav_iQvYP=qYx]wf`vVUlbUoCIM>PnOxVFxmEhSvLsq@yOuxwMmvmwZQNupoahn?pv@ikbest@lAxKN]XuXNc@jtTvsQn`xMGhqn<OETYXaX_uo=LNStt^=PwipraxtiV]<VX`TVUtuyPguxHarxxNCtt>uypLTUyorqqe]qc`L``W[`o@pwMqxetT@iL^\\LCxMGMUatoiipxQt?IK_DS]uKDhQQAPWmyNiveUshtwdh\\VW\\BFltfiiWdYYyPNvpgcPQyHQaIAqa__lXx?_wTh[cGmtxywvwSVfpOlcPtj_xYnwUfxj`mW`ujYtF_cogaNqmlgcLnbtI\\sWt\\goYQyPhxUFr?Q`^wjrVhGQvGYfun]kik=xmEXuf`pFHovAnsvu:?vqOn>WvXOt;qiQodw?fwo[:amch]ZyiyHg_PvEY[`w[X>biFmQxwMQoQIa?YqC@_KahZ@tDfd?w`XQrgitpxwsYuvOo\\QgGq\\>ItY^jO>]Rhkkip<NqnXxHYdKQs=O]vNp]x_Iy]fPafA_OQ^>N]lA_GOj`_a^>dQyftP_mIsfatAhiwV`UqsT^gQ>\\lV\\`PkGf[nWnhpx>>]PFn[q^NhlnwsRN`QN^FvhNpnbXg<a\\_ggQo_c`ywQmiilexnn^u@IcL>ify\\EvwCXyMhix_\\KooCxeTqhWP`N`umipVqs]qdZvcV_ZsQvC>o[>llytOGtWArmGkpYeDyK[iXcBKAfSwC[cWMQHfotkUWfIU[aGK?fcUD=Yf;KrouuVmVyTK?Ml@XT@xMPuYSXoRytmlYUpwUXOvHQ`QvipP=poB@NGtKbeUD`vdTqE]MRDL@=KymOYauLpTQTQ?`vudOruRyyQi]QVLXQ\\qe]wUewj=PF<v`Upm`W[Eo:YtjYnjPo_iR<XN>AXphVa=Mcg_@Hkn@im^fi?taWZCvqNyhlwuhP]Z@y@GfmirRYZXx^>^]JWrNQuSaZwWyGiu_oi^>b[GoTXlWiu@QjJ`dbWeEO[a>nZxemQfVpq;>vVAhbqldn`^ok<@pfo\\Ef\\GxbEGteWniWhU`tRAwaXbfPxk^e?pj^@i:OqjV]iaouXpPa`Wpwxx`gy^wq`po]fX]AiyUo^=nghFxuoaW?kspfOWkUYqdyqUAnXgcP_]SawTfn@OhQVd@_hGImgVdUpsF^uNyqW_lWgc`wmmvcoPqwYv[vkgV[?xhpQhahd]ijYolOa]N_[ogkqgfeXj;goPylF^qVv_n@jTfun>^fasGqaUv`]@wFqpJoddwwrqhM>e:gkCph^wmYFq?icOHi?IjSvZM_vlotvfZOQgMq_<vgna\\NXg<yZq^aqovhwkdHff``pG_d_y^veox`lQuwFm=yrtp^Cnbmxsp_ehAhkX[nI`VyiEAebOin^wbx\\AYyUn^iagCW]ufs^naSOZ;Wx>^k\\WsO?yUvtUomoffxfirQlgPl?VtK?c@@Z]hwevgbpqaxwFXj`VmofcV>yUVxIF`WqDAYGQcaUyVMBB]YWyfB[cc_RAOvIIyXeCiYbcQsAytOkr_SBUEVq]HYyvliXk?uE;vpUVcEfKwrKsCFaukcHf?dIiFYmfdIrEKhCGxocD<?YWWWNuHaGXiCti]ERuT?=IYQwUwsUutrQRPCUcoHnIW^eJXqPXIxZYSl=nyDPS`XWtPm\\YEdTUlYretTtvmlnUxPsUs@upYQQk<QT`wElKJ=KMaSstnnPLlqvjIXl]oltJIHQrtYU<mPmlXdRvXXGaMBdj=pk`pN=DTyTymqsodmRlr<Ho;TleDy_It?`YE<Ojmq>YkCdwElpOAxd<nLas<uvYMm>yJl<x@QVRLrapQNeTsArIPJpQx<dKvyLDYyAiLliroTSHhKwdJ]veNPxeqrNoqrpnqikDQpVhpvHBadqexToDMIFGSe@wTaYYN?BY_dtUrpiDx;fk_g@GRaCbYMCsaT]IL@\\soDlOAJM@Y@mJxTMvpYsHwWUNyxKCtW;twn<x>QSGQtYHvLAXtuvmhUEDyAmJhxXoqpyitnpoShquPkixXNMpadL``solYuLo<EYDmJ\\youHW`UM=amv\\JjpNvhtd<XodujHYqxQttT=tYo`ysPQMhJNaSRdr]MPjpW_<PKyKGxN=`wTiq><yxuTulJ\\muqDjemLP@R@tsCAJS@qglt>locyYtxWS<xXEjQtSUdq^YKh\\pitvyDMrtLaqvTMV^AjlPjZ<L<pPPiymEjkUJdMtqpTqERRLmtin?mtMqRnesqANNLPjiTJiqepRFiP;yNwqOCdJG\\lhtp=XK@UKblJllP;FwKYkoqbD>ox@l`>_l>d@A_FX\\JG_Ywali[B^cW`seak[QmIhcuV_<w^Oqt;a[=YnCh_Inud>an>dAvmG?`rO]pPo\\>^>gbw_vyxsjposOndOpZFs:qkbiq[nj?WpWo`V?mb?v=Au]nZG>o;yrlQx;NvhaaJXqmX`Lwmf`lw_^VNakirMp`npf=iml@rWhqnIsiFZfvfJftWNv`>rjO^Rf^qpqif\\a@g@XbdF^DV`pQ`_>mc`ur?]CAvnqe@njXxreaZSpj;WlEfqCFkd_u\\A]eOv?V_OXaJH\\;Foua]=asUnsAxr;NmxwdHqPMWh_wHERjYB[cF]aVQUb^QeJ=egOvXkt^yrFsWOit>WX;?CN]EtorVYSouidSh@ICtWcZGVHwhs;CO]u[GYDqxrOwxSFoic]wEAErloyBmH:qH>UBUYE`[cFub:mxsyBiwV`ORwOhuMHG=hf=iD_sseSq_VFmb;GE^MHuuuVuFReVGCI<eso[U[GSgEiG_U<uC_Cx_]w\\wheuW<]fXobusGdmIWixPeba;rGmXkAusYs<gCaks@EydKwHihv?ixOvZEhqAhmOb<CH\\We]gc=uXYYH:SDfgEA_dludeMynGuHYVwQt>gDyKXmKDv[BvAs:;X@khayrGKeN;x_KRQAh[cCqeB;_FlOuC;IEiHyQb:;DtoEbobHYrNgxSqTRUDh=c;;T[QsEmTVAXWihL]cdMGrsFhGVV[R:oRX]H_KdkmViuspOeWOCHiREcDRMIa?UrAWV[GwGEYWbfAGbqy>wtrGCbgveUYN=c[;fRWvGWSwSRqOXNybdwXW?G@_U]sdokG>osXQC;kIV=fCYiWSxC?CBaIs]frkbW=Cb;YAErvuvFiriOcw_XmSRFkfsWD;cTGSBFUHScFPwcs;soSCYEEmyEkmh`QYA[th;EiKt<uRiAbDKhYUGL]HtqBN=sSYD\\cF]wW:ecuatU]hwsBwcU=aXuGCbQEdAyIocyCFfCcVUF\\Ew_QEYkFA;to[H_ugmEiBog[OWy[S=Ab\\OG`_HxiX>UROWVLYu\\oS=QiIOxGiY?mT@gGNcTYUsAYB\\GfTovTarv[R=uTbyh\\sGsaXGiH;chuGhTkfdevqer:?WbOy@]swQSwOIDSgmiUIYFT[wsUW_wFNCdLpRattj<RJiPNeS@@Q:utJet]xvyALGln[ew^mr^QJtpj`TyDLjhHnqAqelrZ=RKitm<Xr`RJAWU<TCxLwUx_Lj@UuSPLuYSVIp?Pndtp>Yw_<r:PnbdYM@kIuy]Esa=n:PY^_b=wdJijKO\\<WnO?bkfqiVne?oT`ZK>jhHbcgvTV`@FtJi\\WW]F@gbQyJasAOeb_`Ii`]`wqWne?oD@xL@]SfkKAihQpKPeEfn[?b_xglW_?qrUPoIIfl`tBAgt^lONcPYfCybOaogwx^xdGVgNXg\\vh@x\\ZN]d`gep]BwkFpb;f\\yyv^G[LHrXxpXooSHrFaZng[O_kgi]ng[OGrjfZ=NkdhvaovwOjb?otXvqhe\\HgrIfMFxQngsxd;>WofiGVtkHXaT__dHKYXCtJeVO?YLMg^mELut>cdXgdfCeHgUMWhiQDZKEdOcdMr^[c`Wr:Od@ihB;umqvl;fLGEyCS]Gw:iHh@KtpjXewOHSUaPW\\x[Pwgqk\\LMdPXuxKYykb=wods;tu<]W[@WlIqw`ysDkqlugapJEUeYyYHJ>UROauhTm_pk@pmemlNAqvpjOAPoew[QnuPlaTtH=Rklxq<LGdLGTNiXRl<umqthttnip<mOn]PCtvvAlEdQZ`nwlpQIlemvslNZlLdXWP]tixRd\\xd=ypqo_xQ\\lLT=R_uMoIYKDoJUJDIwllvZqXVUP_\\KS@tWUvityqLP[Pq`<XIXJ\\Dnupjk=qH@SbpoS@K;=S>TYXQuspNtXTNLJ[Pa<Y]x?eAVtWWx?^k=o]nwtb^qHV^I^c=Qqn_\\J?k@FtZN]>KvAld]WRItsXkwhoG]VK<R>elg]rK]jLQvT@lLTwmpxWmuF]jjiU]UyyXlB@TTXWxdVOyl>=SH\\KalT]IKS<wB=TfPMleSW<rmlw`ESOQP:TuliVbyLo`PfEKJHYtIjF\\Q;lK[mVhej`UX>YVU\\QItu?hsB\\vi@W\\LYrynY@J@DXbXo[qoAmxfyKRMPZyqQ\\QIxt]dl`Qq@<rIhMcHQdaTKEKqeQD<lDUX:eK>hOSawE@uedYgqWyQjJMmDQ][qtqYqiGfYgnyqbtio??_g>gJ?uw`bn>k@AttyawHxlNcngpJPo:>Z:>ZCgbH_bhPbZO6J Fig. 2. Chaikin's curve

Unfortunately Goldman's method fails for fractal generation of a linear segment. In [5] and [6] we suggested how to obtain IFS for generation of a linear segment in a fractal way. Basing on Chaikin's approach we can find suitable matrices LS and MS that are used for definig IFS that an arbitrary linear segment generates fractally. Instead of the ratios 3/4 and 1/4 , as it was in the case of quadratic curves, for a linear segment we need the ratios 1/2 and 1/2. The IFS have the form: TUZOV3RLVWI8b2I8Uj1NRExDZE5SZkJAS3I6YlpDSXVdQFNCTlNET0U8eXl5eUFCbEVVTm1aO05iQ19iO19kck91Z3lsW1FsPUBeRl5qU091cXlpX2lxVD9eYE5sOmhtO0lxY2lteElyaz54a0BsX05fSnBxZlFzY3FkdllrREhzP1loZ0haSF5cXGl2aHJueWNYX1hoYFt3d29ZaHRoXFxmR2hoZ19LeWJHQXljPmxnPnFrYWVgQGBUQ3BFcm9nRmJVeVJHeEpvYndvZHRraXU9RD1DdXBMTmBAUmVBd2FNUUZJWFhBWFpsbHU8WT5hb2tgd2Y9bmZ0bWBEUFhRdWY9bWpZUExUSlVNc1ZobUdAbnNMc3VxVlJkVmhtTVFZb0ZkVz1JcGZBTFJNWWZhc1tcXHg7WFZQQWxWRFhsYUxBYXhTQXBhQWpbPExPQXJqbGxCPG5idU5JYW11dFFlcVNJRHFRQHg+YUxIPHBWeE5cXHhUWlFSWXl4U21Ublxcb2JdWERgTT1BeGZpdWFxVlZVUHc9b21NeHZteWVkS0pZa1xcQVF2SU9KeW9bSFBudFNtPW9mXFxOQ1lUZHh4aklVTV14XUxTX2xWdFRKZ3RXSlBUXXhub1VsZGF0b2luWGVraWRyZE1LXnFzRDxtUEhSUmVyRXVxTHFPc3lTSm1RQHRxQT1Yd3FvckBXSFhTcEhKcXlZWGlvcnlLYmx5O0hwbWhzW2RtS0BUP0o6YG5cXHROXFx0VFs8UDsyOw==whereTUZOV3RLVWI8b2I8Uj1NRExDZE5yX3JMS3I6Ykx1V3N5OlNCTlNET0U8eXl5eUFCbEVVTm1aOz5oQ19iO19kck91Z1FtVVFcXDxAYEZeYlNvbnl5eUVJXFxBRnFOSHdfZmNjZ29yaGlKdnJoZndTQFxcXFxvZDtublt2cnZGbGVJW1ZvY2NYbHFZY3VhZVZ5cHNpeHJnZztRbElfc1BZbj5YZmJQXFxXUGY+WHhgTmRCaXBBSHVZeWVaT1xcc1d4O0h3bFh3Ql5lXFxOcHZfXmBGbEFoXVJwb0R5X1ZxXnQ/aXdRYV9eYnZmaE5PblBHeExnd1Jva09Xc09RW2VQbGVIXVBoa0VZb1VQZF1OZWVfYjpgdU9OXFxzWWdCaGVub1o9T25Mb2Y7UGlvPmxsXnhxcWBzQHM/b19sR2xFYGVeUWtsR3NsbnBxRnNIYWJ2WHZwb2w/X2hiaF5oeXBbdmpXcGRKZ2k/eXRcXG51ZGdocWlwYHBddmhuaWhoUXFuUFZ5RE9kbXBnZ0FoTUNFUUJvd3hvb3d0bXU7a2ZEd1Rlc2ZgUUJDPUN3aUJmSXZlZVZyc1V1V1VKZUZVdUZZVVlaQUZxWWZdY1lnU3lDS1hLVztMWFdpck9wcD5QUXlBbU5FUVVsTExlWXdQc3lsVk5tdnlZUVxccWxKdXZZVFBkcHhzaUpNUHdJdHdBZW15XWtiUHE+WVNhQWt2ZHR2bE5QdUtoVEtQaGtlTHZzYU94SVBgUHBPaVReZFdVcHN5bG5PXXhJYHdtPXNvbHNqZUpxSVNpeHZFdFJIeExAZFlYeXg/bE9ZZHVDUVhrRXlHSzpgblxcdE5cXHRUWzxQOzI7 and MFNWtKUb<ob<R=MDLCdNr`rLKr:BN=V;]>SBNSDOE<yyyyABlEUNmZ;njC_b;_drOugQmEv[<A^F^jgQtyyyQXw=v`;GiNPxJ`gq`ba_vVqqV>duGsBpgEq]Ui[fwf]WvDfovvu@yoIYvpvfkigqfcXqusXp>iqNGkvvoByqC@]Rq[EYb@F^M@_RvkFvtII_Kylmvki`r^N[JY^=iseiktHkhNdYaq?NlXOcdFtBwuNwhgxbeIrpN[gXZUA`nVaU?dUgn?_vdyb>pmG^d[ih>q_lIvbVdCgdJWpTFdSoyyp]f?f=V]RAr@n`jhh]WlwHkKxraxve`tOAyyHs@Aesn_Ugo\\iq;PumypEGp_Op>ougPsHiZOqlD^lNobEar[gepo_bppZ?hNwxBo^OvjX?sEfmHfmn`VgxZASlAFIQvxEFfSDoODdkyGYRx[UysunEGj;TkysqqWKWTaIsn;fsUyNmRhKGEKe]Uxa_TnchgceqeGSII_OVQWXYeBgYXPas_cVEkchSIaiTBqsbId:qBp;@=VuYXxchoWcyIsaOiXcXHmIW[YRQxcibqIWMmIWovUGFceWeQfGQVmwvMUtPYeMwSGAXByfbYR:qXsmHh_uyShXwhVeuqKUrIibYyO_gk_F<[vvm_:>ZCgbH_bhPbZO6JMFNWtKUb<ob<R=MDLCdNFZTj:hK>H:DmRhuPr:<O`Lo\\jyyyyk;Hj;LJ<Lj=Xj=`j>`j?hj?lJ@ljAxjA@kB@KCDKCLKDLkEXkE\\KF\\kGhkGpkHpkIxkI<LJ<lKHlKPlLPlMXlM\\LN\\lOhlOplPplQxlQ<MR<MSDMSPmTPmUXmU`mV`mWhmWlMXlMYtMY<NZ<n[Hn[Pn\\Pn]Xn]\\N^\\n_hn_pn`pnaxna@ob@ocHocPodPoeXoe\\Of\\OgdOgpohpoixoi<Pj<pkHpkLPlLpmXpm\\Pn\\pohpolPplpqxpq<Qr<QsDQsPqtPQuTQu`qv`qwhqwlQxlQytQy@r:Ar;Ir;Qr<Qr=Yr=ar>ar?ir?mR@mrAyrA=SB=sCIsCQsDQsEYsEasFaSGeSGqsHqsIysIAtJAtKItKQtLQtMYtMatNaTOeTOqtPqtQytQAuRAUSEUSQuTQuUYuU]UV]UWeUWquXqUYuUY=VZ=v[Iv[MV\\Mv]Yv]av^aV_eV_qv`qvayvaAwbAwcIwcMWdMWeUWeawfaWgeWgqwhqWiuWi=Xj=xkIxkQxlQXmUXm]Xn]XoeXoqxpqxqyxq=Yr=ysIysMYtMYuUYu]Yv]ywiywqyxqyyyyy?:;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZQOclw\\:b==ULCTJcDXoXuuT<cD?id?uUSiCBKV=?FRHK]lOBPLJETNLvDPvt]TJdJGLNLDRjEJ?EuOhr]HTM@Mm@nMHPCHuFmV^MvBar^LM]]k@iOk=P<XWn<RN=Kd]Lq\\L?dRdMqU]RJyKnQxeXqPxXIIldpwhPyWmuxxwxyxxeyxisjFb@W_^Td=dT[f@Cg;MWJOe`CHMgVGewSmHP;VJSsHkRdMEZYdS;XyQS\\usquD;Ar^UYx?CSmsIWsJ=GyigCMIOUBe?rCgROaH>egdySOOYQwGFAxAqbmCFxsDFAt@we;SBeeuPqhE=Y=wSNAx>YWR=iwYEm]yrkXXaFOaISiVJOe@?iuaVqIrW]f`GB\\kIhwwR_EyQIfyrMcF]ATPktJ=SGKT]CIdMVraXkKHXEujIcZWTbOE\\[EdOX\\CTo[FF;ygcukYhZYtvEiVkE_iHcugtsd:Ax=yXJOeaEDZux;;rfgcHsdj;fTSF;mt_UblUdYoyGGSgKujUFN[wuOgXqGa]S_EwbOe@?GpsYv[UmARPQwVKYGYDjgC@aWlsGQUIiSRd=sjYug]yeQvEwtrMUaeHv=y]UUp=GLUf@]U;WU]KWZqc?qeF[CbwfWofHCfWMcysRd=FvsXFmsmwsT=YvQbNAdKkxSAuWKTqQeXaw<KtMYe`iWY[VVEuRcEEErwGcQMFFAeRUfWmDY[TMWB;gSvOxCaEBqSKOhQaSwOiuGwQseu_dxORloUmCftGE`=vlud_Yc^OSGabiOeQkWkWs;Cg@[dcGygoDH?eyurdiEcKg`]UoAcPoCSuXmiwaiFcOef_gPwccYxeut\\CscoFTEfCiE>WfEMCGsw?qRXcvUwyqutwKeVshymeOUimOt\\ovtixJesD=svgY_qHqgFO;T?owiEWt=W>yewSdkOx`SF_eDjKdKeS\\SuGUc]wTa[xAGELETGurLeWYieG?y>MDdGCgcEZgwckuYSEhMG?GHu=GEouu]hwcv\\AvVAg]OvaoFxgdAySTueNMHnaSFAfwQVqQEPQyPoGagFNgWs=FlugkybZeeHYUMIwxUWkUg`CYoUEWwVVYhgWTPGcReHigraCwxIYPEE\\mXLGhf=InQeeYwt=SHQs>WycmF@Qyx[thYStAE@YwZaWxCXm?IwoVxmxZcIA]YS=xyIRNGBoUgv_TIqDl[Du;EKMg@OcWeYK;eeKiXKGOgh\\?tDyDjgX>?wbGIBSTb_hmCYQAWkgBNSWmUbHUR;II;UU?eV=Ib`uNdEwN=u[@QixU>]LJmPnqqqMys]rOhpSMvC<OIQjr`WDhswtXCHLEPW<YK`Lt^mJhTujHSV`UbHvwpn:lYaXuZelohyCEXJULQyOIPXEHsgivPul?=voloWpq]xubxrkpYA]JiiP[uVr@ylXtkanvpO<xTGhsolWXAPpiU@=vdHXxppNXm@XpaArj\\ovXwKdXfppOXw`dvHMyqqVPXu;hwsMU:QocisHInQAmgqrPuRfTXNUj?hp?uRqUOupqM]PUuJ=tsptjHaxTQXt]Xcqv`QoyXtFalyMvZiJ;qsptwHDnNQuDpOUhnXqswxPMEyqPUgyPFuoXUx?HMJEsqqnftnPAus<pNuW_ATiAPKqsUQxmxLEDPUlULEpEPnImXh`yKeqw<wPQqF<nAIqADkaqxVawMDyjtVxmmjaq`\\J@QTfeWf\\rcxpCAtYevo<VH<`\\PhIi[`I[IYgl__boikYo`pgOopT>w?>``yeZwvGa]XWnv_i`xbS_nJQZDypno^GImkvirfy^pqnYfpfiUiyX?w^VqlGphijlYplXiOOrlHv;HnH^dUYqbwsHnjknlOhoYayPaiHiqUH^IoekyqXIbJGuj?f[fx;glKXaOYmkOkGYtE_cdN]rxg`olhWdq_k]Wrk>xLfxpV`@@_ixibFaq`nJaq[`q`QrWyacwv<flUx_R?eRhe=`twwsx@mn?oI?u;Gfbpft>yLogKYZZXd:PhuffnaclyfHF\\tytoNZ>AquVi;ig\\YgGi]LHyOoqxxpia[\\NmNNo\\HkJXZRyfn@iRAa?DOafwOyvcIaGepWxT[sJwYfGS?KWYwyGsuOUvECHNOUT]vaEBTwtUCEjcUYcsRKbTwDccc]_rpeBJ]BdEX<aVRSd>YWdDtcmNMIMZirsHUTemmlJ@ExTYWRMMYYVTLTSxJXmYe]pwdOBIw`un_]qoQMi=SPYqGdtNeNIdet^]rWoTHeYNieIar`lKxq]Ijoa`Ehptnmhgi=HtK_`VWdmVvppiMH\\]ypQ`uOhZeY`C?ulqpHwyCose?oNXlHyqdYnsX_uVsgPZIXwOsqIVSWrCmEdoRisCXgy_iXjIfkCFGauGUy[]Hhwblof;AYWwsgYucUroIfc_T`OwGeqYpOlHvNLPBAtkasQ`T=hTNqo]lSQHx\\lUPptcxQiIjHIjiMTfEXpynOaqLes\\TLWpQqIuPMXjpUFEyTMLRpuvMON\\QZEnpyN?xhJIfPv\\q_uuh^kqjT@iWngyyZUHjFymUXsCXZgpacncmQmHx\\<fpEIg\\ygO_]qf]moxInghvvyGeQXwiqdkIw`Wk@wxkXkNhxZqdhvu[Grogrtoa^i```opVgNxnoYfEPi_N`VP`=vwN_oEov;w[ONZMh`y>y_gv_@qR^fRg[LakiWrlfsUq[l`ZmqskqjsxwMY\\P@]Uqj]YuHOa^HuKXfEocVymnOx@HjPv\\xPuHYtpyh]QujaZspbpH`uWic>uFPuf?duxp_OsGHg=YZxPg]HrKW[gy`NppEvq[?e`?^lQgQV^TAhP^xfGyZh^h@ZxHcFHvk`dmVsaqv=HdVQgkHfOa]JHb@Oeta`oikTVn?^d?W[cfcTxauXt?YlkN\\m>pdG[jHngnrMv^igwLgePaygXl=f`gonyxgNatUhjjysF@voVxZxdmwpmi];^w<I]]WfR^q\\_jvpvTikcyaqfxT?g[oeOwxZymHH]wPfuvalavoVjA^lyyxEVmRWasGhGybONx[que>jQ>ctYo\\vtrWf=ycaFwDaD=y=GwJcWRQEd]B>?dXoSmiBbiwEAxDSV[gX`GFmQePegCSC>gRYCI<kCQGbi=EsWHtGgHOd=SEXCTncxSwUtwCcSIjEx??tw_yugXK]edeBQ[C`UrXysgUYtOtVIU_SxmeUI[y>yuDMWG[RG=d]UvoED\\SWqKbOwyy=by?VFUgGKt=eG:oIGOCsoc:kf[_CyGVdmWJSEh]hj;S@qtM;vMqWA;xWKxTGCUAIQeSj[ciwuY?tWKeg;EraBRkrsEfjMI[Wv]gFiudTmDYGGqucTQB<ArnwgwqsmUgTGgWSwhyCH?H?SdjiBF?XJKC;[uteCeAgT=in[wdqCjWEoiBBKCmsXOGXcOEqetfMeBet\\WYmgtIwV?_if;F[aRoiS[ag@OU^sE:aFrKGmIv?gWNSDOqev[DtICAOeR=WnsSDOhQaCfQCZEgTIe\\OioEC;_S\\[sQkCqCdH?d>iCoCed_C;Qghgr;odFwbdqI[OhDUDJKuIUYlwSm=uvEERqfv[WqAyPOrWgR`?r<[cy=G^sGHgX[ar@WBNYrncwYsYmaBLcIuQGqCV:ACN[DOUVdIvdOcbOiCKimaTLGTwSC<eBwKW`AGH=gPIEcggUcyblvoxRL_iD`t:qnIftUHhZyoXq]jAeknZdGZdorDnlUNeKHa=n^wnsHNi^N`]Fr@GqeAof`nZfebpe=WkThZJiqxFonXdlnmZpuoahcAsjNdqHvpVxdyi?ww=Qhq?jAWggv]yvd`>ujOnp^gXoeMAf@WwOapwIyXhiD`Z?_byW]ApppiwwX[;njTOanQ_`xvovsyVtHwd=gsGYn?V\\sV`xx\\rA`WIrC^tOx]WNg\\ArUH\\tNybpZ^fw:HjRn^=I\\bHiGq`Rn[<wr=gt^_n:gfxop;nlIOvfOa:@^[I^EyoyqsYPiGacdNo[gw??bGHsKIt;aoMaeoikJ>];GoAPxinmPx_vOgfiswFeihydi\\Wxx@gqAw^ZPdQOuOXk[>nRY\\oWiWNZp>ddHndge@HjRNZDodqX\\JNp;njuGtZp^eP^vYcK>`mQsK`yjgxHAxVykFAuY^mb_yd@uUy`LfrZNmUIbJGyb?jcO`e`n_yuN^q`owtIjAFbN@tu>np@rJFl`NatwZNA_Uil\\fn@wu:PmR_hD_ocIpXHZo>yypm\\hoV^xNg[oxuAV\\WPgtYwrY\\<QljQ]KVbiftrGgoibhG_cp^_glmG[dpjBhqjn\\=NkB@x\\Asy?^o^br^rf_\\oh[eqaDy\\ppepo]u?_\\nqmF]<>cGFdnvd@hf[f`^_ZfQ\\sOsaXoQa[y?xonf=abfFy[v]JP]\\nqQVhWGoQGnJHpFivVNd=hlqpdRpgP^_;NgnOdGflo_u?GoTHljV[Jg`>?uo?aC`tZh[ewZsVmC>oTN\\fWsS^dGFq\\q_Y>k;fb@wZyqiLFn@Wu:PkhFxXo^KNgnhmcFpqakenxJ_b^omkAiU>`Z@[R_q@_rqxykfv]A_ZpebqkNAlkNw<xo>@oKQ_Z@uHhfbP]\\nb:?\\ZPgJNecpbHQ^o^f>paRvkNAlkycsxln@oKaue_igwq@gtJF\\^I^knfnnn@iiEWechxJO]@^t\\i`WOkkXx`NvU>]cFqvpb=Inf@Z`g```i>_csapDOiSAp@oeM^mmF_QFo[NbKGyJOiFAgN>dKxp@OcJH]f`qoQdGaZpO`ZPgB@fOvpKg`>_kAHtR>d>_nPnsrQ\\cH^\\qpIq^CPujWcMFxaqoeP_A^dGF[YIrr^wvHkBX_hIy`Nd=_j?_xcGs``iGyvoVnDXk^VsFWxdHcYYxan\\D^fVfoH^ljf`GqZN``xXk:hlqfs:Pk_gi^fcGW`dHhxvsZPgBIuXIicWk`HmTNoPQo:Fo[N^mhiyhqBg[svnSyvw?[u?_L^qdNdYAlrwvWabIflvPqUG^Zf`>_jSwfEgmTNpQa_fphxw[mHf\\YpswbiIf[@bH>_PqxtNd=_n@h`cHb`?plqtHYgtFeZv_^Og\\pyjIbX?s;f^gIpwpvNW`=I^DFg[No`fopg`lPbHqjenpfNy?Fg[fZNOc;xtU?sDF`Z`]EAt=XmJgsbAb^ajfP]yVlW@sCFdonyuhwKP\\lPgRWye@cCFDAHRcbF?V_igWOW<UUGeR<=cddxxxK`Ir^etvUK=HlILpd]svLxJxKnHVWMK?\\JLUZ;OgJL=yKbpr@hMV`ogeLgUjIPJ>dTdLjD<lNxNdHmrmrl@sPQy=@WR`p_=Tqxk\\HlNPyVxSShx?APFMQnYVfljj\\SI^[<gkAoeMWZjPrOo]mwd@>nr`l:Pic?f=qv:Fcaw]DArW>vjNamW^dOp=PqxpvXxhINpK@laHoCNpmYevO[R_n]H^Xfot_fepy;VcP^yx_ZqY\\o@sDgiXWZZYnXFoc>hfvq=vxVQhKhltAjdIZeQdNy[\\n_pOpd@a@`nvnce?Z`FeXygypcwAgkG_MIi_NllajT^ZPxlBg[LWronyIvg[qgaiaYayT@Z:>Z:Fc?oc>oo<?f<3< Fig. 3. Linear segmentSimilarly as for quadratic curve for linear segments we need three control points ! The segment joins points Q0 and Q1. A point P1 is chosen as arbitrary one. P1 should be such that all three points were not to be colinear. In Fig. 3 a linear segment with all its characteristic points are presented. From the above considerations it is easily seen that for quadratic polynomials and for linear segments we have the same approach. Namely, we use Chaikin's geometric method.

For every contour that can be splitted into a finite number of linear and quadratic segments we can give a collection of IFS's that describe fractally the whole contour. In this worksheet similarly as in [5] and [6] we demonstrate a number of examples of fractal modeling of contours using both probabilistic and deterministic methods. The data for modeled contours have been obtained with the help of the program [3].

We start with probabilistic modeling. It works faster in comparison to deterministic modeling.. For details consult http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657).

restart; with(plots):with(linalg):DEFINITION OF HOMOGENOUS POINT IN R2P:=(x,y)->[x,y,1];PROCEDURE FOR CREATING TWO FUNCTIONS f1,f2 FROM GIVEN COEFFICENTS: a,b,c,d,e,fMakeMapFromCoeffs:=proc(a,b,c,d,e,f) local x, y; unapply([a*x+b*y+e,c*x+d*y+f],x,y); end: GENERATOR OF UNIFORM RANDOMLY DISTRIBUTED NUMBERS 1 or 2 d:=rand(1..2): CHAIKIN'S MATRICES FOR GENERATING QUADRATIC BEZIER CURVELC:=[[3/4,1/4,0],[1/4,3/4,0],[0,3/4,1/4]]:MC:=[[1/4,3/4,0],[0,3/4,1/4],[0,1/4,3/4]]:PROBABILISTIC PROCEDURE FOR GENERATING OF QUADRATIC BEZIER CURVE WITH THE USAGE OF CHAIKIN'S ALGORITHM (three non-colinear points Q0,P1,Q2 and n - the number of iterations should be given).Starting point is chosen as Q0 i.e. it belongs to the attractor! BezChaProb:=proc(Q0,P1,Q2,n) local L, M, PP, LC, MC, P0, P2, LP, MP, i, l, z, dd, x, y, f1, f2: LC:=[[3/4,1/4,0],[1/4,3/4,0],[0,3/4,1/4]];MC:=[[1/4,3/4,0],[0,3/4,1/4],[0,1/4,3/4]]; P0:=2*Q0-P1; P2:=2*Q2-P1; PP:=[P0,P1,P2];LP:=evalm(inverse(PP)&*LC&*PP);MP:=evalm(inverse(PP)&*MC&*PP);f1:=MakeMapFromCoeffs(LP[1,1],LP[2,1],LP[1,2],LP[2,2],LP[3,1],LP[3,2]); f2:=MakeMapFromCoeffs(MP[1,1],MP[2,1],MP[1,2],MP[2,2],MP[3,1],MP[3,2]);x[0]:=Q0[1]: y[0]:=Q0[2]: z[0]:=[x[0],y[0]]: for i from 0 to n do dd:=d(): if dd=1 then z[i+1]:=evalf(f1(z[i][1],z[i][2])): else z[i+1]:=evalf(f2(z[i][1],z[i][2])): fi: od: i:='i': l:={seq(z[i],i=0..n)}: pointplot(l,symbol=CROSS,axes=none,color=black): end: AN EXAMPLE - BEZIER CURVE FRACTALLY GENERATED IN A PROBABILISTIC WAYBezChaProb(P(-1,3),P(4,1),P(1,-3),1000); CHAIKIN'S MATRICES FOR GENERATING OF A LINEAR SEGMENT LCS:=[[1/2,1/2,0],[1/2,2/2,0],[0,1/2,1/2]]:MCS:=[[1/2,1/2,0],[0,1/2,1/2],[0,1/2,1/2]]:PROBABILISTIC PROCEDURE FOR GENERATING OF THE LINEAR SEGMENT JOINING POINTS Q0 and Q2 USING CHAIKIN'S ALGORITHM. AN ADDITIONAL POINT P1 IS NEEDED (three points Q0,Q2 and P1 should be non-colinear and the number of iterations - n should be given).Starting point is chosen as Q0 i.e. it belongs to the attractor! SeChaProb:=proc(Q0,P1,Q2,n) local L, M, PP, LCS, MCS, P0, P2, LP, MP, i, l, z, dd, x, y, f1, f2: LCS:=[[1/2,1/2,0],[1/2,1/2,0],[0,1/2,1/2]];MCS:=[[1/2,1/2,0],[0,1/2,1/2],[0,1/2,1/2]];P0:=2*Q0-P1; P2:=2*Q2-P1; PP:=[P0,P1,P2];LP:=evalm(inverse(PP)&*LCS&*PP);MP:=evalm(inverse(PP)&*MCS&*PP);f1:=MakeMapFromCoeffs(LP[1,1],LP[2,1],LP[1,2],LP[2,2],LP[3,1],LP[3,2]); f2:=MakeMapFromCoeffs(MP[1,1],MP[2,1],MP[1,2],MP[2,2],MP[3,1],MP[3,2]);x[0]:=Q0[1]; y[0]:=Q0[2]: z[0]:=[x[0],y[0]]: for i from 0 to n do dd:=d(): if dd=1 then z[i+1]:=evalf(f1(z[i][1],z[i][2])): else z[i+1]:=evalf(f2(z[i][1],z[i][2])): fi: od: i:='i': l:={seq(z[i],i=0..n)}: pointplot(l,symbol=CROSS,axes=none,color=black): end: AN EXAMPLE - A SEGMENT JOINING POINTS [-1,3] and [1,-3] FRACTALLY GENERATED IN A PROBABILISTIC WAY. THE POINT [4,1] HAS BEEN ADDED. SeChaProb(P(-1,3),P(4,1),P(1,-3),500); "BOX ELDER" - ANIMATIONdisplay(seq( display( BezChaProb(P(75,156), P(100,121), P(158,126),n), SeChaProb(P(158,126),P(100,100), P(170,15),n), BezChaProb(P(170,15), P(177,12), P(176,27),n), BezChaProb(P(176,27), P(167,80), P(166,128),n), BezChaProb(P(166,128), P(212,115), P(250,150),n), SeChaProb(P(250,150),P(100,100), P(252,158),n), BezChaProb(P(252,158), P(268,174), P(275,196),n), BezChaProb(P(275,196), P(252,194), P(242,199),n), SeChaProb(P(233,197),P(100,100), P(242,199),n), SeChaProb(P(223,199),P(100,100), P(233,197),n), BezChaProb(P(223,199), P(208,199), P(169,148),n), BezChaProb(P(169,148), P(171,164), P(201,188),n), SeChaProb(P(201,188),P(100,100), P(216,237),n), SeChaProb(P(205,273),P(100,100), P(216,237),n), SeChaProb(P(198,275),P(100,100), P(205,273),n), SeChaProb(P(170,314),P(100,100), P(198,275),n), BezChaProb(P(170,314), P(160,298), P(138,288),n), BezChaProb(P(138,288), P(112,255), P(129,202),n), BezChaProb(P(129,202), P(149,184), P(157,156),n), BezChaProb(P(157,156), P(145,197), P(98,210),n), SeChaProb(P(88,204),P(100,100), P(98,210),n), BezChaProb(P(88,204), P(75,211), P(55,209),n), BezChaProb(P(55,209), P(79,161), P(75,156),n)), n=[0,3,10,20,30,50,100,200,300,500]),scaling=constrained,insequence=true); "DUCK" - ANIMATIONplottools[rotate](display( seq(display( SeChaProb(P(160,315),P(100,100), P(186,312),n), SeChaProb(P(186,312),P(100,100), P(205,289),n), SeChaProb(P(205,289),P(100,100), P(240,274),n), SeChaProb(P(240,274),P(100,100), P(176,267),n), BezChaProb(P(176,267), P(253,197), P(233,149),n), SeChaProb(P(233,149),P(100,100), P(192,111),n), SeChaProb(P(192,111),P(100,100), P(211,92),n), SeChaProb(P(211,92),P(100,100), P(237,95),n), SeChaProb(P(237,95),P(100,100), P(237,79),n), BezChaProb(P(237,79), P(221,82), P(208,63),n), SeChaProb(P(208,63),P(100,100), P(189,108),n), BezChaProb(P(189,108), P(179,114), P(150,85),n), SeChaProb(P(150,85), P(100,100), P(157,50),n), SeChaProb(P(157,50),P(100,100), P(202,53),n), SeChaProb(P(202,53), P(100,100), P(192,31),n), BezChaProb(P(192,31), P(170,34), P(160,15),n), SeChaProb(P(160,15),P(100,100), P(144,44),n), SeChaProb(P(144,44), P(100,100), P(141,88),n), SeChaProb(P(141,88),P(100,100), P(93,114),n), BezChaProb(P(93,114), P(80,136), P(125,191),n), SeChaProb(P(125,191),P(100,100), P(154,210),n), BezChaProb(P(154,210), P(119,277), P(160,315),n)), n=[0,3,5,10,50,100,200,300,500]),scaling=constrained,insequence=true),-Pi/8); "BUTTERFLY"n := 200:display( BezChaProb(P(154,15), P(129,60), P(90,88),n), SeChaProb(P(90,88), P(100,100), P(94,113),n), SeChaProb(P(94,113), P(100,100), P(72,107),n), BezChaProb(P(72,107), P(24,118), P(27,172),n), SeChaProb(P(27,172), P(100,100), P(40,198),n), SeChaProb(P(40,198), P(100,100), P(110,207),n), BezChaProb(P(110,207), P(114,210), P(79,228),n), BezChaProb(P(79,228), P(65,237), P(114,217),n), SeChaProb(P(114,217), P(100,100), P(87,278),n), SeChaProb(P(87,278), P(100,100), P(104,302),n), BezChaProb(P(104,302), P(122,317), P(154,315),n), BezChaProb(P(154,315), P(184,310), P(187,273),n), SeChaProb(P(187,273), P(100,100), P(210,288),n), BezChaProb(P(210,288), P(255,268), P(304,270),n), BezChaProb(P(304,270), P(270,202), P(159,193),n), SeChaProb(P(159,193), P(100,100), P(169,178),n), SeChaProb(P(169,178), P(100,100), P(150,180),n), BezChaProb(P(150,180), P(189,98), P(169,25),n), SeChaProb(P(169,25), P(100,100), P(154,15),n), scaling=constrained); "PUPPY"n:=200:puppy:=display( SeChaProb(P(155,15), P(100,100), P(181,79),n), SeChaProb(P(181,79), P(100,100), P(228,85),n), SeChaProb(P(228,85), P(100,100), P(248,82),n), SeChaProb(P(248,82), P(100,100), P(275,99),n), SeChaProb(P(275,99), P(100,100), P(304,108),n), SeChaProb(P(304,108),P(100,100), P(295,129),n), SeChaProb(P(295,129), P(100,100), P(254,146),n), SeChaProb(P(254,146),P(100,100), P(228,178),n), SeChaProb(P(228,178), P(100,100), P(254,216),n), SeChaProb(P(254,216), P(100,100), P(257,254),n), SeChaProb(P(257,254), P(100,100), P(243,271),n), SeChaProb(P(243,271), P(100,100), P(222,271),n), BezChaProb(P(222,271), P(213,254), P(231,245),n), BezChaProb(P(231,245), P(202,210), P(199,219),n), BezChaProb(P(199,219), P(190,187), P(176,312),n), SeChaProb(P(176,312), P(100,100), P(155,315),n), BezChaProb(P(155,315), P(141,300), P(161,283),n), SeChaProb(P(161,283), P(100,100), P(144,233),n), SeChaProb(P(144,233),P(100,100), P(120,228),n), SeChaProb(P(120,228), P(100,100), P(94,248),n), BezChaProb(P(94,248), P(56,257), P(47,242),n), BezChaProb(P(47,242), P(53,210), P(27,198),n), SeChaProb(P(27,198), P(100,100), P(30,152),n), BezChaProb(P(30,152), P(42,126), P(68,108),n), SeChaProb(P(68,108), P(100,100), P(79,137),n), SeChaProb(P(79,137),P(100,100), P(126,97),n), SeChaProb(P(126,97), P(100,100), P(126,172),n), BezChaProb(P(126,172), P(146,178), P(149,108),n), SeChaProb(P(149,108), P(100,100), P(164,91),n), BezChaProb(P(164,91), P(138,24), P(155,15),n), scaling=constrained):plottools[rotate](puppy,(11/8)*Pi); "DOG"n:=200:dog:=display( BezChaProb(P(87,30), P(88,50), P(75,79),n), SeChaProb(P(75,79), P(100,100), P(67,85),n), BezChaProb(P(67,85), P(64,124), P(96,136),n), SeChaProb(P(96,136), P(100,100), P(117,135),n), BezChaProb(P(117,135), P(117,167), P(139,186),n), BezChaProb(P(139,186), P(136,251), P(166,282),n), SeChaProb(P(166,282), P(100,100), P(136,314),n), BezChaProb(P(136,314), P(148,320), P(179,291),n), BezChaProb(P(179,291), P(206,292), P(233,277),n), BezChaProb(P(233,277), P(242,265), P(238,253),n), SeChaProb(P(238,253), P(100,100), P(233,247),n), SeChaProb(P(233,247), P(100,100), P(253,236),n), BezChaProb(P(253,236), P(265,216), P(254,204),n), BezChaProb(P(254,204), P(267,195), P(263,181),n), SeChaProb(P(263,181), P(100,100), P(252,168),n), SeChaProb(P(252,168), P(100,100), P(243,166),n), SeChaProb(P(243,166), P(100,100), P(237,176),n), BezChaProb(P(237,176), P(211,172), P(196,140),n), SeChaProb(P(196,140), P(100,100), P(171,128),n), BezChaProb(P(171,128), P(161,113), P(161,88),n), BezChaProb(P(161,88), P(173,68), P(163,42),n), SeChaProb(P(163,42), P(100,100), P(156,38),n), BezChaProb(P(156,38), P(131,51), P(126,38),n), BezChaProb(P(126,38), P(121,43), P(114,33),n), BezChaProb(P(114,33), P(109,31), P(130,27),n), SeChaProb(P(130,27), P(100,100), P(136,20),n), BezChaProb(P(136,20), P(145,6), P(87,30),n),scaling=constrained): plottools[rotate](dog,(5/4)*Pi);"COCK"n:=200:cock:=display( BezChaProb(P(28,274), P(43,233), P(51,254),n), BezChaProb(P(51,254), P(45,225), P(78,225),n), SeChaProb(P(78,225), P(100,100), P(101,236),n), BezChaProb(P(101,236), P(139,210), P(133,175),n), BezChaProb(P(133,175), P(45,198), P(31,102),n), BezChaProb(P(31,102), P(40,24), P(115,15),n), SeChaProb(P(115,15), P(100,100),P(78,32),n), SeChaProb(P(78,32), P(100,100),P(48,79),n), SeChaProb(P(48,79), P(100,100),P(69,56),n), SeChaProb(P(69,56), P(100,100),P(104,67),n), BezChaProb(P(104,67), P(124,30), P(177,32),n), BezChaProb(P(177,32), P(142,38), P(124,65),n), SeChaProb(P(124,65), P(100,100),P(142,56),n), SeChaProb(P(142,56), P(100,100),P(153,70),n), BezChaProb(P(153,70), P(179,65), P(174,85),n), SeChaProb(P(174,85), P(100,100),P(197,79),n), BezChaProb(P(197,79), P(153,102), P(179,102),n), BezChaProb(P(179,102), P(174,117), P(194,102),n), BezChaProb(P(194,102), P(185,123), P(211,126),n), BezChaProb(P(211,126), P(197,123), P(197,140),n), BezChaProb(P(197,140), P(232,149), P(220,161),n), SeChaProb(P(220,161), P(100,100),P(264,166),n), SeChaProb(P(264,166), P(100,100),P(287,155),n), BezChaProb(P(287,155), P(273,178), P(302,181),n), SeChaProb(P(302,181), P(100,100), P(290,219),n), BezChaProb(P(290,219), P(278,201), P(258,193),n), BezChaProb(P(258,193), P(177,324), P(115,295),n), SeChaProb(P(115,295), P(100,100),P(115,315),n), SeChaProb(P(115,315), P(100,100),P(72,312),n), BezChaProb(P(72,312), P(25,292), P(28,274),n), scaling=constrained): plottools[rotate](cock,(3/2)*Pi); "MICKEY" - ANIMATIONplottools[rotate](display( seq(display( SeChaProb(P(137,16), P(100,100),P(131,35),n), SeChaProb(P(131,35), P(100,100),P(89,43),n), SeChaProb(P(89,43), P(100,100),P(84,51),n), BezChaProb(P(84,51), P(101,74), P(131,72),n), SeChaProb(P(131,72), P(100,100),P(135,90),n), SeChaProb(P(135,90), P(100,100),P(126,96),n), BezChaProb(P(126,96), P(115,125), P(142,154),n), BezChaProb(P(142,154), P(97,170), P(112,217),n), SeChaProb(P(112,217), P(100,100),P(105,224),n), SeChaProb(P(105,224), P(100,100),P(110,238),n), SeChaProb(P(110,238), P(100,100),P(101,252),n), BezChaProb(P(101,252), P(85,238), P(65,241),n), SeChaProb(P(65,241), P(100,100),P(57,248),n), BezChaProb(P(57,248), P(51,281), P(81,291),n), BezChaProb(P(81,291), P(112,306), P(130,282),n), SeChaProb(P(130,282), P(100,100),P(133,271),n), SeChaProb(P(133,271), P(100,100),P(130,253),n), SeChaProb(P(130,253), P(100,100),P(126,241),n), SeChaProb(P(126,241), P(100,100),P(141,254),n), BezChaProb(P(141,254), P(146,269), P(141,279),n), SeChaProb(P(141,279), P(100,100),P(141,296),n), SeChaProb(P(141,296), P(100,100),P(147,310),n), BezChaProb(P(147,310), P(170,324), P(201,302),n), BezChaProb(P(201,302), P(225,283), P(209,252),n), SeChaProb(P(209,252), P(100,100),P(199,246),n), SeChaProb(P(199,246), P(100,100),P(163,267),n), BezChaProb(P(163,267), P(155,266), P(155,252),n), SeChaProb(P(155,252), P(100,100),P(166,242),n), SeChaProb(P(166,242), P(100,100),P(164,228),n), SeChaProb(P(164,228), P(100,100),P(174,215),n), SeChaProb(P(174,215), P(100,100),P(192,207),n), SeChaProb(P(192,207), P(100,100),P(196,191),n), SeChaProb(P(196,191), P(100,100),P(188,164),n), BezChaProb(P(188,164), P(216,155), P(225,135),n), SeChaProb(P(225,135), P(100,100),P(237,147),n), SeChaProb(P(237,147), P(100,100),P(249,151),n), BezChaProb(P(249,151), P(271,146), P(274,123),n), SeChaProb(P(274,123), P(100,100),P(263,104),n), SeChaProb(P(263,104), P(100,100),P(252,101),n), SeChaProb(P(252,101), P(100,100),P(230,111),n), SeChaProb(P(230,111), P(100,100),P(223,88),n), SeChaProb(P(223,88), P(100,100),P(244,81),n), BezChaProb(P(244,81), P(254,68), P(250,57),n), BezChaProb(P(250,57), P(242,41), P(224,41),n), SeChaProb(P(224,41), P(100,100), P(211,48),n), BezChaProb(P(211,48), P(203,60), P(204,72),n), BezChaProb(P(204,72), P(176,57), P(152,84),n), SeChaProb(P(152,84), P(100,100),P(152,49),n), SeChaProb(P(152,49), P(100,100),P(146,39),n), SeChaProb(P(146,39), P(100,100),P(147,15),n), SeChaProb(P(147,15), P(100,100),P(137,16),n)), n=[0,3,5,10,20,50,100,200,300]), scaling=constrained,insequence=true),(9/10)*Pi); "MOUSE"n:=200:mouse:=display( BezChaProb(P(153,15), P(119,81), P(135,93),n), BezChaProb(P(135,93), P(99,115), P(115,119),n), BezChaProb(P(115,119), P(111,113), P(139,95),n), SeChaProb(P(139,95),P(100,100), P(145,105),n), BezChaProb(P(145,105), P(125,123), P(137,155),n), BezChaProb(P(137,155), P(127,159), P(127,185),n), SeChaProb(P(127,185), P(100,100), P(111,191),n), SeChaProb(P(111,191), P(100,100), P(97,201),n), SeChaProb(P(97,201), P(100,100), P(93,221),n), BezChaProb(P(93,221), P(67,225), P(73,233),n), BezChaProb(P(73,233), P(61,237), P(87,257),n), SeChaProb(P(87,257), P(100,100),P(85,265),n), BezChaProb(P(85,265), P(65,259), P(51,275),n), SeChaProb(P(51,275), P(100,100),P(59,287),n), SeChaProb(P(59,287), P(100,100),P(73,293),n), SeChaProb(P(73,293), P(100,100),P(91,287),n), SeChaProb(P(91,287), P(100,100),P(93,271),n), BezChaProb(P(93,271), P(99,249), P(113,249),n), SeChaProb(P(113,249), P(100,100), P(125,277),n), SeChaProb(P(125,277), P(100,100),P(115,299),n), BezChaProb(P(115,299), P(129,313), P(153,315),n), SeChaProb(P(153,315), P(100,100),P(151,299),n), SeChaProb(P(151,299), P(100,100),P(133,291),n), SeChaProb(P(133,291), P(100,100),P(137,263),n), SeChaProb(P(137,263), P(100,100),P(153,259),n), SeChaProb(P(153,259), P(100,100),P(163,225),n), SeChaProb(P(163,225), P(100,100), P(145,193),n), SeChaProb(P(145,193), P(100,100),P(151,169),n), BezChaProb(P(151,169), P(205,151), P(199,165),n), SeChaProb(P(199,165), P(100,100),P(205,171),n), BezChaProb(P(205,171), P(173,195), P(185,195),n), BezChaProb(P(185,195), P(211,179), P(203,173),n), SeChaProb(P(203,173), P(100,100),P(225,171),n), SeChaProb(P(225,171), P(100,100),P(279,139),n), BezChaProb(P(279,139), P(249,103), P(205,117),n), SeChaProb(P(205,117), P(100,100),P(187,93),n), BezChaProb(P(187,93), P(179,25), P(153,15),n),scaling=constrained): plottools[rotate](mouse,(9/10)*Pi);

Deterministic modeling is very time consuming! Especially for contours with large number of linear segments. To obtain good results of modeling we need about 10 iterations. Convergence is slower and is different than in http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651.restart;with(plots):with(linalg):DEFINITION OF HOMOGENEOUS POINT IN R2P:=(x,y)->[x,y,1];POINT TRANSFORMATIONTransPoint := proc(t, p) [t[1]*p[1]+t[2]*p[2]+t[5], t[3]*p[1]+t[4]*p[2]+t[6]] end:POLYGON TRANSFORMATIONTransPolygon := proc(t,polygon) local i; [seq(TransPoint(t,polygon[i]), i=1 ..nops(polygon))] end:ITERATED FUNCTION SYSTEM (n - the number of iterations, list of transformations and initial polygon should be given)IFS := proc(n, ListTrans, polygon) local i, j, k, s, seqpoly: seqpoly := polygon; for j to n do s := NULL; for i to nops(ListTrans) do s := s, seq(TransPolygon(ListTrans[i], op(k, [seqpoly])), k=1 .. nops([seqpoly])) od; seqpoly := s od; polygonplot([seqpoly],axes=none, color=green, scaling=constrained, thickness=3) end: CHAIKIN'S MATRICES FOR GENERATING QUADRATIC BEZIER CURVELC:=[[3/4,1/4,0],[1/4,3/4,0],[0,3/4,1/4]]:MC:=[[1/4,3/4,0],[0,3/4,1/4],[0,1/4,3/4]]:PROCEDURE FOR GENERATING QUADRATIC BEZIER CURVE USING CHAIKIN'S ALGORITHM (three non-colinear points Q0,P1,Q2, starting polygon and n - the number of iterations should be given)GenChaBez:=proc(Q0,P1,Q2,polygon,n) local LC, MC, PP, LP, MP,P0, P2, chaikin: LC:=[[3/4,1/4,0],[1/4,3/4,0],[0,3/4,1/4]];MC:=[[1/4,3/4,0],[0,3/4,1/4],[0,1/4,3/4]]; P0:=2*Q0-P1; P2:=2*Q2-P1; PP:=[P0,P1,P2];LP:=evalm(inverse(PP)&*LC&*PP);MP:=evalm(inverse(PP)&*MC&*PP);chaikin:=[[LP[1,1],LP[2,1],LP[1,2],LP[2,2],LP[3,1],LP[3,2]],[MP[1,1],MP[2,1],MP[1,2],MP[2,2],MP[3,1],MP[3,2]]]; IFS(n,chaikin,polygon); end:EXAMPLE - QUADRATIC BEZIER CURVE poly:=[[0,0],[1,0],[1,1],[0,1]]:for n from 0 by 3 to 9 do GenChaBez(P(-1,0),P(0,1),P(1,0),poly,n); print(Iteration=n); od;CHAIKIN'S MATRICES FOR GENERATING LINEAR SEGMENTSLCS:=[[1/2,1/2,0],[1/2,1/2,0],[0,1/2,1/2]]: MCS:=[[1/2,1/2,0],[0,1/2,1/2],[0,1/2,1/2]]: PROCEDURE FOR GENERATING OF A LINEAR SEGMENT (JOINING Q0 and Q2 POINTS) USING CHAIKIN'S ALGORITHM.ADDITIONAL POINT P1 IS NEEDED.(three non-colinear points Q0,P1,Q2, starting polygon and n - the number of iterations should be given)GenChaSe:=proc(Q0,P1,Q2,polygon,n) local LCS, MCS, PP, LP, MP, P0 ,P2, chaikin: LCS:=[[1/2,1/2,0],[1/2,1/2,0],[0,1/2,1/2]];MCS:=[[1/2,1/2,0],[0,1/2,1/2],[0,1/2,1/2]]; P0:=2*Q0-P1; P2:=2*Q2-P1; PP:=[P0,P1,P2];LP:=evalm(inverse(PP)&*LCS&*PP);MP:=evalm(inverse(PP)&*MCS&*PP);chaikin:=[[LP[1,1],LP[2,1],LP[1,2],LP[2,2],LP[3,1],LP[3,2]],[MP[1,1],MP[2,1],MP[1,2],MP[2,2],MP[3,1],MP[3,2]]]; IFS(n,chaikin,polygon): end:EXAMPLE - A SEGMENT for n from 0 by 3 to 9 do GenChaSe(P(1,0),P(1,2),P(2,1),[[0,0],[1,0],[1,2]],n); print(Iteration=n); od;"DUCK" - ANIMATIONpoly:=[[0,0],[200,0],[200,200]]:display(seq( display( GenChaSe(P(160,315),P(100,100), P(186,312), poly,n), GenChaSe(P(186,312),P(100,100), P(205,289),poly, n), GenChaSe(P(205,289),P(100,100), P(240,274),poly,n), GenChaSe(P(240,274),P(100,100), P(176,267),poly,n), GenChaBez(P(176,267), P(253,197), P(233,149),poly,n), GenChaSe(P(233,149),P(100,100), P(192,111),poly,n), GenChaSe(P(192,111),P(100,100), P(211,92),poly,n), GenChaSe(P(211,92),P(100,100), P(237,95),poly,n), GenChaSe(P(237,95),P(100,100), P(237,79),poly,n), GenChaBez(P(237,79), P(221,82), P(208,63),poly,n), GenChaSe(P(208,63),P(100,100), P(189,108),poly,n), GenChaBez(P(189,108), P(179,114), P(150,85),poly,n), GenChaSe(P(150,85), P(100,100), P(157,50),poly,n), GenChaSe(P(157,50),P(100,100), P(202,53),poly,n), GenChaSe(P(202,53), P(100,100), P(192,31),poly,n), GenChaBez(P(192,31), P(170,34), P(160,15),poly,n), GenChaSe(P(160,15),P(100,100), P(144,44),poly,n), GenChaSe(P(144,44), P(100,100), P(141,88),poly,n), GenChaSe(P(141,88),P(100,100), P(93,114),poly,n), GenChaBez(P(93,114), P(80,136), P(125,191),poly,n), GenChaSe(P(125,191),P(100,100), P(154,210),poly,n), GenChaBez(P(154,210), P(119,277), P(160,315),poly,n)),n=0..8), insequence=true,scaling=constrained); "BUTTERFLY"poly:=[[0,0],[200,0],[200,200]]:n:=8:display( GenChaBez(P(154,15), P(129,60), P(90,88), poly,n), GenChaSe(P(90,88), P(100,100), P(94,113), poly,n), GenChaSe(P(94,113), P(100,100), P(72,107), poly,n), GenChaBez(P(72,107), P(24,118), P(27,172), poly,n), GenChaSe(P(27,172), P(100,100), P(40,198), poly,n), GenChaSe(P(40,198), P(100,100), P(110,207), poly,n), GenChaBez(P(110,207), P(114,210), P(79,228), poly,n), GenChaBez(P(79,228), P(65,237), P(114,217), poly,n), GenChaSe(P(114,217), P(100,100), P(87,278), poly,n), GenChaSe(P(87,278), P(100,100), P(104,302), poly,n), GenChaBez(P(104,302), P(122,317), P(154,315), poly,n), GenChaBez(P(154,315), P(184,310), P(187,273), poly,n), GenChaSe(P(187,273), P(100,100), P(210,288), poly,n), GenChaBez(P(210,288), P(255,268), P(304,270), poly,n), GenChaBez(P(304,270), P(270,202), P(159,193), poly,n), GenChaSe(P(159,193), P(100,100), P(169,178), poly,n), GenChaSe(P(169,178), P(100,100), P(150,180), poly,n), GenChaBez(P(150,180), P(189,98), P(169,25), poly,n), GenChaSe(P(169,25), P(100,100), P(154,15), poly,n), scaling=constrained);

Below are given some examples in which to obtain good results we need to perform 10 or even more iterations. It can be observed that process of shortening of linear segments of a given contour is rather slow. Convergence of the method can be increased by individual choise of starting polygon to every segment.poly:=[[0,0],[200,0],[200,200]];n:=9; "PUPPY"puppy:=display( GenChaSe(P(155,15), P(100,100), P(181,79), poly,n), GenChaSe(P(181,79), P(100,100), P(228,85), poly,n), GenChaSe(P(228,85), P(100,100), P(248,82), poly,n), GenChaSe(P(248,82), P(100,100), P(275,99), poly,n), GenChaSe(P(275,99), P(100,100), P(304,108), poly,n), GenChaSe(P(304,108),P(100,100), P(295,129), poly,n), GenChaSe(P(295,129), P(100,100), P(254,146), poly,n), GenChaSe(P(254,146),P(100,100), P(228,178), poly,n), GenChaSe(P(228,178), P(100,100), P(254,216), poly,n), GenChaSe(P(254,216), P(100,100), P(257,254), poly,n), GenChaSe(P(257,254), P(100,100), P(243,271), poly,n), GenChaSe(P(243,271), P(100,100), P(222,271), poly,n), GenChaBez(P(222,271), P(213,254), P(231,245), poly,n), GenChaBez(P(231,245), P(202,210), P(199,219), poly,n), GenChaBez(P(199,219), P(190,187), P(176,312), poly,n), GenChaSe(P(176,312), P(100,100), P(155,315), poly,n), GenChaBez(P(155,315), P(141,300), P(161,283), poly,n), GenChaSe(P(161,283), P(100,100), P(144,233), poly,n), GenChaSe(P(144,233),P(100,100), P(120,228), poly,n), GenChaSe(P(120,228), P(100,100), P(94,248), poly,n), GenChaBez(P(94,248), P(56,257), P(47,242), poly,n), GenChaBez(P(47,242), P(53,210), P(27,198), poly,n), GenChaSe(P(27,198), P(100,100), P(30,152), poly,n), GenChaBez(P(30,152), P(42,126), P(68,108), poly,n), GenChaSe(P(68,108), P(100,100), P(79,137), poly,n), GenChaSe(P(79,137),P(100,100), P(126,97), poly,n), GenChaSe(P(126,97), P(100,100), P(126,172), poly,n), GenChaBez(P(126,172), P(146,178), P(149,108), poly,n), GenChaSe(P(149,108), P(100,100), P(164,91), poly,n), GenChaBez(P(164,91), P(138,24), P(155,15), poly,n), scaling=constrained):plottools[rotate](puppy,(11/8)*Pi); "DOG"poly:=[[0,0],[200,0],[200,200],[0,200]];n:=9;dog:=display( GenChaBez(P(87,30), P(88,50), P(75,79), poly,n), GenChaSe(P(75,79), P(100,100), P(67,85), poly,n), GenChaBez(P(67,85), P(64,124), P(96,136), poly,n), GenChaSe(P(96,136), P(100,100), P(117,135), poly,n), GenChaBez(P(117,135), P(117,167), P(139,186), poly,n), GenChaBez(P(139,186), P(136,251), P(166,282), poly,n), GenChaSe(P(166,282), P(100,100), P(136,314), poly,n), GenChaBez(P(136,314), P(148,320), P(179,291), poly,n), GenChaBez(P(179,291), P(206,292), P(233,277), poly,n), GenChaBez(P(233,277), P(242,265), P(238,253), poly,n), GenChaSe(P(238,253), P(100,100), P(233,247), poly,n), GenChaSe(P(233,247), P(100,100), P(253,236), poly,n), GenChaBez(P(253,236), P(265,216), P(254,204), poly,n), GenChaBez(P(254,204), P(267,195), P(263,181), poly,n), GenChaSe(P(263,181), P(100,100), P(252,168), poly,n), GenChaSe(P(252,168), P(100,100), P(243,166), poly,n), GenChaSe(P(243,166), P(100,100), P(237,176), poly,n), GenChaBez(P(237,176), P(211,172), P(196,140), poly,n), GenChaSe(P(196,140), P(100,100), P(171,128), poly,n), GenChaBez(P(171,128), P(161,113), P(161,88), poly,n), GenChaBez(P(161,88), P(173,68), P(163,42), poly,n), GenChaSe(P(163,42), P(100,100), P(156,38), poly,n), GenChaBez(P(156,38), P(131,51), P(126,38), poly,n), GenChaBez(P(126,38), P(121,43), P(114,33), poly,n), GenChaBez(P(114,33), P(109,31), P(130,27), poly,n), GenChaSe(P(130,27), P(100,100), P(136,20), poly,n), GenChaBez(P(136,20), P(145,6), P(87,30), poly,n),scaling=constrained): plottools[rotate](dog,(5/4)*Pi);poly:=[[0,0],[200,0],[200,200],[0,200]];n:=8; "COCK"cock:=display( GenChaBez(P(28,274), P(43,233), P(51,254), poly,n), GenChaBez(P(51,254), P(45,225), P(78,225),poly,n), GenChaSe(P(78,225), P(100,100), P(101,236),poly,n), GenChaBez(P(101,236), P(139,210), P(133,175),poly,n), GenChaBez(P(133,175), P(45,198), P(31,102),poly,n), GenChaBez(P(31,102), P(40,24), P(115,15),poly,n), GenChaSe(P(115,15), P(100,100),P(78,32),poly,n), GenChaSe(P(78,32), P(100,100),P(48,79),poly,n), GenChaSe(P(48,79), P(100,100),P(69,56),poly,n), GenChaSe(P(69,56), P(100,100),P(104,67),poly,n), GenChaBez(P(104,67), P(124,30), P(177,32),poly,n), GenChaBez(P(177,32), P(142,38), P(124,65),poly,n), GenChaSe(P(124,65), P(100,100),P(142,56),poly,n), GenChaSe(P(142,56), P(100,100),P(153,70),poly,n), GenChaBez(P(153,70), P(179,65), P(174,85),poly,n), GenChaSe(P(174,85), P(100,100),P(197,79),poly,n), GenChaBez(P(197,79), P(153,102), P(179,102),poly,n), GenChaBez(P(179,102), P(174,117), P(194,102),poly,n), GenChaBez(P(194,102), P(185,123), P(211,126),poly,n), GenChaBez(P(211,126), P(197,123), P(197,140),poly,n), GenChaBez(P(197,140), P(232,149), P(220,161),poly,n), GenChaSe(P(220,161), P(100,100),P(264,166),poly,n), GenChaSe(P(264,166), P(100,100),P(287,155),poly,n), GenChaBez(P(287,155), P(273,178), P(302,181),poly,n), GenChaSe(P(302,181), P(100,100), P(290,219),poly,n), GenChaBez(P(290,219), P(278,201), P(258,193),poly,n), GenChaBez(P(258,193), P(177,324), P(115,295),poly,n), GenChaSe(P(115,295), P(100,100),P(115,315),poly,n), GenChaSe(P(115,315), P(100,100),P(72,312),poly,n), GenChaBez(P(72,312), P(25,292), P(28,274),poly,n), scaling=constrained): plottools[rotate](cock,(3/2)*Pi); "MICKEY"poly:=[[0,0],[200,0],[200,200],[0,200]];n:=8;miki:=display( GenChaSe(P(137,16), P(100,100),P(131,35),poly,n), GenChaSe(P(131,35), P(100,100),P(89,43),poly,n), GenChaSe(P(89,43), P(100,100),P(84,51),poly,n), GenChaBez(P(84,51), P(101,74), P(131,72),poly,n), GenChaSe(P(131,72), P(100,100),P(135,90),poly,n), GenChaSe(P(135,90), P(100,100),P(126,96),poly,n), GenChaBez(P(126,96), P(115,125), P(142,154),poly,n), GenChaBez(P(142,154), P(97,170), P(112,217),poly,n), GenChaSe(P(112,217), P(100,100),P(105,224),poly,n), GenChaSe(P(105,224), P(100,100),P(110,238),poly,n), GenChaSe(P(110,238), P(100,100),P(101,252),poly,n), GenChaBez(P(101,252), P(85,238), P(65,241),poly,n), GenChaSe(P(65,241), P(100,100),P(57,248),poly,n), GenChaBez(P(57,248), P(51,281), P(81,291),poly,n), GenChaBez(P(81,291), P(112,306), P(130,282),poly,n), GenChaSe(P(130,282), P(100,100),P(133,271),poly,n), GenChaSe(P(133,271), P(100,100),P(130,253),poly,n), GenChaSe(P(130,253), P(100,100),P(126,241),poly,n), GenChaSe(P(126,241), P(100,100),P(141,254),poly,n), GenChaBez(P(141,254), P(146,269), P(141,279),poly,n), GenChaSe(P(141,279), P(100,100),P(141,296),poly,n), GenChaSe(P(141,296), P(100,100),P(147,310),poly,n), GenChaBez(P(147,310), P(170,324), P(201,302),poly,n), GenChaBez(P(201,302), P(225,283), P(209,252),poly,n), GenChaSe(P(209,252), P(100,100),P(199,246),poly,n), GenChaSe(P(199,246), P(100,100),P(163,267),poly,n), GenChaBez(P(163,267), P(155,266), P(155,252),poly,n), GenChaSe(P(155,252), P(100,100),P(166,242),poly,n), GenChaSe(P(166,242), P(100,100),P(164,228),poly,n), GenChaSe(P(164,228), P(100,100),P(174,215),poly,n), GenChaSe(P(174,215), P(100,100),P(192,207),poly,n), GenChaSe(P(192,207), P(100,100),P(196,191),poly,n), GenChaSe(P(196,191), P(100,100),P(188,164),poly,n), GenChaBez(P(188,164), P(216,155), P(225,135),poly,n), GenChaSe(P(225,135), P(100,100),P(237,147),poly,n), GenChaSe(P(237,147), P(100,100),P(249,151),poly,n), GenChaBez(P(249,151), P(271,146), P(274,123),poly,n), GenChaSe(P(274,123), P(100,100),P(263,104),poly,n), GenChaSe(P(263,104), P(100,100),P(252,101),poly,n), GenChaSe(P(252,101), P(100,100),P(230,111),poly,n), GenChaSe(P(230,111), P(100,100),P(223,88),poly,n), GenChaSe(P(223,88), P(100,100),P(244,81),poly,n), GenChaBez(P(244,81), P(254,68), P(250,57),poly,n), GenChaBez(P(250,57), P(242,41), P(224,41),poly,n), GenChaSe(P(224,41), P(100,100), P(211,48),poly,n), GenChaBez(P(211,48), P(203,60), P(204,72),poly,n), GenChaBez(P(204,72), P(176,57), P(152,84),poly,n), GenChaSe(P(152,84), P(100,100),P(152,49),poly,n), GenChaSe(P(152,49), P(100,100),P(146,39),poly,n), GenChaSe(P(146,39), P(100,100),P(147,15),poly,n), GenChaSe(P(147,15), P(100,100),P(137,16),poly,n), scaling=constrained): plottools[rotate](miki,(9/10)*Pi); "MOUSE"poly:=[[0,0],[200,0],[200,200],[0,200]];n:=8;mouse:=display( GenChaBez(P(153,15), P(119,81), P(135,93),poly,n), GenChaBez(P(135,93), P(99,115), P(115,119),poly,n), GenChaBez(P(115,119), P(111,113), P(139,95),poly,n), GenChaSe(P(139,95),P(100,100), P(145,105),poly,n), GenChaBez(P(145,105), P(125,123), P(137,155),poly,n), GenChaBez(P(137,155), P(127,159), P(127,185),poly,n), GenChaSe(P(127,185), P(100,100), P(111,191),poly,n), GenChaSe(P(111,191), P(100,100), P(97,201),poly,n), GenChaSe(P(97,201), P(100,100), P(93,221),poly,n), GenChaBez(P(93,221), P(67,225), P(73,233),poly,n), GenChaBez(P(73,233), P(61,237), P(87,257),poly,n), GenChaSe(P(87,257), P(100,100),P(85,265),poly,n), GenChaBez(P(85,265), P(65,259), P(51,275),poly,n), GenChaSe(P(51,275), P(100,100),P(59,287),poly,n), GenChaSe(P(59,287), P(100,100),P(73,293),poly,n), GenChaSe(P(73,293), P(100,100),P(91,287),poly,n), GenChaSe(P(91,287), P(100,100),P(93,271),poly,n), GenChaBez(P(93,271), P(99,249), P(113,249),poly,n), GenChaSe(P(113,249), P(100,100), P(125,277),poly,n), GenChaSe(P(125,277), P(100,100),P(115,299),poly,n), GenChaBez(P(115,299), P(129,313), P(153,315),poly,n), GenChaSe(P(153,315), P(100,100),P(151,299),poly,n), GenChaSe(P(151,299), P(100,100),P(133,291),poly,n), GenChaSe(P(133,291), P(100,100),P(137,263),poly,n), GenChaSe(P(137,263), P(100,100),P(153,259),poly,n), GenChaSe(P(153,259), P(100,100),P(163,225),poly,n), GenChaSe(P(163,225), P(100,100), P(145,193),poly,n), GenChaSe(P(145,193), P(100,100),P(151,169),poly,n), GenChaBez(P(151,169), P(205,151), P(199,165),poly,n), GenChaSe(P(199,165), P(100,100),P(205,171),poly,n), GenChaBez(P(205,171), P(173,195), P(185,195),poly,n), GenChaBez(P(185,195), P(211,179), P(203,173),poly,n), GenChaSe(P(203,173), P(100,100),P(225,171),poly,n), GenChaSe(P(225,171), P(100,100),P(279,139),poly,n), GenChaBez(P(279,139), P(249,103), P(205,117),poly,n), GenChaSe(P(205,117), P(100,100),P(187,93),poly,n), GenChaBez(P(187,93), P(179,25), P(153,15),poly,n),scaling=constrained): plottools[rotate](mouse,(9/10)*Pi);

In this worksheet we demonstrated that it is possible to determine another (in comparison to [5] and [6] ) collection of IFS's that characterize a given contour fractally. The approach used here avoids analytic representation of curves, is purely geometric and it ensures to treat in the same way both quadratic and linear segments. Experiments from the worksheet and from [5], [6] showed that contours are generated fractally in the progressive way and iterations converge with different speed to the attractor i.e. to the modeled contour. Additionaly, we observed that probabilistic approach gives results much quicker in comparison to deterministic method.

[1] Chaikin G., An algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (1974), 346-349. [2] Goldman R.; The fractal nature of B\351zier curves, Proceedings of the Geometric Modeling and Processing 2004, April 13-15, Beijing, China, 2004, 3-11. [3] Goluch P.; Characterization of contour 2D with the help the highest curvature points, BSc Diploma, Institute of Computer Science, Silesian University, Sosnowiec, Poland (2003), (in Polish). [4] Joy K., On-Line Computer Graphics Notes, http://graphics.cs.ucdavis.edu/GraphicsNotes/Graphics-Notes.html. [5] Kotarski W., Lisowska, A. On fractal modeling of contours, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651. [6] Kotarski W., Lisowska, A. probabilistic approach to fractal modeling of shapes, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657. [7] Riesenfeld R., On Chaikin's algorithm, IEEE Computer Graphics and Applications 4, 3 (1975), 304-310. The undersigned (the "Author") has developed an Application (the "Application") using Maple and wishes to submit the Application to Maplesoft, a division of Waterloo Maple Inc. for publication by Maplesoft or its agents. 1. The Author grants to Maplesoft a non-exclusive, perpetual License and right to use all or any part of the Application, to modify or adapt the Application for the promotion of Maplesoft. 2. The Author represents that the Application is the original work of the Author, the Author has the right to grant a License to Maplesoft with respect to the Application, no other party has any right, title or interest in or to the Application, and that the use of the Application by Maplesoft will not infringe any intellectual property rights of any third party. 3. The Author agrees to execute such documents and take such other action as Maplesoft may request in order to give effect to the intention of this License Agreement. This License Agreement shall be binding upon and inure to the benefit of the Author, and all heirs, executors, administrators, successors and permitted assigns of the Author. Wieslaw Kotarski & Agnieszka Lisowska11th of March 2005