{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 261 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 262 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 273 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 274 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 279 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 40 "Bezier curves and Bernstein poly nomials " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C. , Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 26.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "load interpolation and \+ function approximation procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The \+ Maple m-file " }{TEXT 279 10 "fcnapprx.m" }{TEXT -1 37 " contains the \+ code for the procedure " }{TEXT 0 6 "bezier" }{TEXT -1 25 " used in th is worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a \+ Maple session by a command similar to the one that follows, where the \+ file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Introduction " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 725 " The development of free-form curves and surfaces for computer graphics began in the French auto industry in the late 60's with the work of \+ P. de Casteljau, an engineer with the Citroen corporation, and P. Bezi er at Renault. Bezier created the UNISURF CAD system for designing car s which utilized his curve theories. P. de Casteljau\222s research was earlier than Bezier's, but was never published, so Bezier get most of the credit. These men were pioneers in Computer Aided Geometric Desig n (CAGD) for the auto industry, which replaced the use of hand drawn F rench-curve templates in the design of auto bodies. Bezier curves are \+ based on Berstein Polynomials which had been developed by the mathemat ician Berstein much earlier." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 4 "From" }{TEXT -1 2 ": " }{URLLINK 17 "Digital Sc ulpture" 4 " http://www.sculpture.org/documents/webspec/magazine/wseng lis.htm" "" }{TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT -1 971 " \"Duri ng the early sixties at the Renault corporation, where I was an engine er at the time, I went to see my supervisor to tell him that I had fou nd a new mathematical method for drawing curves that would replace all previous rough calculations and other lathed shapes and models. He sa w my project, looked at me, and said, 'Monsieur Bezier, if your thing \+ worked, the Americans would already be using it.\"\205Over eighty-five years old, Pierre Bezier, with acutely mocking eyes, is still laughin g at his beginnings in an atmosphere of typical French reticence that \+ unfortunately hasn't left us. Like Monsieur Jourdain with his prose, a ll of us in \"computer arts\" use his theory without knowing it: the f amous Bezier Curves that have permitted the emergence of computer draw ing. Dear Pierre Bezier, since then the Americans are also using your \+ work, you are renown worldwide, and amongst other things we owe you fo r the birth of what you named, \"computer assisted sculpture\"\n" }} {PARA 0 "" 0 "" {TEXT 262 4 "From" }{TEXT -1 3 ": " }{URLLINK 17 "Pla net Photoshop" 4 "http://www.planetphotoshop.com/PeteBauer16.html" "" }{TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT -1 1716 "PETER BAUER ON PHO TOSHOP\n\nWorking with Vectors, Part 1\nBy Peter Bauer\n\nLearning how to manipulate paths is the key to using Photoshop 6's new Shape tools . Photoshop users who haven't upgraded, read along! We'll be working a lot with the Pen tool in this first installment.\n\nThe line between \+ vector and raster/bitmap programs is blurring more and more. First Ill ustrator 9 got transparency, then Photoshop 6 got vector type and the \+ Shape tools. To get the most out of both vector shapes and the Pen too l, you should really understand paths and how they work. In this week' s column, we'll look at the theory of paths as we know them, a bit abo ut their history, and how to create paths with the Pen tool.\n\nThere \+ are two terms that are directly tied to paths. The term vector is used to define art crated with paths. (Illustrator, for example, is a vect or art program.) The term relates directly to paths in Photoshop throu gh the direction lines. A vector is a quantity that has both magnitude (the length of the direction line) and direction (its angle from the \+ anchor point). The second term is Bezier curve. Pierre Etienne Bezier \+ (the first \223e\224 should carry the French accent aigu), a long-time employee of the auto maker Renault, developed the system which we use today. Bezier (1910-1999) saw these curves as a way to transfer auto \+ body design information to the factory. While it took years for him to convince Renault to accept the system, it has found its way into many facets of life. Bezier curves are even at the heart of the desktop pu blishing revolution, being integral to Adobe's PostScript page descrip tion language. Vector art is based upon and created with Bezier curves , but you can call them \223paths.\224\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Quadratic \+ Bezier curves " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 129 "Let A, B and C be 3 points in the coordi nate plane, which are not in the same straight line, as shown in the f ollowing animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 411 323 323 {PLOTDATA 2 "6%-%(ANIMATEG6<7 *-%%TEXTG6$7$$!\"#!\"\"$\"\"!F/Q\"A6\"-F(6$7$$\"\"#F/$\"$:$F,Q\"BF1-F( 6$7$$\"$D'F,F.Q\"CF1-%'CURVESG6%7%7$F.F.7$F5$\"\"$F/7$$\"\"'F/F.-%'COL OURG6&%$RGBG$\")#)eqk!\")$\"))eqk\"FQFR-%*THICKNESSG6#F6-FA6'7%FDFEFD- FL6&FNF/F/F/-%&STYLEG6#%&POINTGFT-%'SYMBOLG6#%'CIRCLEG-FA6'7$FDFE-FL6& FN$\"*++++\"FQF.F.-Fgn6#%%LINEGFTFjn-%+AXESLABELSG6%Q!F1F[p-%%FONTG6#% (DEFAULTG-%%VIEWG6$F_pF_p7.F'F2F:-F(6$7$$!+++++5!#5$\"++++W=FipQ\"PF1- F(6$7$$\"++++SB!\"*$\"+++!G)GFbqQ\"QF1-F(6$7$$\"++++_DFip$\"++++/8FipQ \"RF1F@-FA6'7%7$$\"3=+++++++!)!#>$\"3%**************>\"!#=7$$\"39+++++ +g@!#<$\"3))************zGF[s7$$\"37++++++K;Fgr$\"3%************RI#Fgr FZFfnFTFjn-FA6'7$FarFhrFaoFeoFTFjn-FA6%7SFD7$$\"3Y;>kP]1*[$!#?$\"3?_U2 ()evE_F\\t7$$\"3cq^#ovbt_'F\\t$\"3+#\\C&zl5n(*F\\t7$$\"3I\\:v6;)p%**F \\t$\"3)\\`uk_&\\'[\"Fdr7$$\"3Idx)[a@#R8Fdr$\"3oGnLNux)*>Fdr7$$\"3maMu )Q+Co\"Fdr$\"3mO'p()zXx]#Fdr7$$\"31!H[D;K3+#Fdr$\"3II^Nd9%)yHFdr7$$\"3 K]#Gc^93L#Fdr$\"3E\\_6Q'ReY$Fdr7$$\"3QuvB:gOsEFdr$\"3GwsG>SkoRFdr7$$\" 30$*)H:\")3J,$Fdr$\"3u@.TIf(*oWFdr7$$\"3AnTIhK!RO$Fdr$\"3],v3'*Ht#)\\F dr7$$\"3SlSAES8tOFdr$\"3N/yKJ!)\\MaFdr7$$\"3*o(**3Gd`@SFdr$\"3yr+tb^BU fFdr7$$\"3e-w9pvmrVFdr$\"3g$>dDRe6X'Fdr7$$\"3YAZN/&o$4ZFdr$\"33Ne$p9[2 %pFdr7$$\"3waf2G[F;]Fdr$\"3QP@x!)zh%Q(Fdr7$$\"3mg/*yL::Q&Fdr$\"3\"*>'G j_C:\"zFdr7$$\"3?Xn,ti<\"p&Fdr$\"3Ql(\\4a?rN)Fdr7$$\"37r_Z[!z;0'Fdr$\" 3c(=u&fTdu))Fdr7$$\"37)zkCW[5P'Fdr$\"3G/cgK#==L*Fdr7$$\"3v.`B1=t@nFdr$ \"3')*3%H')QjK)*Fdr7$$\"3;TGC6?%f0(Fdr$\"3/[r7o)p3.\"Fgr7$$\"3i*=Ez1V \\S(Fdr$\"3)Q9AY!fX!3\"Fgr7$$\"3?SNV)Fdr$\"3qi5N[ahD7Fgr7$$ \"3))R()3K\"4bu)Fdr$\"3%zPt=%)>(p7Fgr7$$\"3uoB2oCZ%3*Fdr$\"3*)*GylNusJ \"Fgr7$$\"3f3sTZN%\\V*Fdr$\"3aP[x(=;jO\"Fgr7$$\"33330H^4y(*Fdr$\"3,eZG ])3UT\"Fgr7$$\"38g*e&Hz.65Fgr$\"3w>J#QDn/Y\"Fgr7$$\"3)orT&GK'z/\"Fgr$ \"3S\\[P,pt6:Fgr7$$\"3)*H)\\N!*p63\"Fgr$\"3g50N0?sd:Fgr7$$\"36K!H$4Sl; 6Fgr$\"3#Q!H,UUt1;Fgr7$$\"3k7#>DlL)[6Fgr$\"3[iB%\\Ip5l\"Fgr7$$\"3UET(p `US=\"Fgr$\"3u?w2N^X*p\"Fgr7$$\"3%QB!=$4)><7Fgr$\"3[)Hf4@-\\u\"Fgr7$$ \"3+#fkk\"H)=D\"Fgr$\"3yBi5fVK#z\"Fgr7$$\"3'G:rr0@eG\"Fgr$\"3:Ul[=ggQ= Fgr7$$\"3K%)eKd,Q@8Fgr$\"33YB_!frp)=Fgr7$$\"35GWQ_elb8Fgr$\"3>;n%)=#pM $>Fgr7$$\"3!*)\\k9Fgr7$$\"3Ni'o-WabU\"Fgr$\"3s7 SpX6#z-#Fgr7$$\"3s*Q8gxtvX\"Fgr$\"3OJ)fRCA52#Fgr7$$\"37cC#4&4I%\\\"Fgr $\"3cKEBeFL?@Fgr7$$\"3.U\"3(znz%zfD#FgrF^s-FL6&FNF. FcoF.FTFhoF`p7.F'F2F:-F(6$7$$!*++++#Fip$\"++++))HFipF\\q-F(6$7$$\"++++ +DFbq$\"+++glFFbqFeq-F(6$7$$\"++++oTFip$\"++++;MFipF]rF@-FA6'7%7$$\"3- +++++++;Fgr$\"3!**************R#Fgr7$$\"3%)************>BF[s$\"3!)**** ********fFF[s7$$\"3&)***********zK$Fgr$\"3#************fT%FgrFZFfnFTFj n-FA6'7$F`dlFedlFaoFeoFTFjn-FA6%7SFD7$$\"3kIVB<3<\")pF\\t$\"3p+(H[&*QW /\"Fdr7$$\"3K,M1wX`18Fdr$\"35'z4=/J-&>Fdr7$$\"37(Gn8vj=*>Fdr$\"3iR\")* ew)elHFdr7$$\"3e&p\"Ho#Fdr$\"3u9pMh$[T)RFdr7$$\"3A^rI[t%=P$Fdr$\"3 _X&y])=N%*\\Fdr7$$\"3zE)fo*Gi6SFdr$\"3H@0U*>2y#fFdr7$$\"3s+I^-A8vYFdr$ \"3)o*4Yi(p6*oFdr7$$\"3Wlph(enCO&Fdr$\"3#R5\\rO\"3%)yFdr7$$\"3)=d>hIX( [gFdr$\"3W>\")o.())Fdr7$$\"3W***\\&)pgev'Fdr$\"3I0+NkMI\"))*Fdr7$$ \"3!)GHc6;pztFdr$\"3u@6`QD(o2\"Fgr7$$\"3-1*fB(=5$3)Fdr$\"3oG?H1zVw6Fgr 7$$\"3c4/fw')f!z)Fdr$\"3axG-i9\"Fgr$\"3/1*zBfruk\"Fgr7$$\"3<3,RhbN>7Fgr$\"3 Cv'Hodbyu\"Fgr7$$\"3K_#>Vx)>%G\"Fgr$\"3XUA/*Q'RO=Fgr7$$\"3q@T\\keXb8Fg r$\"3oNw^n#)>L>Fgr7$$\"3$e8(\\],UB9Fgr$\"3!>f3:\\W]-#Fgr7$$\"3Q3Q]g)[W \\\"Fgr$\"3Jv&)[o5`?@Fgr7$$\"3[\\FSH*y(f:Fgr$\"3\\_`V1z.Fgr$\"3i\\$*4$e)RnEFgr7$$\"3%GK?wwa*y>Fgr$\"3/K!R^Z 5%eFFgr7$$\"3kSeB)e/q/#Fgr$\"3myCH?$[h%GFgr7$$\"3]no;UboA@Fgr$\"3>)R* \\Zl>VHFgr7$$\"3O_E`^))z!>#Fgr$\"3sT?Sxo1IIFgr7$$\"3SGhJ<*ROE#Fgr$\"3m :;0)ysC7$Fgr7$$\"30Po^W>Q$Fgr7$$\"38o$e)RewTDFg r$\"37&*[U(p[1Z$Fgr7$$\"3rVz,Azp6EFgr$\"3moh%RhWqb$Fgr7$$\"3QQNIzj-&o# Fgr$\"3=&Q*3()\\9ZOFgr7$$\"3MyV??EwbFFgr$\"3!['oQ\"p&eLPFgr7$$\"3\\hY_ CX@GGFgr$\"3!p,EC:U;#QFgr7$$\"3^$)zS?-=+HFgr$\"3x^gxr#G'3RFgr7$$\"3kfN 0s!4k'HFgr$\"3QC$Ry#*f#))RFgr7$$\"3OdJ-@CVUIFgr$\"3#)H0$*eRYUFgr7$ $\"3?(GzI\")oHD$Fgr$\"3)*Q@wF=@GVFgr7$$\"3S++++++GLFgrF]elFeblFTFhoF`p 7.F'F2F:-F(6$7$$\"*++++'Fip$\"++++KTFipF\\q-F(6$7$$\"++++gEFbq$\"+++S[ EFbqFeq-F(6$7$$\"++++[eFip$\"++++O`FipF]rF@-FA6'7%7$Fcdl$\"3')******** *****f$Fgr7$$\"3)*************zCF[s$\"38++++++SEF[s7$$\"3I++++++)3&Fgr $\"3_++++++OjFgrFZFfnFTFjn-FA6'7$F\\fmF_fmFaoFeoFTFjn-FA6%7SFD7$$\"3[C x(QtJw/\"Fdr$\"3eEoO)4!Hl:Fdr7$$\"3y_E9,qRh>Fdr$\"3-T?d^6c?HFdr7$$\"3] &Rw0x'\\\"*HFdr$\"3>83F=(zsV%Fdr7$$\"3<:)*)R9r5.%Fdr$\"3Ce0.yF6cfFdr7$ $\"3q*3\"py3Mo]Fdr$\"3oGn#*e#=)fuFdr7$$\"3$)4Y$H?sB.'Fdr$\"3Ksh>Es*o%) )Fdr7$$\"3e_UlgI&H.(Fdr$\"3HCPIP!*fF5Fgr7$$\"3Wr\"Qrr/.2)Fdr$\"3-[&eV? JY<\"Fgr7$$\"3ZO!pP[4p5*Fdr$\"3K*GpV'yT?8Fgr7$$\"3!)\\P6\"Fgr$\"3!Q-&>1#o-g\"Fgr7$$\"3))y4GV)p%=7Fgr $\"35kq:sJkYf;^#H9F gr$\"3&4DU-g-H.#Fgr7$$\"3Hf$oKo@N_\"Fgr$\"3?A\\p*on%f@Fgr7$$\"3D995Kx) ej\"Fgr$\"3Eddplz:4BFgr7$$\"3`q]r(R08t\"Fgr$\"3#*)ya[hy_V#Fgr7$$\"3YVx ipHcU=Fgr$\"3Kpnh#[%>\"e#Fgr7$$\"3r<$=-z\"GT>Fgr$\"3^X]M(pV'4FFgr7$$\" 39tVq:%Fgr7$$\"3T^a(3%p;CK Fgr$\"3EXOPQ*yVH%Fgr7$$\"3dn%[R%o))GLFgr$\"3D%fahhMqT%Fgr7$$\"3=)GhRsd 4W$Fgr$\"3QMh6Qc@ZXFgr7$$\"3c7HnADuUNFgr$\"3Kh7[>K[kYFgr7$$\"3;Qrw/DFa OFgr$\"3I&e)pB2$>z%Fgr7$$\"3o/@i%=a%fPFgr$\"3O'oLw\"p76\\Fgr7$$\"39G8= q(['pQFgr$\"3d9g&\\,t\\.&Fgr7$$\"3#RPSXfIw(RFgr$\"3+z)yjy:`:&Fgr7$$\"3 _gH$fmQ44%Fgr$\"3q;6q*=?0G&Fgr7$$\"3T_)fMI?.?%Fgr$\"3&eW?wT\\.S&Fgr7$$ \"3J)[!=f]V7VFgr$\"3-O&e9TB@_&Fgr7$$\"3bhzTSt(QU%Fgr$\"3[:hCy87UcFgr7$ $\"3\"y]?\"))e]EXFgr$\"3Yy%Q;08!H_3#Fdr7$$\"39s-#4&GK*p$Q&Fdr$\"3ydwQ& oqY\"zFdr7$$\"3Wr_*)z4)=x'Fdr$\"3.%=9.#\\9/**Fdr7$$\"3ASEx!3!3j!)Fdr$ \"3E3#oP:6O<\"Fgr7$$\"38.?0!4xUS*Fdr$\"3s)R%)p9I?O\"Fgr7$$\"3i>,QSxez5 Fgr$\"3KT'f[gLbb\"Fgr7$$\"3pGyWM,w=7Fgr$\"3;9llU0/Z+SFgr7$$\"3o/&efu2$*[\"Fgr$\"3t'[Ch!oj8@Fgr7$$\"3QiR%4a KEj\"Fgr$\"3'Q6oJJR[I#Fgr7$$\"3Sjhj]^-xFgr$\"3)[L(4rblxEFgr7$$\"3OF:#4S)yV?Fgr$\"3.=aB$y6?%GFgr7$$ \"3OOZi'Rsa>#Fgr$\"3a!zDh-6g.$Fgr7$$\"3V#zE3avVK#Fgr$\"3#Ri>:7L\"*>$Fg r7$$\"3)HVg&H,zuCFgr$\"3S,(=L8uuQ$Fgr7$$\"3*QN3EFgr$\"3gp*o\"[P#H b$Fgr7$$\"3_&[w>u\\`v#Fgr$\"3KV02'\\#3LPFgr7$$\"3EW&))RHnd*GFgr$\"3toV ./!3L!RFgr7$$\"3x+>o3(QF/$Fgr$\"3y+V&R8SUFgr7$$\"3*p!*QNV)eCLFgr$\"3H\"G$Q>$>?T%Fgr7$$\"3u**4Y**z@xMFg r$\"3+-qh(e'4*e%Fgr7$$\"3)\\)>MFx[5OFgr$\"3oWS(R9F?u%Fgr7$$\"31)y:pNY[ v$Fgr$\"3#zj_sJDf!\\Fgr7$$\"3CLv'Q+XW!RFgr$\"3m*R(R'*>'Q2&Fgr7$$\"3A\" H\"[-DD^SFgr$\"3,Fhb[?zO_Fgr7$$\"3ohL%H4CP>%Fgr$\"3)\\\"*p6*=:$R&Fgr7$ $\"3))pumCuS_VFgr$\"3s\"f(*R2%GlbFgr7$$\"3mxsz!)QV&\\%Fgr$\"3+p\"3;AD' =dFgr7$$\"359XEHug[YFgr$\"3Mjk?#zi4)eFgr7$$\"3q+uISx\"yy%Fgr$\"3z)zx!H y#o-'Fgr7$$\"3fCgeN*f/%\\Fgr$\"3mH>Cx6&\\='Fgr7$$\"3STP)G=7X3&Fgr$\"3? v([LT\\CL'Fgr7$$\"3'RZLuqJbB&Fgr$\"3e\"e*p6tH&['Fgr7$$\"3eT%QZ2>OQ&Fgr $\"3ptYyN&>Mj'Fgr7$$\"3]`T@(4(y'Fgr7$$\"3_Z3:-*G$*o& Fgr$\"3Kfua(RgP$pFgr7$$\"3!)y>VvxRVeFgr$\"35kSqD\"yB3(Fgr7$$\"31+')H+e k'*fFgr$\"3`0U5l/SGsFgr7$$\"3hNU@CU'y8'Fgr$\"3=%Hd`h\"QhtFgr7$$\"3:&Hf (=p=+jFgr$\"3B<@s(o\"Q7vFgr7$$\"3sp-LjfzXkFgr$\"33'>4!y3;YwFgr7$$\"3nw s0T$)[,mFgr$\"3Gq\"G))4puy(Fgr7$$\"3I\\rJ+Q$4v'Fgr$\"3^b&[I8N9#zFgrF^h nFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++AFip$\"++++?kFipF\\q-F(6$7$$\"++++ !)HFbq$\"++++9CFbqFeq-F(6$7$$\"+++++%*Fip$\"+++++')FipF]rF@-FA6'7%7$$ \"3A+++++++SFgr$\"3w**************fFgr7$$\"3#)*************z#F[s$FddlF [s7$$\"3-+++++++))Fgr$\"3k*************f*FgrFZFfnFTFjn-FA6'7$F]ioFbioF aoFeoFTFjn-FA6%7SFD7$$\"3qc9xgKdZ>lnC[)Fdr$\"3P\"4Cp=LFB\"Fgr7$$\"3 _\"RPIlu.,\"Fgr$\"3sDyQ8]af9Fgr7$$\"3]D12H/\"*y6Fgr$\"39B\")G0.T#p\"Fg r7$$\"3M.1wk&=RN\"Fgr$\"3[*==#Q`6J>Fgr7$$\"3BKZ#))3#3H:Fgr$\"3/.e-;\\! p;#Fgr7$$\"3H\\Pf1e+5$Q$yph#Fgr7$$\"3#3%\\A!G)z]?Fgr$\"3_x^KHj-^GFgr7$$\"390!pGK(4LAFgr$\" 3S%)HR\"e)y$3$Fgr7$$\"3x/o)3I'[4CFgr$\"3L&eRt#z'fI$Fgr7$$\"3G`l&o>o-d# Fgr$\"3.S.$>k;f]$Fgr7$$\"3#R[f!Q9?iFFgr$\"31[:#e&4VTPFgr7$$\"3A'>ve&)H a#HFgr$\"3t7WP7^.RRFgr7$$\"3fw\")=Tq.;JFgr$\"3mqa$*GXpmTFgr7$$\"3'zK& \\z]T&G$Fgr$\"3k8S^TlBmVFgr7$$\"3ee#)eN4'>Z$Fgr$\"3qB_tX=.$e%Fgr7$$\"3 ;+r5)[)G]OFgr$\"3W,(yJ*oR(y%Fgr7$$\"3l-)[J+uq$QFgr$\"3#Qf`b6s%)*\\Fgr7 $$\"3b$=n7a[#4SFgr$\"3s\\%)pyOO!>&Fgr7$$\"3a27d1ml&>%Fgr$\"3;wjGb4E&R& Fgr7$$\"3)zZuXWX+R%Fgr$\"3aklx)*=!eg&Fgr7$$\"3o]=A!p%*)fXFgr$\"3oXW$o$ )yry&Fgr7$$\"36&eZo^)*Ru%Fgr$\"3wWsXM\\3\")fFgr7$$\"3o?Ia[x\"\\$\\Fgr$ \"3[O4P9ICzhFgr7$$\"3$z,FE)\\SA^Fgr$\"37Z*=r*>(4P'Fgr7$$\"3>.S(*QpZ/`F gr$\"3]!*zd&RuWb'Fgr7$$\"3S?Ha$*pS2bFgr$\"3gP7Pa>\"fv'Fgr7$$\"3)o2z?'* R/p&Fgr$\"3jiFw$pQ[$pFgr7$$\"3'>!eAL!*e')eFgr$\"3)pfA.D9P7(Fgr7$$\"33! )pkzM(\\1'Fgr$\"3Af!fN(3$HH(Fgr7$$\"3%>;`Vm&pgiFgr$\"3-90%ptEeZ(Fgr7$$ \"3Iy\"RG[BbW'Fgr$\"3IhC)R+7fk(Fgr7$$\"3F-[h^YTRmFgr$\"3,&fbwe@;#yFgr7 $$\"3Y\\@hiLmHoFgr$\"3u_N;ieN\"*zFgr7$$\"3N7r9L9cHqFgr$\"3'fleI,wo;)Fg r7$$\"3!fOxiT)yAsFgr$\"36/z;J'=QL)Fgr7$$\"3)H8zKn-6U(Fgr$\"30+E;&H1C]) Fgr7$$\"3h%*)\\+g&[=wFgr$\"3C?.NKbYn')Fgr7$$\"3%>uM.3%[+yFgr$\"3*4x&** =cE<))Fgr7$$\"3VJZRY*4)4!)Fgr$\"3U2e\"e@[n)*)Fgr7$$\"3#*3-PR')p(>)Fgr$ \"35o$*)=/Sj8*Fgr7$$\"3%f*>l'**=()R)Fgr$\"3K=Sai1w$H*Fgr7$$\"3QWbCf)*y \"f)Fgr$\"3(Fgr7$$\"3'*************fHF[s$\"3!)************zAF[s7$$\"3%** *********>v5F[s$\"3/+++++S%4\"F[sFZFfnFTFjn-FA6'7$FjjpF][qFaoFeoFTFjn- FA6%7SFD7$$\"3_(*3^N,+)4#Fdr$\"3x1tY$>qB7$Fdr7$$\"3z61dC[OKRFdr$\"3uk \")GO)4C\"eFdr7$$\"3-#e0BS+_+'Fdr$\"3IbK3`)Qz!))Fdr7$$\"3Zf#ff\"=O-\") Fdr$\"3!HA7spc\"z6Fgr7$$\"32OkZ*35+-\"Fgr$\"3M\"pq0\"z$HZ\"Fgr7$$\"3aV QMWSw8I#Fgr7$$\"3H9w]6olT=Fgr$\"3#o:xW)4,!e#Fgr7$$ \"31**\\p?KUg?Fgr$\"3K/]\"f+'RjGFgr7$$\"3)\\j1Wv:SD#Fgr$\"3o%4!y-GH5JF gr7$$\"3!\\\"R7hq'HZ#Fgr$\"3Ac#G1A/_Q$Fgr7$$\"3Yn8$*[e*Qp#Fgr$\"3%y*e? 8\"4!eOFgr7$$\"3e**pnvT#y!HFgr$\"3:.!p*o'Ry\"RFgr7$$\"3xOM2r5'H5$Fgr$ \"3')*ozdE#=^TFgr7$$\"3$pl0k&[2OLFgr$\"3zHIyaxTDWFgr7$$\"3*=GgGMoW`$Fg r$\"3Aa\">ue%)\\l%Fgr7$$\"3Iu4^/PImPFgr$\"3kxqYpc&)=\\Fgr7$$\"3GsK(GNl C(RFgr$\"3O!=!Qx?e\\^Fgr7$$\"39$4ZCj\"o*>%Fgr$\"3MA(eco_'*R&Fgr7$$\"3$ RAu/VTqT%Fgr$\"3&*HtdP0zMcFgr7$$\"3DvU`W&p[k%Fgr$\"3#e<(R;V-xeFgr7$$\" 3#\\uCmt=]&[Fgr$\"3+od7`\\n'4'Fgr7$$\"3cRDY0Kp#3&Fgr$\"3a!Q7OJ(fIjFgr7 $$\"3x\\syn?G?`Fgr$\"35`#Qcnz-d'Fgr7$$\"3ml%>bm:!GbFgr$\"3g,;W#\\)=wnF gr7$$\"3#H_g]!pK`dFgr$\"3qK%=o'er&*pFgr7$$\"3[]>q!4Nr)fFgr$\"3'*[T*)R- R>sFgr7$$\"310He/K*o@'Fgr$\"3P%G^-sW^V(Fgr7$$\"3ojD79&e,W'Fgr$\"3\"zIK Er55k(Fgr7$$\"3'o!=]Zc;*o'Fgr$\"3'Ge%\\zDEmyFgr7$$\"3OsQz(30R\"pFgr$\" 3#zP=E.vc1)Fgr7$$\"3ca^%e`-\\:(Fgr$\"3sRXY7+Zv#)Fgr7$$\"3q];pS(4UP(Fgr $\"3sV]#HN#zi%)Fgr7$$\"35a&o5pz\\h(Fgr$\"3sTVz-ubk')Fgr7$$\"31<%)[%3)[ UyFgr$\"3IYZ`*o9:&))Fgr7$$\"3s9`s-wH\"3)Fgr$\"3,cS#G%e%R/*Fgr7$$\"3l$ \\h\"eMw:$)Fgr$\"33:b^lZ7H#*Fgr7$$\"3qV=t8>Fi&)Fgr$\"3DkW!Qjc)>%*Fgr7$ $\"3a2%Re%))p+))Fgr$\"3b$y\"[=T_+'*Fgr7$$\"3%>&>s_(\\b/*Fgr$\"3'\\9M)> z?#y*Fgr7$$\"3'o%=nRnR*G*Fgr$\"3ugW)*zlJf**Fgr7$$\"3DJ?[caO9&*Fgr$\"3L \"Rbi]O>,\"F[s7$$\"3#RT3K\\jKx*Fgr$\"3Xwuj=B'*H5F[s7$$\"3G5$\\*y#y0+\" F[s$\"3%*o?NyuzX5F[s7$$\"3;()y.3$pa-\"F[s$\"3]Qj=%>#Qi5F[s7$$\"3Oe8<' \\*Q\\5F[s$\"3!\\#fe\"*p'z2\"F[s7$$\"3;+++++?v5F[sFe[qFeblFTFhoF`p7.F' F2F:-F(6$7$$\"+++++QFip$\"++++3()FipF\\q-F(6$7$$\"+++++LFbq$\"+++gz@Fb qFeq-F(6$7$$\"+++!3K\"Fbq$\"+++g46FbqF]rF@-FA6'7%7$$\"3a+++++++cFgr$\" 3o*************R)Fgr7$$\"36++++++?JF[sFir7$$\"3%***********zw7F[s$\"3, +++++g47F[sFZFfnFTFjn-FA6'7$Fd\\rFi\\rFaoFeoFTFjn-FA6%7SFD7$$\"3O)QXW3 J([CFdr$\"3lLQmYCdROFdr7$$\"3yKLW[4[\"f%Fdr$\"3_-+n\\)y*pnFdr7$$\"3+)f #ePJq9qFdr$\"3'4ADd1-]-\"Fgr7$$\"3%>5+&zjRo%*Fdr$\"3;q*\\^`!*4P\"Fgr7$ $\"3AFH%32yC>\"Fgr$\"3.>7(HmG5r\"Fgr7$$\"3P?m[9b\\@9Fgr$\"3!y8S!=+XA?F gr7$$\"3AE%y0#o#*f;Fgr$\"3/AZwnx,TBFgr7$$\"3&f6k1)3!z!>Fgr$\"3w^w].oJm EFgr7$$\"3Evk\\-V[c@Fgr$\"3mu0,[(ej)HFgr7$$\"3\"\\T!R_gk8CFgr$\"3ib(G) )yr6J$Fgr7$$\"3q>z\\#H$QTEFgr$\"3'*Ri]*>!e$f$Fgr7$$\"3)[)3k$))Q\"**GFg r$\"3E[t2()HP2RFgr7$$\"3k]K#)G2UfJFgr$\"3k]-`$e]!=UFgr7$$\"39Q\"Q`DU;T $Fgr$\"3y(e&)fzqK^%Fgr7$$\"3%y,fDWFgr$\"3NAN\"\\bdRk&Fgr7$$\"3;5v27Z]pYFgr$\"3Dpuwb.'H!fFgr7$$\" 34))HbK=^Q\\Fgr$\"3GR5%e,XH='Fgr7$$\"37;**3@h-'>&Fgr$\"3%[DIs$*)[XkFgr 7$$\"3q>$QGLDhY&Fgr$\"3SV][wd>:nFgr7$$\"32Y.N%4Har&Fgr$\"3)>'*[/ul!fpF gr7$$\"3O,H@I#)p&)fFgr$\"3^&HhVRG!=sFgr7$$\"3A6$*4py#zE'Fgr$\"3[l??=*H D[(Fgr7$$\"3EG[B`1&[^'Fgr$\"3c8bz5h04xFgr7$$\"3y*fa:_JGy'Fgr$\"37+iL9 \"=)\\zFgr7$$\"3-@VMIq4hqFgr$\"3)f.nHn.V>)Fgr7$$\"3Ea*[$orrMtFgr$\"3^O J&z@5$H%)Fgr7$$\"3RX!*Q=)o2g(Fgr$\"3!G'GLU3w_')Fgr7$$\"3ECTa'Q$o(*yFgr $\"3SEwO\\fL'*))Fgr7$$\"3Eg;%zDHe;)Fgr$\"3+;] Kd,7L+*Fgr$\"3I3M![>%)*Fgr$\"3HcSeCEnM5F[s7 $$\"3?Mo*e%[s85F[s$\"3s'\\fg!Rga5F[s7$$\"3-(p$3>gIU5F[s$\"354*[fo(Qt5F [s7$$\"3YVgP,Rnr5F[s$\"3mp=(**Hy@4\"F[s7$$\"3\"fW;>#z$45\"F[s$\"3qi1!3 O&R56F[s7$$\"3e4cEN3&z7\"F[s$\"3XrJg%*zwE6F[s7$$\"3;M+#fva!f6F[s$\"3I) *)R*f15X6F[s7$$\"3e%*)p8%[+(=\"F[s$\"3?;.H5l5h6F[s7$$\"3.\"zB#f5%p@\"F [s$\"3OF'yOHlx<\"F[s7$$\"3mE@x5\\sX7F[s$\"3))>O8hmJ$>\"F[s7$$\"3;+++++ !oF\"F[sF_]rFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++YFip$\"++++_)*FipF\\q-F (6$7$$\"++++gMFbq$\"+++Si?FbqFeq-F(6$7$$\"+++!3_\"Fbq$\"+++g07FbqF]rF@ -FA6'7%7$$\"39+++++++kFgrFiio7$$\"3!)************zKF[s$\"3/++++++S?F[s 7$$\"3!***********z%[\"F[s$\"34+++++g08F[sFZFfnFTFjn-FA6'7$F^^sFa^sFao FeoFTFjn-FA6%7SFD7$$\"3gH\\d2hw*z#Fdr$\"3;7_FxC'e:%Fdr7$$\"3'4U9q\\g;D &Fdr$\"3pQn&*)edVs(Fdr7$$\"3Y)ew=Itm-)Fdr$\"3+BqV<+Zo6Fgr7$$\"3q7$[z(* *)Q3\"Fgr$\"3)G1b@s$[h:Fgr7$$\"3;U9H42ll8Fgr$\"3+uc7Wa+Z>Fgr7$$\"39Ad( 4t\\&G;Fgr$\"3SLG2j6U*H#Fgr7$$\"3f,3-#\\gC!>Fgr$\"3S&fP>2X#fEFgr7$$\"3 5Xr=sBb(=#Fgr$\"3Ll&Qa`Of-$Fgr7$$\"3i98zhXctCFgr$\"37dgi1#[fQ$Fgr7$$\" 3])**z;Iu'pFFgr$\"3e2+'*=8`]PFgr7$$\"3/&o+l=$4KIFgr$\"3UYz\\G0%o1%Fgr7 $$\"3V\\exZPJHLFgr$\"3LbCnGE`nHOFgr$\"3qTgO#*H\"Rw%F gr7$$\"3t?-()R0%4#RFgr$\"3wR$*Q38E#4&Fgr7$$\"3[wFNag)p=%Fgr$\"3Ir;%*3e z&Q&Fgr7$$\"3ryA$Q#[D0XFgr$\"3;jJ]g>4HdFgr7$$\"3k,0kKy\\wZFgr$\"3[(\\y +*f-:gFgr7$$\"3_KsFgr7$$\"3UN41o8%3I'Fgr$\"3\"**>av0)Fgr7$$\"34l1 ^[G)HB(Fgr$\"3*\\+oki_DM)Fgr7$$\"3fRzO`'*R?vFgr$\"3c\"='*=p\"y&e)Fgr7$ $\"3&z\")HjO7D$yFgr$\"3'pa5qn\"RV))Fgr7$$\"3YM,ZnN!o:)Fgr$\"3n)f*e8L)R 5*Fgr7$$\"3al^#R(o(eZ)Fgr$\"3w0XA!\\oMN*Fgr7$$\"3I[Mx^yI'y)Fgr$\"3[e'z YyC(*e*Fgr7$$\"32x)p1@gH8*Fgr$\"3Ao.*R1Kh%)*Fgr7$$\"3yUC_sC@Y%*Fgr$\"3 dnK/J;725F[s7$$\"3y^!eqBDDy*Fgr$\"3\"[e#)[W*fI5F[s7$$\"3,F'*y#Q#*)35F[ s$\"3;>6jh!z80\"F[s7$$\"3t4W$QEpD/\"F[s$\"3[rnH0%eN2\"F[s7$$\"3!*HoG>x Uu5F[s$\"3=5&RX.9R4\"F[s7$$\"3@*QtDN1z5\"F[s$\"3=K)z!H%fY6\"F[s7$$\"3* *4(Gs5839\"F[s$\"3^pQJIgTM6F[s7$$\"3=hc)o5\\a<\"F[s$\"3;;IMHBaa6F[s7$$ \"3i0q7Xs)*37F[s$\"3I$)*=ao(Qt6F[s7$$\"3Je%Rc/nMC\"F[s$\"3gD;[`J6#>\"F [s7$$\"3ysF&QIVyF\"F[s$\"38#oT;mP,@\"F[s7$$\"35%p&)oF\"f48F[s$\"3!z\"H %p-@iA\"F[s7$$\"3U^qV9i;Y8F[s$\"3%p%))[X)*3W7F[s7$$\"39u()>^&\\!z8F[s$ \"3QyO++6cf7F[s7$$\"3l5HA`rG99F[s$\"3Ko7tk`_v7F[s7$$\"3OfosH_=[9F[s$\" 3!>U>wh\"H!H\"F[s7$Fg^s$\"3()**********f08F[sFeblFTFhoF`p7.F'F2F:-F(6$ 7$$\"+++++aFip$\"+++g*4\"FbqF\\q-F(6$7$$\"++++?OFbq$\"+++?X>FbqFeq-F(6 $7$$\"+++?F>F[s7$$\"3/+++++?*p\"F[s$\"31++ +++S#Q\"F[sFZFfnFTFjn-FA6'7$Fh_tF[`tFaoFeoFTFjn-FA6%7SFD7$$\"3Y>&**[?0 6:$Fdr$\"3%)Q9I&GS7n%Fdr7$$\"3LwQGqM!H\"fFdr$\"3;r$[T0Ybn)Fdr7$$\"3ifv =&*36T!*Fdr$\"3AKPWSx>68Fgr7$$\"3%H$3f,sQ@7Fgr$\"3=,vAeij]#[+G&R:Fgr$\"3WbS.a#o3=#Fgr7$$\"3J[6k/)*fO=Fgr$\"3WaldOZStDFgr7$$ \"3#o#))e![Wj9#Fgr$\"3W>NtC9UtHFgr7$$\"3%\\NC%>u(*oCFgr$\"3/MpA+cB!Q$F gr7$$\"31K@R*e(*Gz#Fgr$\"3]-OKg$z(yPFgr7$$\"3OZPcoz]GJFgr$\"3]e(3jfu9= %Fgr7$$\"3_G\\TOa9EMFgr$\"3O7_v*yt+`%Fgr7$$\"3A3)GNl\"\\jPFgr$\"3@vNTX Jo:\\Fgr7$$\"3`wb4]&\\Y5%Fgr$\"3#*pKrRjf&H&Fgr7$$\"3!zCt#H!>dV%Fgr$\"3 !pD!=17\"[l&Fgr7$$\"3FL_Tj\"=$QZFgr$\"3%=Ia#GP9vfFgr7$$\"3\\FF\"HPh05& Fgr$\"3Y;=En$z([jFgr7$$\"33McVN))[4aFgr$\"39)4$pp\\Y?A 7x&Fgr$\"3(Q#4bgN)H,(Fgr7$$\"3ki['zo]N4'Fgr$\"3!zS0,9:)>tFgr7$$\"31ef] U2]\\kFgr$\"3qD@)p9W&\\wFgr7$$\"3#>+nqu!p!z'Fgr$\"3a\"**)H-+!o&zFgr7$$ \"3m>@?]w,\\rFgr$\"3+UORukRq#)Fgr7$$\"3#Qn0pHq,[(Fgr$\"3#p(Hy`e3_&)Fgr 7$$\"3I82HdNhRyFgr$\"3Cdy7BZ<\\))Fgr7$$\"3G58-1qW:#)Fgr$\"3WpgV+yM]\"* Fgr7$$\"3_)z=fmiYa)Fgr$\"3R.OuN_O1%*Fgr7$$\"3JwhQR%pB!*)Fgr$\"3Ct9%[bO kn*Fgr7$$\"3S'Qz?qaUF*Fgr$\"3yM=wh\"H%[**Fgr7$$\"3lO:J@BPS'*Fgr$\"3))Q lq`>w?5F[s7$$\"35pdF9cx'***Fgr$\"3Gps'RD!>X5F[s7$$\"3I1z)>h*\\R5F[s$\" 3%4GOB4l:2\"F[s7$$\"3K@O:ta]v5F[s$\"3[N\"R\\7#f%4\"F[s7$$\"3kf^cVM=96F [s$\"3;@XI7jZ=6F[s7$$\"3W@cSl,W\\6F[s$\"3j(\\Y\\j7F[s$\"3a7/O[=p-7F[s7$$\"33Oj3f)R9I\"F[s$\"3%4*4u$pU@A\"F[s 7$$\"3tk'RV(**RT8F[s$\"3r/5BL4!=C\"F[s7$$\"3amQriN6!Q\"F[s$\"3w+%e-T_+ E\"F[s7$$\"3?RC13W$*>9F[s$\"3c#o7DND!y7F[s7$$\"3alhx>elf9F[s$\"3o.:_g& e^H\"F[s7$$\"3!oX30(yN'\\\"F[s$\"3CIYF.cH58F[s7$$\"3(G*=([sg'Q:F[s$\"3 s@VGv)HpK\"F[s7$$\"31[fV3Crw:F[s$\"3Eb@\\Z7;T8F[s7$$\"3DY_.!f2vh\"F[s$ \"3OhUM2Cmb8F[s7$$\"3Ecb.`/xc;F[s$\"3%4LV5'=*)o8F[s7$$\"3!)**********> *p\"F[sFc`tFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++iFip$\"++++97FbqF\\q-F(6 $7$$\"++++!y$Fbq$\"++++G=FbqFeq-F(6$7$$\"++++S>Fbq$\"++++S8FbqF]rF@-FA 6'7%7$$\"3U+++++++!)Fgr$FfrF[s7$$\"33+++++++OF[s$\"3/+++++++=F[s7$F^`t $\"3%*************R9F[sFZFfnFTFjn-FA6'7$FbauFfauFaoFeoFTFjn-FA6%7SFD7$ $\"3qf\">kP[F]$Fdr$\"3;Fgr7$$\"3.XXVd*4Tr\"Fgr$\"3Eljp#4dD158eFgr7$$\"3m>saBx( f&\\Fgr$\"3RT$e$*[?4?'Fgr7$$\"3QX&f2NjeH&Fgr$\"3?g8s7C&ea'Fgr7$$\"3r,Y !*)H7Iq&Fgr$\"3M#>'GVO.ZpFgr7$$\"31]u;5SY]gFgr$\"3[^w\\HtDzsFgr7$$\"3W 0FvuycdkFgr$\"3A\")=u!o2pl(Fgr7$$\"3Qyzk/tb?oFgr$\"3yeg0Y;H$)zFgr7$$\" 3ININ_%f;A(Fgr$\"3?&*3%z%4&GL)Fgr7$$\"3J)RGCoqjg(Fgr$\"3M.[rnETd')Fgr7 $$\"3tv=EzTl5!)Fgr$\"39uV@7dU()*)Fgr7$$\"3'4SN<9,XQ)Fgr$\"37+Qzz^r#G*F gr7$$\"3il\"='fQ_!z)Fgr$\"3+/b9r**)Gf*Fgr7$$\"3!oCJ;M?`@*Fgr$\"3pdi5Sa \"f!**Fgr7$$\"3=1u)3pRwe*Fgr$\"38yPBu13<5F[s7$$\"3utOsSFS#***Fgr$\"3!z *Gyu_*[/\"F[s7$$\"3C3sTV]MT5F[s$\"3wu$[<7kF2\"F[s7$$\"3)p!30^.#G3\"F[s $\"3+zv%eLw\"*4\"F[s7$$\"3+,'*e5s@B6F[s$\"3N'>J2THR7\"F[s7$$\"3?orT6\" z$o6F[s$\"3!\\\\[FD*[]6F[s7$$\"31)H)\\.cB47F[s$\"3M0^]0\\_t6F[s7$$\"3Y ?.H`v9`7F[s$\"3_Q!H,h`s>\"F[s7$$\"3WD@>+g>$H\"F[s$\"3!QiB>E#)z@\"F[s7$ $\"3)\\ETx()fsL\"F[s$\"3q0ix7J!)R7F[s7$$\"3%zL-3&z)*y8F[s$\"3y%)H4QE`f 7F[s7$$\"30@fkZH)GU\"F[s$\"3)yBibEC$z7F[s7$$\"3EE:r\">sgY\"F[s$\"3C@a' [h_yH\"F[s7$$\"3bX)e#[ud6:F[s$\"3&GYBxrzjJ\"F[s7$$\"3yzU%=(\\ob:F[s$\" 3nhrYg=QL8F[s7$$\"3N')\\k))f2,;F[s$\"33S]1(*[\"*\\8F[s7$$\"3=CmopaPY;F [s$\"3/G,Wd!eaO\"F[s7$$\"3/(*Q8;1D)o\"F[s$\"3!*3$)fB<**y8F[s7$$\"3Z@F[s$\"3%***********Ry9F[sFZFfnFTFjn-FA6'7$FhbvF[c vFaoFeoFTFjn-FA6%7SFD7$$\"3M]Q8AcpaQFdr$\"3SY%)fL#f#*p&Fdr7$$\"36!*y\" 4pz&QsFdr$\"3MLYA;_$o0\"Fgr7$$\"3gUg)o$)Qx5\"Fgr$\"3ir=M+DV%f\"Fgr7$$ \"3]G&3)*HAx\\\"Fgr$\"3u:Wdx$>\\7#Fgr7$$\"3SK\"Hrc'R*)=Fgr$\"3b,E6g>DU EFgr7$$\"3iu4]BvobAFgr$\"3lvq**o\"4C6$Fgr7$$\"3XF=54M;QEFgr$\"3?;X>w7i *e$Fgr7$$\"3M?8/\"=[r.$Fgr$\"3SPgPG<(G2%Fgr7$$\"3+/<^\\>KQMFgr$\"3\")) )['HvmTa%Fgr7$$\"33YP6b:faQFgr$\"3Tj(eER7\"=]Fgr7$$\"35dy(R+xUA%Fgr$\" 3PHk14\"fkU&Fgr7$$\"3k9()))*ecQk%Fgr$\"3[fQL/o&f(eFgr7$$\"3G&)fo'y$yo] Fgr$\"3[W?%*\\eU;jFgr7$$\"3aO@pAmr\"[&Fgr$\"3O%fBz:*eInFgr7$$\"3K;W6F[s7$$\"3^w9:Wq$R?\"F[s$\"3_qbW&***eq6F[s7$$\"3a\"F[s7$$\"3ytZN>X$**H\"F[s$\"3,zcj(o&Q@7F[s7$ $\"3#RF'GQ;\"eM\"F[s$\"3Qx6us*>RC\"F[s7$$\"3u(G\")G&[9&R\"F[s$\"3#p8c$ Q8$pE\"F[s7$$\"3MQ\"\\r))f,W\"F[s$\"3i%e-tiZoG\"F[s7$$\"3gIpQHCr*[\"F[ s$\"3\"*3#RCt.wI\"F[s7$$\"3^iT\"=1hm`\"F[s$\"3^6v5aH;E8F[s7$$\"3Ek:AY7 2'e\"F[s$\"3)zI&o!ocXM\"F[s7$$\"3$)zU50,rM;F[s$\"3yfro$zX:O\"F[s7$$\"3 %H?V'G:)fo\"F[s$\"37!R?Goy#y8F[s7$$\"3#\\C=DZ,dt\"F[s$\"3Ql_/OgP$R\"F[ s7$$\"3U+rQ(y\"*oy\"F[s$\"3R)pQry\"y29F[s7$$\"3[[Te`A+Q=F[s$\"3wavR_h. @9F[s7$$\"3Y:?w8&p_)=F[s$\"3o`R\"zQ4BV\"F[s7$$\"3&)\\]\\,*)zR>F[s$\"3B ^[hnC;W9F[s7$$\"35#=N^c#*)))>F[s$\"3AaWz:#)z`9F[s7$$\"3Ml&*4#\\r:/#F[s $\"3#RI^sSnIY\"F[s7$$\"3WV[r7cJ#4#F[s$\"3kpaq3#o4Z\"F[s7$Facv$\"3;++++ +Sy9F[sFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++yFip$\"+++!GW\"FbqF\\q-F(6$7 $$\"+++++TFbq$\"+++g$f\"FbqFeq-F(6$7$$\"+++![Q#Fbq$\"+++g(R\"FbqF]rF@- FA6'7%7$FiioF\\bu7$$\"3$*************>RF[s$\"31++++++g:F[s7$$\"3!)**** ******z!Q#F[s$\"3/+++++g(\\\"F[sFZFfnFTFjn-FA6'7$FbdwFcdwFaoFeoFTFjn-F A6%7SFD7$$\"3N\"fV?%p%p?%Fdr$\"3hE#pQP+>@'Fdr7$$\"36ZCGQH,.zFdr$\"3=m_ ^!)p*4:\"Fgr7$$\"3$GBA\\#G#*47Fgr$\"3!=ILs`R\\t\"Fgr7$$\"3F.PQu,cO;Fgr $\"39\"*)[3'*\\+J#Fgr7$$\"3mWd!R$yQl?Fgr$\"3CmFGcGxpGFgr7$$\"3yu`po^sm CFgr$\"3quQ\"z-IuP$Fgr7$$\"3S.o/\\$)4')GFgr$\"3/!ff[xW;*QFgr7$$\"3;w5U &*Q*QK$Fgr$\"30snt\"z37T%Fgr7$$\"3md/.#G8Ww$Fgr$\"30G'3>*Hs;\\Fgr7$$\" 3Q&**zZZT=A%Fgr$\"3F;+m6p!QU&Fgr7$$\"3aSli@jNGYFgr$\"3mx.7n6hfeFgr7$$ \"31gc\\?O/!4&Fgr$\"3?AI^Y*z!QjFgr7$$\"39pasN/%zb&Fgr$\"3#>fBG,sb!oFgr 7$$\"3[(*zqEd$H,'Fgr$\"3d8g(=@M+.\"F[s7$$\"3&y*) \\=L3P-\"F[s$\"392.0P7Ah5F[s7$$\"3'e,&=u\\-u5F[s$\"3[_\\Wj/m$4\"F[s7$$ \"3h**[ra%Hn7\"F[s$\"3,,`l6o.E6F[s7$$\"3a'y2ydtH<\"F[s$\"3_SmPkj7`6F[s 7$$\"3.4U#G!)*HB7F[s$\"3)HPFz!*R7=\"F[s7$$\"3?!y!3pvqv7F[s$\"3qfw:&RO! 47F[s7$$\"3CiJV6tQF8F[s$\"3j80]KH+N7F[s7$$\"3aD!\\w7vxP\"F[s$\"3;BH0o= (*e7F[s7$$\"3KA2!e$e;M9F[s$\"31Ly*pRaUG\"F[s7$$\"39\\v^xNB&[\"F[s$\"3! =NZmKxdI\"F[s7$$\"3vh!QBMv,a\"F[s$\"3m:e)p\\4vK\"F[s7$$\"3egmFE=L!f\"F [s$\"31=+F[s$\"3,8F*p$\\.S9F[s7$$\"3u!y)G/=Qx>F[s$\"3qdOtAgi^ 9F[s7$$\"3*)Q(o9BF[ s$\"3]*pVHJWW\\\"F[s7$Fidw$\"3#)**********f(\\\"F[sFeblFTFhoF`p7.F'F2F :-F(6$7$$\"*+++g)Fbq$\"+++?d:FbqF\\q-F(6$7$$\"++++gUFbq$\"+++Sw9FbqFeq -F(6$7$$\"+++!oh#FbqF]dwF]rF@-FA6'7%7$$\"3/++++++S5F[sFfdw7$$\"33+++++ +!3%F[sF\\bu7$$\"3-+++++!3i#F[sF[ewFZFfnFTFjn-FA6'7$FhexF[fxFaoFeoFTFj n-FA6%7SFD7$$\"31#Q[hL-&fXFdr$\"3nb[b\"HHOs'Fdr7$$\"3-s`M5'4&o&)Fdr$\" 3j)Q'R<(R[C\"Fgr7$$\"3U?,'eb`BJ\"Fgr$\"39R'>aM1Z(=Fgr7$$\"3='4.E$\\%ex \"Fgr$\"3U72>$*)RQ\\#Fgr7$$\"3'4Qkxv$3UAFgr$\"3mdo?\"yz^4$Fgr7$$\"3))* 4m5ne(yEFgr$\"3(3q,`Jj%ROFgr7$$\"3&*Gu61OQNJFgr$\"3U8x9Ath*=%Fgr7$$\"3 o7]^lJT7OFgr$\"3%='\\&z=DUu%Fgr7$$\"3)**=bGQdF4%Fgr$\"3yIW$f#4_#G&Fgr7 $$\"3w5//7o*=f%Fgr$\"3no(yy<&3@eFgr7$$\"3A.n=&*zxN]Fgr$\"3y)))Ru:OFG'F gr7$$\"3%3g?FpL-a&Fgr$\"3]+#QQ'R>)y'Fgr7$$\"3!eZ%fQM#=0'Fgr$\"3_wl@9\" R0G(Fgr7$$\"3)=![fN]j\\lFgr$\"3/*f:7l/1u(Fgr7$$\"3NLO[#Qxd+(Fgr$\"3-+ \"\\q0Vh9)Fgr7$$\"39y([$Q8B`vFgr$\"3]nO&po(>8')Fgr7$$\"3S!*H)fc%H@!)Fg r$\"3%4.^+Zgf**)Fgr7$$\"3ss?B&RB2d)Fgr$\"3kzP!3^CjU*Fgr7$$\"3#o3b'p;^h !*Fgr$\"3!*RZ.>w\"Rz*Fgr7$$\"3\\ImP,Ez/'*Fgr$\"3L6qBH+G=5F[s7$$\"3O*R+ y4!o75F[s$\"3#>!)[&>z!R0\"F[s7$$\"3%pD&zCDjn5F[s$\"33HU6)*Gi*3\"F[s7$$ \"3x#GV]YL&=6F[s$\"3E_,_B!367\"F[s7$$\"3-qaB\"F[s$\"3kdHL:`#f=\"F[s7$$\"3sr*HS[&)GG\"F[s$\"36&3g Q!zs77F[s7$$\"3#3IU9+c$Q8F[s$\"3S(4t<#HLS7F[s7$$\"3+#3N:_]hR\"F[s$\"3Y `Z*HY(Qn7F[s7$$\"3'R'e*G:rJX\"F[s$\"3M3C,Z^T#H\"F[s7$$\"3*eUYbR$z3:F[s $\"3-A2ho^F:8F[s7$$\"3_9]vgI2r:F[s$\"3#o&\\$3Q&4R8F[s7$$\"3$R7#>@9]F;F [s$\"3;GOAnp4f8F[s7$$\"3wT1m@!R#)o\"F[s$\"3Uv!=g3))*y8F[s7$$\"3u\"puv \"=rVF[s$\"3+#=&p/:#3W\"F[s 7$$\"3+zC>uE+%)>F[s$\"3qhDUC>)GX\"F[s7$$\"3/\\u!)3&pu/#F[s$\"3I_w#[:PS Y\"F[s7$$\"3PKo\\[(p!4@F[s$\"3A.&4LceLZ\"F[s7$$\"3KF+NRgas@F[s$\"3!y\" *\\QgZ9[\"F[s7$$\"3t%RSLAxfB#F[s$\"3Q;)Gk8G!)[\"F[s7$$\"3='GE]w&o%H#F[ s$\"3+U6_[$4G\\\"F[s7$$\"3)[]*y&GpCO#F[s$\"3S'[JpV)z'\\\"F[s7$$\"3+'fn 9JXNU#F[s$\"358s>:u,*\\\"F[s7$$\"3M[nTlF8*[#F[s$\"3-b(*R$H!)**\\\"F[s7 $$\"3e()*4)*QgBb#F[s$\"3=O+7/da*\\\"F[s7$F_fxF_dxFeblFTFhoF`p7.F'F2F:- F(6$7$$\"*+++S*Fbq$\"+++gr;FbqF\\q-F(6$7$$\"++++?WFbq$\"+++?f8FbqFeq-F (6$7$$\"+++?bGFbqFcbvF]rF@-FA6'7%7$$\"36++++++?6F[sF^cv7$$\"3A++++++SU F[sFibv7$$\"3'***********>nGF[sFccvFZFfnFTFjn-FA6'7$FjfyF]gyFaoFeoFTFj n-FA6%7SFD7$$\"3N?#[W!=O7\\Fdr$\"3nN`l')fWMsFdr7$$\"3\"[m1rqp]B*Fdr$\" 39,!ooUj$Q8Fgr7$$\"3o0(*pH5.:9Fgr$\"3S$)3!\\#Ht8?Fgr7$$\"3$pqmWdwb\">F gr$\"3ez)*fu\"*GwEFgr7$$\"3KT]qQV[>CFgr$\"3Pw[)[ts%=LFgr7$$\"3O\\JhI!) y\"*GFgr$\"3#>bg6.4&)*QFgr7$$\"3=0PJ!=>gQ$Fgr$\"3Q'))e!=*QN[%Fgr7$$\"3 cJJK\"*fq-RFgr$\"3m01.<4#>2&Fgr7$$\"3Q+f)>DaLU%Fgr$\"3p)HU]bg:k&Fgr7$$ \"3#R*\\*ocdZ'\\Fgr$\"3r@]J\">Z*4iFgr7$$\"3OZ$eY-UlW&Fgr$\"3Qf\\-!3Mep 'Fgr7$$\"3UONc1oU%*fFgr$\"3<#R4j&))HEsFgr7$$\"3%>+$H&zK/b'Fgr$\"3Z+57a rKTxFgr7$$\"3%3b_$\\X\"=4(Fgr$\"3w]B%fZ^4A)Fgr7$$\"3cy`&H)[<)e(Fgr$\"3 ZnQ8-[HU')Fgr7$$\"3Cbj)>xfU=)Fgr$\"30Q4/S%>d7*Fgr7$$\"3+N:YGk?%p)Fgr$ \"3D(R:'=)*G?&*Fgr7$$\"3![I:@2[JH*Fgr$\"3-*3a1h6?'**Fgr7$$\"3PtLkHYZG) *Fgr$\"3\\()p]4DvL5F[s7$$\"3:&>@C$*Q@/\"F[s$\"3O:kV0$zK2\"F[s7$$\"3Emf j!zS\"*4\"F[s$\"3l,@>)3\"H46F[s7$$\"3'4mokF!=f6F[s$\"3U8F[s7$$\"3)=eR&o&)G\"e\"F[s$\"3la7)*Qm#GM\"F[s7$$\"3N=cNOXIU;F[ s$\"3=XJV])*4k8F[s7$$\"3#*\\w@%>c1r\"F[s$\"3K]q9R'3fQ\"F[s7$$\"3k(**4$ p^hsF[s$\"3S-.P%y)pN9F[s7$$\"3qzG4j!*\\n>F[s$\"3+j87b'=(\\9F[s7$$ \"3X;f'eP_7.#F[s$\"3S]A5fzLh9F[s7$$\"3e/g5YhV)4#F[s$\"3O')>e8R&=Z\"F[s 7$$\"3-Dz))Htlk@F[s$\"3KDiLw[_![\"F[s7$$\"3wPte3MbMAF[s$\"3=()zthm*y[ \"F[s7$$\"3[a9!Q_@CI#F[s$\"3)oj&*\\\"pM$\\\"F[s7$$\"3#3%3d#\\%QsBF[s$ \"3Yyu[IlC(\\\"F[s7$$\"36<\"*>4aKUCF[s$\"3#3l-b-U%*\\\"F[s7$$\"3FQCm=J 32DF[s$\"3a&o7[k\"***\\\"F[s7$$\"3ApM\"e0z=e#F[s$\"3i$ffzo#*))\\\"F[s7 $$\"31Xiu0*)H\\EF[s$\"3![Eh:NXj\\\"F[s7$$\"3Ak^H;\\s@FF[s$\"3%*3X\"Q>- ?\\\"F[s7$$\"3C1&))[8q:z#F[s$\"3))zWB#Qsi[\"F[s7$$\"3U+++++?nGF[sFccvF eblFTFhoF`p7.F'F2F:-F(6$7$$\"++++?5Fbq$\"++++'y\"FbqF\\q-F(6$7$$\"++++ !e%Fbq$\"++++U7FbqFeq-F(6$7$$\"+++++JFbqF]auF]rF@-FA6'7%7$FeauFiau7$$ \"3M+++++++WF[sFeau7$Fj\\rF\\buFZFfnFTFjn-FA6'7$F^hzF_hzFaoFeoFTFjn-FA 6%7SFD7$$\"3U5J%pMDbE&Fdr$\"3Bl1?:#Fgr7$$\"3#e`u**4bd0#Fgr$ \"3k#Rw]!yRdGFgr7$$\"3sDxsw&*e(f#Fgr$\"3#=#oJ<%Fgr$\"3q0P'*yfH%R&Fgr7$$\"3U*eA%*)Q?cZFgr$\"35IABz=%Q*fFgr7$$\"3[TPM RPUS`Fgr$\"3Gu(o>&HR!f'Fgr7$$\"3)zYT+T[1'eFgr$\"3e!fv[$\\!*)4(Fgr7$$\" 3%zYC?'Hi_kFgr$\"3?(fERi%R_wFgr7$$\"3KX5#e]oP0(Fgr$\"3gho`Kh$z=)Fgr7$$ \"3SW7)zEu%RwFgr$\"3%)pi0'od[o)Fgr7$$\"3ey*3\" *Fgr7$$\"3Xa``#eKC#))Fgr$\"3QLRRi!3oh*Fgr7$$\"33in(GY-^P*Fgr$\"3Srp8lm 1-5F[s7$$\"3we$p+DfC+\"F[s$\"3GB>aXR1Z5F[s7$$\"3=&zXvmU01\"F[s$\"3W8EO Jl6&3\"F[s7$$\"3!GVH]b4\\7\"F[s$\"3B,\"F[s7$$\"3%\\YS, yvDJ\"F[s$\"3!\\gG.X0xA\"F[s7$$\"35(39f+NK3:F[s$\"3!4+^;P)3:8F[s7$$ \"3OEG;n7_u:F[s$\"3:?:,W.OS8F[s7$$\"3i=()o0)fNk\"F[s$\"3OVQ$4Y>XO\"F[s 7$$\"3d:VOe&R?N\"fW 09F[s7$$\"3uG')=O_\"H&=F[s$\"3_8T$><%pC9F[s7$$\"3Yq6(=#[d?>F[s$\"3;([' Q3J7S9F[s7$$\"3y?K!*\\fY$*>F[s$\"3qO.Hblka9F[s7$$\"3%=G#ocf4g?F[s$\"33 `JqBv1m9F[s7$$\"3\"f%y\"R$4dL@F[s$\"3WikC<8mw9F[s7$$\"3(3Ebv!oI.AF[s$ \"3/;U38V#[[\"F[s7$$\"3Y@LXZW#oF#F[s$\"3[N+R?j[\"\\\"F[s7$$\"3OL4NmvJ \\BF[s$\"3e(>Zf3^h\\\"F[s7$$\"3Q-C$[\"R'eU#F[s$\"3k\"z_#\\j2*\\\"F[s7$ $\"3ZG'\\1R=-]#F[s$\"3c860#******\\\"F[s7$$\"3#*>7&R;(*od#F[s$\"39Sjk- G-*\\\"F[s7$$\"3G/\\/H2e`EF[s$\"3%yG:E^Mh\\\"F[s7$$\"3ao7ICmgCFF[s$\"3 )\\>'4&[&z\"\\\"F[s7$$\"3]d_vx=n1GF[s$\"3WFUB$yP[[\"F[s7$$\"37*=Lukq1) GF[s$\"3mJ/qX)=oZ\"F[s7$$\"3?&z')*49>gHF[s$\"3s9'*yl5Sm9F[s7$$\"3J*)4K %y/p.$F[s$\"3gIqGZVia9F[sFbhzFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++6Fbq$ \"+++S+>FbqF\\q-F(6$7$$\"++++SZFbq$\"+++![7\"FbqFeq-F(6$7$$\"+++?^LFbq Fc_tF]rF@-FA6'7%7$$\"3-++++++!G\"F[sF^`t7$$\"3g************fXF[sFi_t7$ $\"3)***********>zLF[sFc`tFZFfnFTFjn-FA6'7$F[i[lF^i[lFaoFeoFTFjn-FA6%7 SFD7$$\"3(30LO'H**=cFdr$\"37Z354FM`#)Fdr7$$\"3+NCZ&p#ej PXC:Fgr7$$\"3HpR31i7@;Fgr$\"3/#4[xRl&*G#Fgr7$$\"3S$eE\"40Q'>#Fgr$\"3*4 D?Yylr.$Fgr7$$\"3uMC$=Z*RwFFgr$\"37&p-&GnrePFgr7$$\"3NAiB@V$3K$Fgr$\"3 :L8H[xj2WFgr7$$\"3!f?$3!GT8*QFgr$\"3_\"Q]dDH#f]Fgr7$$\"3_7>35Bh)[%Fgr$ \"3ZhUvt.N6dFgr7$$\"3od_;&H184&Fgr$\"3CFU])*[ORjFgr7$$\"3YemQH`*)=dFgr $\"3sH+%)fCUipFgr7$$\"3#G2O8:(4yiFgr$\"3i%y\"*>s[>\\(Fgr7$$\"3P&R.\"f@ #[\"pFgr$\"3!y\")*om7[m!)Fgr7$$\"3q8'y,dI=c(Fgr$\"3=jTY\\gO?')Fgr7$$\" 3a#)3[\">9E>)Fgr$\"3=btb\"GBB8*Fgr7$$\"3/SWu=\"4;x)Fgr$\"3u%omxe!oy&*F gr7$$\"3W#y&**p(\\xY*Fgr$\"3Yl7Sbjk35F[s7$$\"3FnG#pE)R15F[s$\"3!*)RJo4 4(\\5F[s7$$\"3o#p'4oc]w5F[s$\"3k@*4X!3A&4\"F[s7$$\"3)G))42yO#R6F[s$\"3 :^.ZFQ[L6F[s7$$\"34xB'z7\"z37F[s$\"39pG\"\\IzK<\"F[s7$$\"3?q;y+z$4nV0o\">F[s$\"3Kj=(y N8$R9F[s7$$\"3+^zm'=]y*>F[s$\"3KZh>z>Xb9F[s7$$\"3IVc()y.Qr?F[s$\"3-qI( *3'HyY\"F[s7$$\"3Y?K#))>H1:#F[s$\"3))Q.`Nk#)y9F[s7$$\"3.T=\\/,5BAF[s$ \"3#oZCl57o[\"F[s7$$\"3GQw`Mw/.BF[s$\"3K'3()**[*Q$\\\"F[s7$$\"3E')RTpl &*yBF[s$\"3AT!elnCv\\\"F[s7$$\"3)pb$\\\\F,fCF[s$\"3!*G$>^s=(*\\\"F[s7$ $\"352:e$Q$)z`#F[s$\"33yaD`0w*\\\"F[s7$$\"3:XEaF5S@EF[s$\"3![1st@wv\\ \"F[s7$$\"3ob8/\\.Y-FF[s$\"3sLfZ%zGcs%HF[s$\" 3uq;Pp3Ao9F[s7$$\"3`r[h^x%o.$F[s$\"3Q(QbFsLYX\"F[s7$$\"37H%Gl`gw6$F[s$ \"3(Hr9w*yVS9F[s7$$\"3fU;\\YA`/KF[s$\"3zs]K4pZF[sFiio7$$\"3/+++++![k$F[sFi^sFZFfnFTFjn-FA6'7$F_j\\lFbj\\lFaoFeoF TFjn-FA6%7SFD7$$\"3qT!=XlkF(fFdr$\"3cyeWOFUh()Fdr7$$\"3]u!eX08T7\"Fgr$ \"39xd#3R?qh\"Fgr7$$\"3lZ'G'3RaC!GXPB Fgr$\"3Db9B8Jf:KFgr7$$\"3!y;>S-9f&HFgr$\"3q&\\U%oxmvRFgr7$$\"3%eC7BD^o `$Fgr$\"3KjKc\\2sdYFgr7$$\"3/Kkl0y-YTFgr$\"3P02`(*z*4M&Fgr7$$\"3GwD..e A%y%Fgr$\"3!=F-95%3BgFgr7$$\"3Z.R@p9mGaFgr$\"3w'GeGhH\"ymFgr7$$\"3^TP- PB<+hFgr$\"3Z#yG\\rNgK(Fgr7$$\"39a@a[#))))p'Fgr$\"3;QNPTa'\\(yFgr7$$\" 3g<.!yRC5Q(Fgr$\"3w^!*f%ye&o%)Fgr7$$\"3Y+dO))*=Y2)Fgr$\"3$>!H!\\!phQ!* Fgr7$$\"3Gl9&)>VB^()Fgr$\"355cWi#[Lc*Fgr7$$\"3Ud<1aeks$*Fgr$\"3Bt9$GY \"*=+\"F[s7$$\"3[jnVL6-75F[s$\"3s4(*[\"foM0\"F[s7$$\"3OE'eO \"F[s$\"3AyXFZMU`7F[s7$$\"3k:+kz'**=W\"F[s$\"3Ca*zN'pf(G\"F[s7$$\"39k[ =gTZ7:F[s$\"3C3aH#oMnJ\"F[s7$$\"3u;j***=,%*e\"F[s$\"3Q]5^$*[vX8F[s7$$ \"31(4ox1*=q;F[s$\"3W3du&z^KP\"F[s7$$\"3?gkL@rCT1aC?*\\R\"F[ s7$$\"3l3e>Cqv==F[s$\"38vDJ#HwhT\"F[s7$$\"3=Kd$)Gpm**=F[s$\"31.G*=%zbN 9F[s7$$\"3YKobgAkz>F[s$\"3'H]H'zn0_9F[s7$$\"3!y%RD'>&zd?F[s$\"31e\")[$ =-dY\"F[s7$$\"3o;clX5YX@F[s$\"3k]J$41#=y9F[s7$$\"3G:MKS=.DAF[s$\"31b(H iR)*p[\"F[s7$$\"3PE!4wjD3J#F[s$\"3i@HZ/BJ*Q#F[s$ \"3usU$G`Kz\\\"F[s7$$\"3obA&\\;HfZ#F[s$\"3)RBVL<.**\\\"F[s7$$\"3#34U9m ,#eDF[s$\"3?EP_\\!R%*\\\"F[s7$$\"3n5nA_5+XEF[s$\"3an)px7^l\\\"F[s7$$\" 3fV'z:ya1t#F[s$\"3An5EyKN\"\\\"F[s7$$\"3$Q1=nuk6#GF[s$\"3y2e4miR$[\"F[ s7$$\"3xMm(*)RZ\"4HF[s$\"3!p4qAK+LZ\"F[s7$$\"3+y1>h^%***HF[s$\"3glz_$Q 20Y\"F[s7$$\"3yxwpqF\"34$F[s$\"3')pp&4Gb`W\"F[s7$$\"3wipe\"4K]<$F[s$\" 3g7\"Rwzn#H9F[s7$$\"3]4BRxmSsKF[s$\"38uI_10G39F[s7$$\"3Cj>.t&o-O$F[s$ \"3m4TI2D?(Q\"F[s7$$\"3]0(4eUZZX$F[s$\"3'H)3UC(HBO\"F[s7$$\"3a\\yC'z[f a$F[s$\"3B_k?QT?O8F[s7$$\"3[+++++![k$F[sF]^tFeblFTFhoF`p7.F'F2F:-F(6$7 $$\"++++g7Fbq$\"+++?H@FbqF\\q-F(6$7$$\"++++g]Fbq$\"++++/*)FipFeq-F(6$7 $$\"+++!G(QFbqF_\\rF]rF@-FA6'7%7$F\\buFir7$$\"3))************z[F[sFg\\ r7$$\"3%)**********z;RF[sF_]rFZFfnFTFjn-FA6'7$Fc[^lFd[^lFaoFeoFTFjn-FA 6%7SFD7$$\"3vy!)f>/%oK'Fdr$\"3ycd?T0fo#*Fdr7$$\"3_]N6MW>\">\"Fgr$\"3J[ $f1*zE4(*yCFgr$\"3 #[+54zzER$Fgr7$$\"3$\\#zGLK8OJFgr$\"3-Bi8P[]!>%Fgr7$$\"3)Gfk0/kQv$Fgr$ \"3Q=iIzh\"[!\\Fgr7$$\"3?2`N[Y1-WFgr$\"3cwS$zy:(=cFgr7$$\"3;?up^Gh\"3& Fgr$\"3fNx!>;(\\HjFgr7$$\"3eH&o:Tp#odFgr$\"3-7WHAg85qFgr7$$\"3K()\\DiZ D%['Fgr$\"3'e.NsrK7o(Fgr7$$\"3Q;(f;q@I7(Fgr$\"3G_3-$4bzC)Fgr7$$\"3aL_6 y'H7&yFgr$\"33*Haw!=>H\"F[s7$$\"3%z%33I:$)Q:F[s$\"3Eduv%H&*pK\"F[s7$$\" 3*)pAcu$>Yh\"F[s$\"35\">8_AhYN\"F[s7$$\"3!eG;4'pE(p\"F[s$\"3'G9^i>-=Q \"F[s7$$\"35C&3Qu.Ty\"F[s$\"3JFWFKl-29F[s7$$\"3G>vwLg^g=F[s$\"3]TuRXD_ E9F[s7$$\"3?qW&pL,R%>F[s$\"3:*eOP'\\+X9F[s7$$\"37b<.!=#)4.#F[s$\"3KMZ] 8@Hh9F[s7$$\"3x9Y#H(R4<@F[s$\"3qbhU\"QlWZ\"F[s7$$\"3k2.rGyF,AF[s$\"3)o 2p.R7Y[\"F[s7$$\"37D;:8yu&H#F[s$\"3)Q7XrT%)G\\\"F[s7$$\"3]&[9i?H:Q#F[s $\"3eTl:q%Hw\\\"F[s7$$\"3KQ1Em_0uCF[s$\"3=&3=s[())*\\\"F[s7$$\"3M'[Amb K(eDF[s$\"3'3aKE!)G%*\\\"F[s7$$\"3o(ph^_:Al#F[s$\"331\\JnB?'\\\"F[s7$$ \"3Zv&RO3U5u#F[s$\"3'4F\")>Vn0\\\"F[s7$$\"3c#y_cN*yMGF[s$\"3#4lT$GN)>[ \"F[s7$$\"3fW`MgSU& \\OlX\"F[s7$$\"3IlaXS&z-7$F[s$\"3_.OVvv%*R9F[s7$$\"3`c(\\q[![=KF[s$\"3 &)H2D#p:-U\"F[s7$$\"3PiY]#\\*y;LF[s$\"3Q9g=iN)yR\"F[s7$$\"3qEQB`S$zS$F [s$\"3!3_)*)pi$\\P\"F[s7$$\"3Wrv3b'[L^$F[s$\"3Y(GPX8ydM\"F[s7$$\"3&GzV pv%\\3OF[s$\"3E@'oZn7rJ\"F[s7$$\"3H&)4%z%p$3r$F[s$\"3AXq26&fQG\"F[s7$$ \"3!eAU(e\"e'4QF[s$\"3eBL2k>V\\7F[s7$$\"3R**********z;RF[sF_]rFeblFTFh oF`p7.F'F2F:-F(6$7$F]au$\"+++gVAFbqF\\q-F(6$7$$\"++++?_Fbq$\"++++KxFip Feq-F(6$7$$\"+++?VTFbqFejpF]rF@-FA6'7%7$$\"3-++++++?:F[sF`[q7$$\"3/+++ +++S]F[sF[[q7$$\"3!)**********>&>%F[sFe[qFZFfnFTFjn-FA6'7$Fc\\_lFf\\_l FaoFeoFTFjn-FA6%7SFD7$$\"3'*oJ()e-A\"o'Fdr$\"35!\\!QBh%[x*Fdr7$$\"3ij) Qh:#Qe7Fgr$\"3O5M3jl>,=Fgr7$$\"3Z(4BCa>@$>Fgr$\"3332t\"Rixp#Fgr7$$\"33 M![%Q!Q4i#Fgr$\"3Q,fl:\\))*Fgr$\"3lh$Q!QwgP5F[s7$$\"3K'>af&eLf 5F[s$\"356u[+lM%3\"F[s7$$\"3NhXQfnlW6F[s$\"3f:jWCJoO6F[s7$$\"3O\\/*>$G &y@\"F[s$\"3M_'GC)[Ey6F[s7$$\"35Fd>tn0/8F[s$\"3T>GOGeXB7F[s7$$\"3)3#zp rDJ\"Q\"F[s$\"3OPiIh`gg7F[s7$$\"3)eW4([:5n9F[s$\"3;j;#=#HG)H\"F[s7$$\" 37_m=x=z\\:F[s$\"3>W+HkS;J8F[s7$$\"3aPN@0%4rj\"F[s$\"31*Qf=bbBO\"F[s7$ $\"3ozk<0'G#=d>)Q\"F[s7$$\"34e@Uu&Hn!=F[s$\"3kENBr*eIT\" F[s7$$\"3#3))ewLf(**=F[s$\"37fLP:&ybV\"F[s7$$\"3o`.WZjl\")>F[s$\"3sR*G D')QCX\"F[s7$$\"3cY7asK1r?F[s$\"3Yhinm2yn9F[s7$$\"3YAh(4*=Zk@F[s$\"3EL ;(e!H]![\"F[s7$$\"3]8MZf#zoD#F[s$\"3yf(z1Et)*[\"F[s7$$\"3Eg%ySL`sM#F[s $\"3c>Y^yR/'\\\"F[s7$$\"35yf:*[5([CF[s$\"31n?$y/f&*\\\"F[s7$$\"3f\"4I\"\\\"F[s7$$\"3[mf;:n!>$GF[s$\"3'>5->2(G#[ \"F[s7$$\"32Tk+OyZFHF[s$\"3]v1$R#)44Z\"F[s7$$\"3UuV,GJF[s$\"3RqTO,&)[Q9F[s7$$\"3'HVkW+sLB$F[s$\"34,n N0p*pT\"F[s7$$\"3#*\\yZtn&eL$F[s$\"3;`k'RbfKR\"F[s7$$\"3*>So5.!pTMF[s$ \"3m%z%\\Y8!fO\"F[s7$$\"318()H[LnZNF[s$\"3thQXT/pN8F[s7$$\"3#)oL)o;ifk $F[s$\"3*[*)\\hGE_I\"F[s7$$\"3me1q%ot'fPF[s$\"3qE!)z1m7n7F[s7$$\"3[JGu$Fgr7$ $\"3U7:2Bco)\\$Fgr$\"3uhaygq$Qh%Fgr7$$\"3uj#)f)=x3>%Fgr$\"3s3_?CV/!R&F gr7$$\"3k4+8&G*==\\Fgr$\"31s*4Y^)*>;'Fgr7$$\"3[a'phh2,OCakFgr$\"3CbGTER(Ql(Fgr7$$\"3;()**\\le$3E(Fgr$\" 3qX+]jzPm$)Fgr7$$\"3A!GHcn:8)zFgr$\"3Wj@6$>`Q'*)Fgr7$$\"3Ea!*fj$\\O!)) Fgr$\"3CQG?*eOFg*Fgr7$$\"3*33/fOU8k*Fgr$\"3Av(G-^H3-\"F[s7$$\"3)y))=Mf (*f/\"F[s$\"3cOLu^zxv5F[s7$$\"3G&[qHX.87\"F[s$\"3KW&)3M\"yF7\"F[s7$$\" 35u^*)Gi/77F[s$\"3%pZ98Uv]<\"F[s7$$\"3]8.S@X$**G\"F[s$\"3[g!*z2jK;7F[s 7$$\"3U#3,>46>B+k9F[s$\"3qBCUad)pH \"F[s7$$\"3)Q@TH#pUb:F[s$\"3rdj<#4&GL8F[s7$$\"3If8(*=;fV;F[s$\"3u@f3'3 IXO\"F[s7$$\"3a$3Q]ILnt\"F[s$\"3')\\d)[txOR\"F[s7$$\"3H$\\F?&=IB=F[s$ \"3&)>v\"\\;QtT\"F[s7$$\"37LR^I!*y<>F[s$\"3Z,#e%=__R9F[s7$$\"3Zl\">$\\ e:$\\\"F[s 7$$\"3+GK??\")**)R#F[s$\"3q:.R 2HF[s$\"3)\\\\%ps)[NZ\"F[s7$$\"3**e]'\\t-],$F[s$\"31A[5(Gd\"e9F[s7$$\" 3I'oU&=*3v6$F[s$\"3eR>PDiYS9F[s7$$\"39$o$ekfwDKF[s$\"3u^*[KK['=9F[s7$$ \"3QQ%z,Y-FL$F[s$\"3%[oh%**4.%R\"F[s7$$\"3_#QNIa:eW$F[s$\"3m^Q*e\\xZO \"F[s7$$\"35&yV!)4zeb$F[s$\"3IY'oyDOKL\"F[s7$$\"3/8mC$zt&pOF[s$\"3Eg,E YVc(H\"F[s7$$\"3%)H)z!QVY$y$F[s$\"3m60w=fxe7F[s7$$\"3!yeNDV;\"*)QF[s$ \"3vNKRYy8?7F[s7$$\"3$)p:BmV1-R1C%F[s$\"3?E/2***\\S2\"F[s7$$\"3kqGz'f ^aN%F[s$\"3L(Q@mZj2-\"F[s7$$\"3a************zWF[s$\"3_)************f*F grFeblFTFhoF`p7.F'F2F:-F(6$7$$\"+++++:Fbq$\"+++SsCFbqF\\q-F(6$7$$\"+++ +SbFbq$\"++++)Q&FipFeq-F(6$7$$\"+++?.ZFbqFagnF]rF@-FA6'7%7$F^cvF\\hn7$ $\"3K++++++g`F[sFign7$$\"3M+++++?rZF[sFahnFZFfnFTFjn-FA6'7$Fh^alFi^alF aoFeoFTFjn-FA6%7SFD7$$\"3-(\\3+;#*3R(Fdr$\"3S]u*>1i%y5Fgr7$$\"3?***)fF m2$R\"Fgr$\"3#=+.di'4%)>Fgr7$$\"3OPV#y9#oS@Fgr$\"3]')p_xD>mHFgr7$$\"3[ !4]&43@1HFgr$\"3GG(\\$=f*e\"RFgr7$$\"3!GM'e.)=5o$Fgr$\"3$4(4u:ALA[Fgr7 $$\"3z%ez$[v(3T%Fgr$\"3\"=Ch$RqL:W)fFgr$\"3Mj)o0M7o@(Fgr7$$\"3@u#o%[)41!oFgr$\"3!*p^4@ aglzFgr7$$\"3WLP^VXL`wFgr$\"3e'zeu?Ejp)Fgr7$$\"3ay7['>waT)Fgr$\"3EehbT ;w1$*Fgr7$$\"3sbzwoP'eG*Fgr$\"33Ihp2wxc**Fgr7$$\"3*[#4%*>OI<5F[s$\"3sC s^?WeMf6F[s7$$\"3SHp4O,:!G\"F[s$\"3\"4@4^SC8@\"F[s7$$\"37XQ +GY\"GO\"F[s$\"3+k%)e\"[#*>D\"F[s7$$\"3=Vfx&Qn-Y\"F[s$\"3))p@s;iT&H\"F [s7$$\"3'*e(p_(4pZ:F[s$\"3/A2z@%p.L\"F[s7$$\"3J)o(\\ZK'[k\"F[s$\"3wMpb 6W&\\O\"F[s7$$\"3\\B4eNXhQT'Hy9F[s7$$\"3Y8.T\"HV>)HF[s$\"3kg!p FHKKY\"F[s7$$\"3[ss4E;C'3$F[s$\"3)4=eHnshW\"F[s7$$\"3Ky*eXe.:?$F[s$\"3 hmI#H,8QU\"F[s7$$\"3h6$[7LN6J$F[s$\"3Wj]IOmB*R\"F[s7$$\"315&)3qU&pU$F[ s$\"3;pWe<2))p8F[s7$$\"3SHyC\"='RTNF[s$\"3u2lDbnbP8F[s7$$\"3))3:2)o&[i OF[s$\"3gra.n#y)*H\"F[s7$$\"3gsK:9lM!y$F[s$\"3o#=Srox)f7F[s7$$\"3,\"R% et<8-RF[s$\"3'o#oa\"p/_@\"F[s7$$\"3&=,[=YiT-%F[s$\"3Okf5%**Rr;\"F[s7$$ \"3(e[!>]oRPTF[s$\"3]U&G1&4n>6F[s7$$\"3_/.o**GZoUF[s$\"3y&3fS3w81\"F[s 7$$\"3O^!HJ>#)oQ%F[s$\"3/YG\"Q_;d+\"F[s7$$\"3B=T@r:N9XF[s$\"3_^k2/q5F% *Fgr7$$\"3bS\"\\BnNvj%F[s$\"3M'zDIjr')y)Fgr7$$\"3W**********>rZF[s$\"3 ;,+++++k!)FgrFeblFTFhoF`p7.F'F2F:-F(6$7$$\"++++!e\"Fbq$\"+++!oe#FbqF\\ q-F(6$7$$\"+++++dFbq$\"++++;UFipFeq-F(6$7$$\"+++!G*\\FbqFgemF]rF@-FA6' 7%7$$\"3-++++++gbF[sF]fm7$$\"3U+++++!)o]F[ sFgfmFZFfnFTFjn-FA6'7$F\\`blF_`blFaoFeoFTFjn-FA6%7SFD7$$\"3IO(o=A%=YxF dr$\"3uy$RM&R\")G6Fgr7$$\"3_AQ.xLeg9Fgr$\"3YL&)*e6o]2#Fgr7$$\"37/v([cL `C#Fgr$\"3w'[ntK(H*4$Fgr7$$\"3%oWn0_<&\\IFgr$\"3%>m(H#**>w3%Fgr7$$\"3C (>$=Tm0kQFgr$\"3_3/X*R8(G]Fgr7$$\"3;KsLlP(=j%Fgr$\"35.$))H=AL'eFgr7$$ \"3b6tS!>:(RaFgr$\"3!e1ydVx!*o'Fgr7$$\"3+;')\\.m*))G'Fgr$\"3+^T]KF%>]( Fgr7$$\"3!\\\"o/k)G#\\rFgr$\"3g`&f3hy0F)Fgr7$$\"3A];7R'Q'[!)Fgr$\"3c_] j)>ey,*Fgr7$$\"3=iZCt!zH&))Fgr$\"37=dEAIkR'*Fgr7$$\"3Kb[b:73s(*Fgr$\"3 `VN8]4))H5F[s7$$\"3'QRu'Qc%42\"F[s$\"3G=o(f_=<4\"F[s7$$\"3KaoZ.Qli6F[s $\"3iP%pZR'=Z6F[s7$$\"3k?w[qD5Z7F[s$\"3mQr$3j\\S>\"F[s7$$\"3YF)*)4[o*[ 8F[s$\"3C<0$e2IaC\"F[s7$$\"3vX5!=:$\\O9F[s$\"35joz.ME&G\"F[s7$$\"3w4.# [lD(R:F[s$\"3oq!RE_PtK\"F[s7$$\"3BA'H*R&yBj\"F[s$\"3gK6Tjjvg8F[s7$$\"3 &*o)yB_5at\"F[s$\"3=$Rj*z3H$R\"F[s7$$\"38X`,F1'[$=F[s$\"3KlRDcND?9F[s7 $$\"3(yui(y\">+%>F[s$\"3$pv6(yy?W9F[s7$$\"3qL*RWRSy.#F[s$\"3'))>!y4pWi 9F[s7$$\"3d#>b/Z)pW@F[s$\"3mBWjHh3y9F[s7$$\"3#Q_pgirrD#F[s$\"3%)G94Wz* )*[\"F[s7$$\"37/&4fp0jN#F[s$\"3!p[r:f-l\\\"F[s7$$\"3Se-F;[lkCF[s$\"3+E #*ek4z*\\\"F[s7$$\"3E)G*H#y()zd#F[s$\"35M@]2]**)\\\"F[s7$$\"311f?klB!p #F[s$\"3!>G#=jD4%\\\"F[s7$$\"3)*pOl(3P,!GF[s$\"3A\"**Q0.na[\"F[s7$$\"3 ^%4>#oRlBHF[s$\"3/;Fu'e;9Z\"F[s7$$\"3*Hw6Urxf.$F[s$\"3o5ZcKmxa9F[s7$$ \"3G^^_zcIdJF[s$\"3&paCav9GV\"F[s7$$\"3'o))Hp3%\\oKF[s$\"3%*Q.r;B<49F[ s7$$\"3y@x%RE49R$F[s$\"3sMoN\\UDz8F[s7$$\"3Vj$F[s$\"3NJ7%)4Jr48F[s7$$\"3s'y$3$[&4aPF[s$\"3kQ'[(od 1p7F[s7$$\"3[7GdRCQ$)QF[s$\"3Xg:y=#*HA7F[s7$$\"3L8j!=-f#4SF[s$\"3/i5yT Q=t6F[s7$$\"3YN<3sRORTF[s$\"3\"QzaBQA)=6F[s7$$\"3SgKg>xwpUF[s$\"3A>-\\ nEyg5F[s7$$\"35i![)>M!3R%F[s$\"3!\\\"e&))fDQ+\"F[s7$$\"3$e'o/&3Z4`%F[s $\"3>VSf!*3xU$*Fgr7$$\"33hSnm4edYF[s$\"3#)o\"yPA*3#o)Fgr7$$\"3+h#)fk%Q Rz%F[s$\"3rf@0LP[PzFgr7$$\"3Tu$fAlWd#\\F[s$\"3/%y=#p8'f=(Fgr7$$\"3_*** *******zo]F[sF[fnFeblFTFhoF`p7.F'F2F:-F(6$7$$\"++++g;Fbq$\"+++?,FFbqF \\q-F(6$7$$\"++++geFbq$\"++++WIFipFeq-F(6$7$$\"+++!))G&FbqF[dlF]rF@-FA 6'7%7$$\"33++++++S=F[sFhdl7$$\"3q************zcF[sFcdl7$$\"3!)******** **zs`F[sF]elFZFfnFTFjn-FA6'7$F`aclFcaclFaoFeoFTFjn-FA6%7SFD7$$\"3iES#z N!y,\")Fdr$\"3I#zAEiu!z6Fgr7$$\"3\\#[Q*ok>G:Fgr$\"3/aXoy0sl@Fgr7$$\"3) *oB$[sJ-N#Fgr$\"3<%*G][=mJKFgr7$$\"33@!H_6rK>$Fgr$\"3!4%HJ:M+eUFgr7$$ \"3uv?'e8*zZSFgr$\"3GsP\">h!)HB&Fgr7$$\"3m.7ZRe'Q&[Fgr$\"3;)Q'3b(za4'F gr7$$\"37RWt=O]-dFgr$\"3A*o'Hp/aYpFgr7$$\"3gqVtI9:&f'Fgr$\"3-\"*oHdCv \"y(Fgr7$$\"3pL8$zk+,](Fgr$\"3!3+1d\\$zo&)Fgr7$$\"3AMPK_\"[nW)Fgr$\"3H 5)Gq$R(4L*Fgr7$$\"3IA(>fICQH*Fgr$\"3OO3CNt\\i**Fgr7$$\"33vfRq,BE5F[s$ \"3gv?,FK)G1\"F[s7$$\"3)[\"3z#Hg]7\"F[s$\"3Xbviuh.D6F[s7$$\"3,zkBRR!=A \"F[s$\"3'Gc!4CXU!=\"F[s7$$\"3.n%))44M4J\"F[s$\"3,+Y)R\\*)oA\"F[s7$$\" 3IoQdj7]=9F[s$\"3;'RyMV#Rx7F[s7$$\"3_9>z#4q4^\"F[s$\"3fdUUu!RhJ\"F[s7$ $\"3a\"=M!**e3?;F[s$\"3Ybu/.Hbc8F[s7$$\"3\"\\6sJ,l!=F[s$\"3MJhi /5hU9F[s7$$\"3wlGm_6oV?F[s$\"3k.9^ReTj9F[s7$$\"3ug8+!p0t9#F[s$\"3M?f94 KTy9F[s7$$\"3cwYIa%[0E#F[s$\"3Arfe$z!=!\\\"F[s7$$\"33(ff6*3zzBF[s$\"3= 57Z8*ev\\\"F[s7$$\"3-Q%\\XhJ\\[#F[s$\"3)oo,Q4i**\\\"F[s7$$\"3g!\\pJC() )*f#F[s$\"3')H:f$Hb$)\\\"F[s7$$\"3yKqBK`Bg/n?$F[s$\"3/3?% )R\">FU\"F[s7$$\"3g&z0vD,dL$F[s$\"3G9E[[wH$R\"F[s7$$\"3M5I$**faRX$F[s$ \"3Ip4&R!yai8F[s7$$\"33#HJJx>Ze$F[s$\"3/FhS'*4[C8F[s7$$\"3L.x;ZT<4PF[s $\"35!*ok'[>WG\"F[s7$$\"38@p<$)3tSQF[s$\"3HQ#>+]X\"Q7F[s7$$\"3A2tol.!3 (RF[s$\"3'z2Q*R!e&)=\"F[s7$$\"3B%HRvz0&3TF[s$\"3V=@8^./K6F[s7$$\"3$f!H +@mhUUF[s$\"3C%G\"z@Z:t5F[s7$$\"3QY'Q()Qq7Q%F[s$\"3whSo=uT35F[s7$$\"31 vbM6,G?XF[s$\"3OmF8*QRqR*Fgr7$$\"35;$3:9O$\\YF[s$\"3YO0v5z,E()Fgr7$$\" 33^7LAV!))z%F[s$\"3j(\\iIQ*G5zFgr7$$\"3Dmti()y*Q$\\F[s$\"3xH!zryo%QrFg r7$$\"33@cz+(*Rz]F[s$\"3?$QJw(Fgr7$$\"3n.Vo8)zJ!pFgr$\"3'*)3Z\\^Ti0)Fgr7 $$\"3YH=7+_A`yFgr$\"3S5Xjv+Dg))Fgr7$$\"3%)z*>J3jw%))Fgr$\"35j+kAMnN'*F gr7$$\"3Bih]%*=,Q(*Fgr$\"3_^\"[!eC`F5F[s7$$\"3!RE)RBDlv5F[s$\"3y3_S\"f %o%4\"F[s7$$\"3s(=!H#eZ'z6F[s$\"3WO%Hr\"f$p:\"F[s7$$\"3g)>$[&4-:G\"F[s $\"3'\\S]>f=?@\"F[s7$$\"3c)\\<$>pQv8F[s$\"3[.vkLa'yD\"F[s7$$\"3-^!\\Q[ [()[\"F[s$\"3@ZG0y9@28F[s7$$\"3?^k(4XXie\"F[s$\"3CZ1Z$\\>YM\"F[s7$$\"3 tevT=\"[8q\"F[s$\"3-Ct%zNiIQ\"F[s7$$\"3bOs*\\R]Z!=F[s$\"38!H3%p+a79F[s 7$$\"3i[`6Bz$)>>F[s$\"3^bR&QEl*R9F[s7$$\"37e(eVLA5.#F[s$\"3AFP_G*)Hh9F [s7$$\"3GS[D^\"*o[@F[s$\"3ezaBEdey9F[s7$$\"39\"f*z,]BeAF[s$\"3#)G7!)fu )**[\"F[s7$$\"3]j+u!G&*zP#F[s$\"3i4)z(H\\[(\\\"F[s7$$\"3p)ffS2^T]#F[s$ \"3W/7UHr***\\\"F[s7$$\"3WX6VM*Gah#F[s$\"3oilI#R2y\\\"F[s7$$\"3sOo*GHP rt#F[s$\"3)>\\4Tvm3\\\"F[s7$$\"3Q?J#>MPX'GF[s$\"31R1.zhqy9F[s7$$\"3_[E $\\I43*HF[s$\"3sa?!)==!>Y\"F[s7$$\"3K-hfaR_9JF[s$\"3=#p6#QE-T9F[s7$$\" 3K*)G!o\\iOD$F[s$\"3TJ8R=j^79F[s7$$\"3_'>q[Rx-Q$F[s$\"3c2%*yLR7#Q\"F[s 7$$\"3&fC_`-Ir^$F[s$\"3jiK%>(4oW8F[s7$$\"33Um5lJiUOF[s$\"3vs+oM\"*H18F [s7$$\"3k&o4@6N9y$F[s$\"3#Q%42aK\\f7F[s7$$\"3To9Q]le8RF[s$\"34$fbg#>$3 @\"F[s7$$\"3_.0w!>>L0%F[s$\"3U!\\=\"))y%F[s$\" 3;E[#)oN.'4\"F[s7$$\"3Y^4(>wbyL%F[s$\"3VYr3k;5H5F[s7$$\"3U]Iu6$>/[%F[s $\"3K.&31)z_tq`F[s$\"3?RT\"yLY&HWFgr7$$\"3,M<9Dt`?bF[s$\"3ivzu\\%*HHMFgr7 $$\"3O**********>$o&F[s$\"3)=+++++SI#FgrFeblFTFhoF`p7+F'F2F:F@-FA6'7%F EFHFHFZFfnFTFjn-FA6'7$FEFHFaoFeoFTFjn-FA6%7SFD7$$\"3e`(>E&[)Q\"))Fdr$ \"3)Q29UHA$z7Fgr7$$\"3%=Jd,oTPm\"Fgr$\"3Ak!G?UogM#Fgr7$$\"3Q!>ZMFo2c#F gr$\"3wF%eY?qT\\$Fgr7$$\"3U@[[b*=@[$Fgr$\"3sLba3$[Zf%Fgr7$$\"3U0fY'4)R %[Bv\")H@(Fgr$\"3DRZX0*4aK)Fgr7$$\"3 J.$=1_-'3#)Fgr$\"3\\$3X1N[\\9*Fgr7$$\"3)[R5:V$Q^#*Fgr$\"3o<)oalc>$**Fg r7$$\"3>3/!R=a&=5F[s$\"3Yv()zvC\"y0\"F[s7$$\"3'>Ni0=va7\"F[s$\"3bVHJV] GD6F[s7$$\"3m7D<2vqM7F[s$\"3qhC[`xT(=\"F[s7$$\"3)>,C\"F[s7$$\"3P;ZZb5YS9F[s$\"3Q]e#)\\u(pG\"F[s7$$\"3[w`\"=95(f:F[s$ \"3jqQb4s)[L\"F[s7$$\"3ybYNE#>Bm\"F[s$\"3#=.O4m/2P\"F[s7$$\"3KT/(HJ7Ny \"F[s$\"37w'Qt)e'oS\"F[s7$$\"3Q()\\S&oMC*=F[s$\"3LP]yLo$RV\"F[s7$$\"3V Z1(*[!=P,#F[s$\"3?f!Q$zJIe9F[s7$$\"3C\\xE]z$48#F[s$\"3__n%zKf7s\\\"F[s7$$\"3=GYbbwzZFF[s$\"3I9h3([Q+\\\"F[s7$$\"3%eH_a'\\S wGF[s$\"3'>6VhMDtZ\"F[s7$$\"3#>bd8\"Q26IF[s$\"3sUt#fpw(e9F[s7$$\"3;Vvc OgfWJF[s$\"3Uotzi`IN9F[s7$$\"3M1]V(pcaF$F[s$\"3py\\%R^#e29F[s7$$\"3a,t &Qi!oAMF[s$\"3A%4GffC5P\"F[s7$$\"3IjV'=4'pcNF[s$\"3E4pS95*HL\"F[s7$$\" 3K-X1$)>f,PF[s$\"3e\"\\1eskpG\"F[s7$$\"3Q$y]Cy*\\MQF[s$\"3Q[w*)3jUS7F[ s7$$\"3o/H)3Gb:)RF[s$\"3![G^B-\"H%=\"F[s7$$\"3d8Yw$G%f@TF[s$\"3'e:c\\P ek7\"F[s7$$\"33/q.*\\2(pUF[s$\"35()*QTF531\"F[s7$$\"35Rq>toA;WF[s$\"3' e#))3aN#\\\"**Fgr7$$\"3]&ynGLK9d%F[s$\"3=JkYw:$[8*Fgr7$$\"3yYn-%4nEs%F [s$\"3\"yf(>zl!4L)Fgr7$$\"3!f;Jl(e5z[F[s$\"3m/]1z_RbuFgr7$$\"38*R\"z'H Eg.&F[s$\"3*))*zveF#Q`'Fgr7$$\"3]bo$3/!y\"=&F[s$\"3%yI%*[2)=ScFgr7$$\" 3g#\\`E&zm]`F[s$\"3'\\?&R&p^3c%Fgr7$$\"3Zf)e<<'Q.bF[s$\"3C4UBZYfXNFgr7 $$\"3/')*H;?Xzm&F[s$\"3B5/g8AB6CFgr7$$\"3jgQ6=57FeF[s$\"3?wT3%zZ`F\"Fg rFHFeblFTFhoF`p-%*AXESSTYLEG6#%%NONEG-%&TITLEG6#%Aquadratic~Bezier~cur ve~animationG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 128 "You can run the animation by clicking on the graphic, and using t he controls in the context bar, or by using the animation menu." }} {PARA 0 "" 0 "" {TEXT -1 46 "Stop the animation at some intermediate s tage." }}{PARA 0 "" 0 "" {TEXT -1 66 "Suppose that the point P divides the line segment AB in the ratio " }{XPPEDIT 18 0 "(1-t)*`:`*t;" "6#* (,&\"\"\"F%%\"tG!\"\"F%%\":GF%F&F%" }{TEXT -1 8 ", where " }{TEXT 268 1 "t" }{TEXT -1 42 " is some number between 0 and 1, that is, " } {XPPEDIT 18 0 "AP*`:`*PB = (1-t)*`:`*t;" "6#/*(%#APG\"\"\"%\":GF&%#PBG F&*(,&F&F&%\"tG!\"\"F&F'F&F+F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 114 "Then the animation is designed so that, at ever stage, t he point Q divides the line segment BC in the same ratio " }{XPPEDIT 18 0 "(1-t)*`:`*t;" "6#*(,&\"\"\"F%%\"tG!\"\"F%%\":GF%F&F%" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "BQ*`:`*QC = (1-t)*`:`*t;" "6#/*(%#BQG \"\"\"%\":GF&%#QCGF&*(,&F&F&%\"tG!\"\"F&F'F&F+F&" }{TEXT -1 64 ", and \+ the point R divides the line segment PQ in the same ratio " }{XPPEDIT 18 0 "(1-t)*`:`*t;" "6#*(,&\"\"\"F%%\"tG!\"\"F%%\":GF%F&F%" }{TEXT -1 12 " , that is, " }{XPPEDIT 18 0 "PR*`:`*RQ = (1-t)*`:`*t;" "6#/*(%#PR G\"\"\"%\":GF&%#RQGF&*(,&F&F&%\"tG!\"\"F&F'F&F+F&" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The curve traced out by the point R is the " }{TEXT 260 22 "quadratic Bezier cu rve" }{TEXT -1 37 " determined by the points A, B and C." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let A, B and C be \+ the points" }{XPPEDIT 18 0 "``(x[0],y[0]),``(x[1],y[1]);" "6$-%!G6$&% \"xG6#\"\"!&%\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1" }{TEXT -1 4 " and" } {XPPEDIT 18 0 "``(x[2],y[2]);" "6#-%!G6$&%\"xG6#\"\"#&%\"yG6#F)" } {TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, for each value of " }{TEXT 275 1 "t" } {TEXT -1 33 " between 0 and 1, P is the point " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "P(x[0*_*1],y[0*_*1]) = ``((1-t)*x[0]+t* x[1],(1-t)*y[0]+t*y[1]);" "6#/-%\"PG6$&%\"xG6#*(\"\"!\"\"\"%\"_GF,F,F, &%\"yG6#*(F+F,F-F,F,F,-%!G6$,&*&,&F,F,%\"tG!\"\"F,&F(6#F+F,F,*&F8F,&F( 6#F,F,F,,&*&,&F,F,F8F9F,&F/6#F+F,F,*&F8F,&F/6#F,F,F," }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 19 "and Q is the point " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Q(x[1*_*2],y[1*_*2]) = ``((1-t)*x[ 1]+t*x[2],(1-t)*y[1]+t*y[2]);" "6#/-%\"QG6$&%\"xG6#*(\"\"\"F+%\"_GF+\" \"#F+&%\"yG6#*(F+F+F,F+F-F+-%!G6$,&*&,&F+F+%\"tG!\"\"F+&F(6#F+F+F+*&F8 F+&F(6#F-F+F+,&*&,&F+F+F8F9F+&F/6#F+F+F+*&F8F+&F/6#F-F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "This means that R is the point \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(x[0*_*1*_*2],y[ 0*_*1*_*2]) = ``((1-t)*x[0*_*1]+t*x[1*_*2],(1-t)*y[0*_*1]+t*y[1*_*2]); " "6#/-%\"RG6$&%\"xG6#*,\"\"!\"\"\"%\"_GF,F,F,F-F,\"\"#F,&%\"yG6#*,F+F ,F-F,F,F,F-F,F.F,-%!G6$,&*&,&F,F,%\"tG!\"\"F,&F(6#*(F+F,F-F,F,F,F,F,*& F9F,&F(6#*(F,F,F-F,F.F,F,F,,&*&,&F,F,F9F:F,&F06#*(F+F,F-F,F,F,F,F,*&F9 F,&F06#*(F,F,F-F,F.F,F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "Thus the " }{TEXT 276 1 "x" } {TEXT -1 20 " coordinate of R is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (1-t)*x[0*_*1]+t*x[1*_*2];" "6#/%\"xG,&*&,&\"\"\"F (%\"tG!\"\"F(&F$6#*(\"\"!F(%\"_GF(F(F(F(F(*&F)F(&F$6#*(F(F(F/F(\"\"#F( F(F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-t)*[(1-t)*x[0]+t*x[1]]+t*[ (1-t)*x[1]+t*x[2]]" "6#/%!G,&*&,&\"\"\"F(%\"tG!\"\"F(7#,&*&,&F(F(F)F*F (&%\"xG6#\"\"!F(F(*&F)F(&F06#F(F(F(F(F(*&F)F(7#,&*&,&F(F(F)F*F(&F06#F( F(F(*&F)F(&F06#\"\"#F(F(F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-t)^2*x[0]+2*(1-t)*t*x[1]+t^2*x[ 2];" "6#/%!G,(*&,&\"\"\"F(%\"tG!\"\"\"\"#&%\"xG6#\"\"!F(F(**F+F(,&F(F( F)F*F(F)F(&F-6#F(F(F(*&F)F+&F-6#F+F(F(" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "x01 : = (1-t)*x0+t*x1;\nx12 := (1-t)*x1+t*x2;\nx012 := (1-t)*x01+t*x12;\ncol lect(%,[x0,x1,x2],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x01G, &*&,&\"\"\"F(%\"tG!\"\"F(%#x0GF(F(*&F)F(%#x1GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x12G,&*&,&\"\"\"F(%\"tG!\"\"F(%#x1GF(F(*&F)F(%#x2GF( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%x012G,&*&,&\"\"\"F(%\"tG!\"\" F(,&*&F'F(%#x0GF(F(*&F)F(%#x1GF(F(F(F(*&F)F(,&*&F'F(F/F(F(*&F)F(%#x2GF (F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&),&!\"\"\"\"\"%\"tGF(\" \"#F(%#x0GF(F(**F*F(F)F(F&F(%#x1GF(F'*&)F)F*F(%#x2GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 284 1 "y" }{TEXT -1 133 " c oordinate is obtained in a similar way so that we have the following p arametric equations for the curve traced out by the point R: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = (1-t)^ 2*x[0]+2*(1-t)*t*x[1]+t^2*x[2], ``],[y = (1-t)^2*y[0]+2*(1-t)*t*y[1]+t ^2*y[2], ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,(*&,&\"\"\"F,%\"tG!\"\"\"\" #&F(6#\"\"!F,F,**F/F,,&F,F,F-F.F,F-F,&F(6#F,F,F,*&F-F/&F(6#F/F,F,%!G7$ /%\"yG,(*&,&F,F,F-F.F/&F=6#F2F,F,**F/F,,&F,F,F-F.F,F-F,&F=6#F,F,F,*&F- F/&F=6#F/F,F,F:" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 269 21 "_____________________" }{TEXT -1 12 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "From the way that the point R sets out from the point A and arrives at the poi nt C, it is intuitively clear that the line segments AB and BC are " } {TEXT 260 10 "tangential" }{TEXT -1 14 " to the curve." }}{PARA 0 "" 0 "" {TEXT -1 29 "We can check this as follows." }}{PARA 0 "" 0 "" {TEXT -1 8 "Firstly," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt=2*(t-1)*x[0]+2*(1-2*t)*x[1]+2*t*x[2]" "6#/*&%#dxG\"\"\"%#dtG! \"\",(*(\"\"#F&,&%\"tGF&F&F(F&&%\"xG6#\"\"!F&F&*(F+F&,&F&F&*&F+F&F-F&F (F&&F/6#F&F&F&*(F+F&F-F&&F/6#F+F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 11 "Similarly, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dt = 2*(t-1)*y[0]+2*(1-2*t)*y[1]+2*t*y[2];" "6#/*&%# dyG\"\"\"%#dtG!\"\",(*(\"\"#F&,&%\"tGF&F&F(F&&%\"yG6#\"\"!F&F&*(F+F&,& F&F&*&F+F&F-F&F(F&&F/6#F&F&F&*(F+F&F-F&&F/6#F+F&F&" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "x01 := (1-t)*x0+t*x1:\nx12 := (1-t)*x1+t*x2:\nx012 := (1-t)*x01 +t*x12:\ndiff(%,t):\ncollect(%,[x0,x1,x2],factor);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*&,&!\"#\"\"\"*&\"\"#F'%\"tGF'F'F'%#x0GF'F'*&,&F)F' *&\"\"%F'F*F'!\"\"F'%#x1GF'F'*(F)F'F*F'%#x2GF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = (2*(t-1)*y[0]+2*(1-2*t)*y[1]+2*t*y[2])/(2*(t-1) *x[0]+2*(1-2*t)*x[1]+2*t*x[2]);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,(*(\"\" #F&,&%\"tGF&F&F(F&&%\"yG6#\"\"!F&F&*(F,F&,&F&F&*&F,F&F.F&F(F&&F06#F&F& F&*(F,F&F.F&&F06#F,F&F&F&,(*(F,F&,&F.F&F&F(F&&%\"xG6#F2F&F&*(F,F&,&F&F &*&F,F&F.F&F(F&&F?6#F&F&F&*(F,F&F.F&&F?6#F,F&F&F(" }{TEXT -1 3 ". " } }{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "t=0" "6#/%\"tG\"\" !" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx= (y[1]-y[0])/(x[1]-x[0])" "6#/*&%#dyG\"\"\"%#dxG! \"\"*&,&&%\"yG6#F&F&&F,6#\"\"!F(F&,&&%\"xG6#F&F&&F36#F0F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "which is the gradient of AB. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When \+ " }{XPPEDIT 18 0 "t=1" "6#/%\"tG\"\"\"" }{TEXT -1 10 ", we have " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx= (y[2]-y[1])/(x [2]-x[1])" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&&%\"yG6#\"\"#F&&F,6#F&F(F&,& &%\"xG6#F.F&&F36#F&F(F(" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "which is the gradient of BC." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 8 "Example " }}{PARA 0 "" 0 "" {TEXT -1 28 "If A, B and C a re the points" }{XPPEDIT 18 0 " ``(0,0), ``(2,3)" "6$-%!G6$\"\"!F&-F$6 $\"\"#\"\"$" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(6,0)" "6#-%!G6$\" \"'\"\"!" }{TEXT -1 46 " respectively, then the parametric equations \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = (1 -t)^2*x[0]+2*(1-t)*t*x[1]+t^2*x[2], ``],[y = (1-t)^2*y[0]+2*(1-t)*t*y[ 1]+t^2*y[2], ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,(*&,&\"\"\"F,%\"tG!\"\" \"\"#&F(6#\"\"!F,F,**F/F,,&F,F,F-F.F,F-F,&F(6#F,F,F,*&F-F/&F(6#F/F,F,% !G7$/%\"yG,(*&,&F,F,F-F.F/&F=6#F2F,F,**F/F,,&F,F,F-F.F,F-F,&F=6#F,F,F, *&F-F/&F=6#F/F,F,F:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "bec ome " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 4*t*(1-t)+6*t^2, ``],[y = 6*t*(1-t), ``]);" "6#-%*PIECEWISEG6$7$/% \"xG,&*(\"\"%\"\"\"%\"tGF,,&F,F,F-!\"\"F,F,*&\"\"'F,*$F-\"\"#F,F,%!G7$ /%\"yG*(F1F,F-F,,&F,F,F-F/F,F4" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 2*t*(2+t), ``],[y = 6*t*(1-t), ``]);" "6#-%*PIECE WISEG6$7$/%\"xG*(\"\"#\"\"\"%\"tGF+,&F*F+F,F+F+%!G7$/%\"yG*(\"\"'F+F,F +,&F+F+F,!\"\"F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "x01 := (1-t)*x0+t*x1:\nx12 := (1-t)*x1+t*x2:\nx012 := (1-t)*x01+t*x12:\nsubs(\{x0=0,x1=2,x2=6\}, %);\nx=factor(%);\ny01 := (1-t)*y0+t*y1:\ny12 := (1-t)*y1+t*y2:\ny012 \+ := (1-t)*y01+t*y12:\nsubs(\{y0=0,y1=3,y2=0\},%);\ny=factor(%);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"tG\"\"\",&F&F&F%!\"\"F&\"\"#*&F %F&,&F)F&*&\"\"%F&F%F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG ,$*&%\"tG\"\"\",&\"\"#F(F'F(F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&%\"tG\"\"\",&F&F&F%!\"\"F&\"\"$*&F%F&,&F)F&*&F)F&F%F&F(F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,$*&,&!\"\"\"\"\"%\"tGF)F)F*F)! \"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "T he picture shows the points " }{XPPEDIT 18 0 "A(0,0), B(2,3)" "6$-%\"A G6$\"\"!F&-%\"BG6$\"\"#\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C(6, 0)" "6#-%\"CG6$\"\"'\"\"!" }{TEXT -1 60 ", together with the curve giv en by the parametric equations:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([x = 2*t*(2+t), ``],[y = 6*t*(1-t), ``]);" "6 #-%*PIECEWISEG6$7$/%\"xG*(\"\"#\"\"\"%\"tGF+,&F*F+F,F+F+%!G7$/%\"yG*( \"\"'F+F,F+,&F+F+F,!\"\"F+F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot([[[0,0] ,[2,3],[6,0]],[2*t*(2+t),6*t*(1-t),t=0..1]],\n color=[brown,green ],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 318 195 195 {PLOTDATA 2 "6'-%'CURVESG6$7%7$$\"\"!F)F(7$$\"\"#F)$\"\"$F)7$$\"\"'F)F(-%'COLOURG6 &%$RGBG$\")#)eqk!\")$\"))eqk\"F8F9-F$6$7SF'7$$\"3e`(>E&[)Q\"))!#>$\"3) Q29UHA$z7!#=7$$\"3%=Jd,oTPm\"FD$\"3Ak!G?UogM#FD7$$\"3Q!>ZMFo2c#FD$\"3w F%eY?qT\\$FD7$$\"3U@[[b*=@[$FD$\"3sLba3$[Zf%FD7$$\"3U0fY'4)R%[Bv\")H@(FD$\"3DRZX0*4aK)FD7$$\"3J.$=1_-'3#)FD$\"3 \\$3X1N[\\9*FD7$$\"3)[R5:V$Q^#*FD$\"3o<)oalc>$**FD7$$\"3>3/!R=a&=5!#<$ \"3Yv()zvC\"y0\"F_p7$$\"3'>Ni0=va7\"F_p$\"3bVHJV]GD6F_p7$$\"3m7D<2vqM7 F_p$\"3qhC[`xT(=\"F_p7$$\"3)>,C\"F_p7$$\"3P;ZZ b5YS9F_p$\"3Q]e#)\\u(pG\"F_p7$$\"3[w`\"=95(f:F_p$\"3jqQb4s)[L\"F_p7$$ \"3ybYNE#>Bm\"F_p$\"3#=.O4m/2P\"F_p7$$\"3KT/(HJ7Ny\"F_p$\"37w'Qt)e'oS \"F_p7$$\"3Q()\\S&oMC*=F_p$\"3LP]yLo$RV\"F_p7$$\"3VZ1(*[!=P,#F_p$\"3?f !Q$zJIe9F_p7$$\"3C\\xE]z$48#F_p$\"3__n%zKf7s\\\"F_p 7$$\"3=GYbbwzZFF_p$\"3I9h3([Q+\\\"F_p7$$\"3%eH_a'\\SwGF_p$\"3'>6VhMDtZ \"F_p7$$\"3#>bd8\"Q26IF_p$\"3sUt#fpw(e9F_p7$$\"3;VvcOgfWJF_p$\"3Uotzi` IN9F_p7$$\"3M1]V(pcaF$F_p$\"3py\\%R^#e29F_p7$$\"3a,t&Qi!oAMF_p$\"3A%4G ffC5P\"F_p7$$\"3IjV'=4'pcNF_p$\"3E4pS95*HL\"F_p7$$\"3K-X1$)>f,PF_p$\"3 e\"\\1eskpG\"F_p7$$\"3Q$y]Cy*\\MQF_p$\"3Q[w*)3jUS7F_p7$$\"3o/H)3Gb:)RF _p$\"3![G^B-\"H%=\"F_p7$$\"3d8Yw$G%f@TF_p$\"3'e:c\\Pek7\"F_p7$$\"33/q. *\\2(pUF_p$\"35()*QTF531\"F_p7$$\"35Rq>toA;WF_p$\"3'e#))3aN#\\\"**FD7$ $\"3]&ynGLK9d%F_p$\"3=JkYw:$[8*FD7$$\"3yYn-%4nEs%F_p$\"3\"yf(>zl!4L)FD 7$$\"3!f;Jl(e5z[F_p$\"3m/]1z_RbuFD7$$\"38*R\"z'HEg.&F_p$\"3*))*zveF#Q` 'FD7$$\"3]bo$3/!y\"=&F_p$\"3%yI%*[2)=ScFD7$$\"3g#\\`E&zm]`F_p$\"3'\\?& R&p^3c%FD7$$\"3Zf)e<<'Q.bF_p$\"3C4UBZYfXNFD7$$\"3/')*H;?Xzm&F_p$\"3B5/ g8AB6CFD7$$\"3jgQ6=57FeF_p$\"3?wT3%zZ`F\"FDF/-F36&F5F($\"*++++\"F8F(-% *THICKNESSG6#F,-%+AXESLABELSG6$Q!6\"F`\\l-%%VIEWG6$%(DEFAULTGFe\\l" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Cubic Bezier curves " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 91 "L et A, B, C and D be 4 points in the coordinate plane, as shown in the \+ following animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 476 370 370 {PLOTDATA 2 "6%-%(ANIMATEG6 <7--%%TEXTG6$7$$!\"#!\"\"$\"\"!F/Q\"A6\"-F(6$7$$\"\"#F/$\"$:$F,Q\"BF1- F(6$7$$\"#_F-$\"$0#F,Q\"CF1-F(6$7$$\"$D'F,F.Q\"DF1-%'CURVESG6%7&7$F.F. 7$F5$\"\"$F/7$$\"\"&F/F57$$\"\"'F/F.-%'COLOURG6&%$RGBG$\")#)eqk!\")$\" ))eqk\"FfnFgn-%*THICKNESSG6#F6-FI6'7(FLFMFPFLFMFL-FW6&FYF/F/F/-%&STYLE G6#%&POINTGFin-%'SYMBOLG6#%'CIRCLEG-FI6'7%FLFMFP-FW6&FY$\"*++++\"FfnF. F.-Fbo6#%%LINEGFinFeo-FI6'7$FLFM-FW6&FYF^p$\")AR!)\\FfnF.F`pFinFeo-FI6 %7SFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLF LFLFLFLFLFLFLFLFLFLFLFLFLFLFLFL-FW6&FYF.F^pF.Fin-%+AXESLABELSG6%Q!6\"F bq%(DEFAULTG-%%VIEWG6$FdqFdq73F'F2F:FB-F(6$7$$!+++++5!#5$\"++++W=F^rQ \"PF1-F(6$7$$\"++++7B!\"*$\"++++kIFgrQ\"QF1-F(6$7$$\"++++S_Fgr$\"++++G =FgrQ\"RF1-F(6$7$$\"++++cMF^r$\"+++S#)>F^rQ\"SF1-F(6$7$$\"+++gzAFgr$\" +++S=GFgrQ\"TF1-F(6$7$$\"+++3'G%F^r$\"+++KcKF^rQ\"UF1FH-FI6'7(7$$\"3=+ ++++++!)!#>$\"3%**************>\"!#=7$$\"36++++++?@!#<$\"3'*********** **fHFhu7$$\"3/++++++S]Fhu$\"3$*************>>Fhu7$$\"3%************fh \"Fdu$\"3-++++++OBFdu7$$\"37+++++!oB#Fhu$\"35+++++S=HFhu7$$\"3#******* ***zgW#Fdu$\"3<+++++#*4MFduF_oFaoFinFeo-FI6'7%F^uFeuF[vF\\pF`pFinFeo-F I6'7$F`vFevFfpF`pFinFeo-FI6%7SFL7$$\"36Us2obfL_!#?$\"3FOT`i_&y$yF[x7$$ \"3=Vf!3j?5z*F[x$\"3Kz![)=(pUY\"Fau7$$\"3'>gpbkU?\\\"Fau$\"3m+A99t*yA# Fau7$$\"3SY^i96#)3?Fau$\"3!p%*z$Gd#[*HFau7$$\"38oJD,%yN_#Fau$\"33(G,>& fNcPFau7$$\"3JXZhw4@,IFau$\"3q\\zTC0$3Y%Fau7$$\"3UN.yJH;'\\$Fau$\"3cQD R?4p)=&Fau7$$\"3*>;M/Xg%3SFau$\"30`n\\LHvRfFau7$$\"3#=>Z6UO&>XFau$\"3? \"Ho%H\\p'o'Fau7$$\"3U&4xJoyc/&Fau$\"3cdT&42NKX(Fau7$$\"35:E_3p7)Fau7$$\"3K+z))\\K+KgFau$\"3a4sqj-j$)))Fau7$$\"3G()GbMg6d lFau$\"3,/HGa_nT'*Fau7$$\"3_H&)Qm;djqFau$\"3GA:Sx,0P5Fdu7$$\"3=0\\0*[J Q_(Fau$\"30I.rj44.6Fdu7$$\"3[74735Fdu$\"3gY8D9=qm9Fdu7$$\" 3].Rjz3Be5Fdu$\"3F=o089HP:Fdu7$$\"3\\Mg0tib56Fdu$\"3Q\"=x)eQx5;Fdu7$$ \"37:G,7Ske6Fdu$\"3e#44k]+\"y;Fdu7$$\"33Z=Y9Cb57Fdu$\"3wN`&)*3d0v\"Fdu 7$$\"3')4RO/P^k7Fdu$\"3]%oiOoQc#=Fdu7$$\"38B[)QyA:J\"Fdu$\"3)RCD%Ry%3* =Fdu7$$\"3yYd5<5Li8Fdu$\"3m)ohn@>6'>Fdu7$$\"3TeBl\"Hh[T\"Fdu$\"3i(*)4S 2YN.#Fdu7$$\"378$f(Q:Hm9Fdu$\"3xBym*>MU5#Fdu7$$\"3:*p-u9!4;:Fdu$\"3#3) >$33rC<#Fdu7$$\"3y#3'>edUr:Fdu$\"3e\">mG7b![AFdu7$$\"3u***z$Gb=@;Fdu$ \"3vldk^!3eJ#Fdu7$$\"3qH'*GsXNu;Fdu$\"3UE(H2))yzQ#Fdu7$$\"3*)zFZ#4pDs \"Fdu$\"3sj2W_\\A`CFdu7$$\"3o2FSu(=`x\"Fdu$\"3xBP@5:RCDFdu7$$\"3hlpK:( ))\\#=Fdu$\"3m#)>hkn>\"f#Fdu7$$\"3)p>ha%f%p(=Fdu$\"3)*Q\")pi\\'3m#Fdu7 $$\"3KrMP;4yF>Fdu$\"3JTjNTy\")GFFdu7$$\"3'o;>[!)R5)>Fdu$\"3o_VYo#)y*z# Fdu7$$\"3&H`%zCCPK?Fdu$\"3ITbZ&*p(z'GFdu7$$\"3E@Apwe!\\3#Fdu$\"3+Nv@nK aPHFdu7$$\"3`:MP+E/P@Fdu$\"3.0wmtsO1IFdu7$$\"3#ptxpM%)\\=#Fdu$\"3S5?'H aj%pIFdu7$$\"3?*py8Oq*RAFdu$\"3U]-(HJ2;9$Fdu7$$\"37(po$Gg=*G#Fdu$\"3zl Y%\\Jyf?$Fdu7$$\"3?L+gMqpTBFdu$\"3O2E&3')\\WF$Fdu7$$\"3y\"*yB!G#*>R#Fd u$\"3d5z/$)*H)RLFdu7$$\"3[+++++3YCFdu$\"3i*********>*4MFduF]qFinF_qFeq 73F'F2F:FB-F(6$7$$!*++++#F^r$\"++++))HF^rFar-F(6$7$$\"++++CCFgr$\"++++ GIFgrFjr-F(6$7$$\"++++!G&Fgr$\"++++cHFhu7$$\"33++++++!3&Fhu$\"33++++++S=Fhu7$$ \"3C++++++kKFdu$\"3E++++++WXFdu7$$\"31+++++?nCFhu$\"37+++++gLGFhu7$$\" 3'**********Rm(\\Fdu$\"3w*********ftW'FduF_oFaoFinFeo-FI6'7%FgilF\\jlF ajlF\\pF`pFinFeo-FI6'7$FfjlF[[mFfpF`pFinFeo-FI6%7SFL7$$\"3sVm(=.ur/\"F au$\"3M2zO+yul:Fau7$$\"3c_'\\0Y\"zf>Fau$\"3gT];#pm@#HFau7$$\"3W%)e]d)e x)HFau$\"3CC8MJw,TWFau7$$\"3\\FYnn#yU-%Fau$\"3aWdMac!H'fFau7$$\"37?Ycn Nfd]Fau$\"3c(>`+dl0Z(Fau7$$\"3mB#Gw?Or,'Fau$\"3cgD]@K8i))Fau7$$\"3C\"= t/dFA,(Fau$\"3UJ=K'er'H5Fdu7$$\"3SYy/-G*H/)Fau$\"33\"eneRit<\"Fdu7$$\" 37w\")***e.@2*Fau$\"3\\vju`%)*QK\"Fdu7$$\"3zv\"fx8QK,\"Fdu$\"3[DL*QB?R Z\"Fdu7$$\"3/unwD/x16Fdu$\"3e3*Hg0lag\"Fdu7$$\"3oLj02]A77Fdu$\"3s4\"3-!>Fdu7$$\"3!4*z\\')fk?9Fdu$ \"3\\4Nmzx]T?Fdu7$$\"3!3N)Rjut8:Fdu$\"3'4$\\c4>Dp@Fdu7$$\"3bdWG\"G)fC; Fdu$\"3C9F^;uW?BFdu7$$\"3![(o)3L_'=Fdu$\"3>SApcWdDFFdu7$$\"3D`4! o#H2K?Fdu$\"3)**\\v9wzv'GFdu7$$\"3EDbK\\o%Q8#Fdu$\"3Rjt$*o[:-IFdu7$$\" 3\\iDp$z\">SAFdu$\"3)4IS!ft*=9$Fdu7$$\"3>'R)*z)y)zL#Fdu$\"3aL#*==/ipKF du7$$\"3T'o<\"GJBrOFdu7$$\"3'4b-!es)Gv#Fdu$\"3^)= nC=?K!QFdu7$$\"3?S&on.e+'GFdu$\"3z%[!)e+1*QRFdu7$$\"3#4-WmI]]'HFdu$\"3 WFX1K.*42%Fdu7$$\"3mIy'o!HxmIFdu$\"3q()3do-<)>%Fdu7$$\"3cg$=jjv)zJFdu$ \"3nO2$HCq'QVFdu7$$\"31@5uhOk\"G$Fdu$\"3'=/i$)zxUY%Fdu7$$\"3)\\%GMU$[/ R$Fdu$\"3!)zXt>]s(f%Fdu7$$\"3Kjpm\\1<*[$Fdu$\"366s[#4b!=ZFdu7$$\"3+-pP 4)Rsf$Fdu$\"3+A))3>M'*[[Fdu7$$\"39?V`Oo0*p$Fdu$\"3)3Z@dEC:(\\Fdu7$$\"3 =A?!fK@c!QFdu$\"3=A`Bf/+*4&Fdu7$$\"3\"o&3'z_T*4RFdu$\"37'ReH&[+B_Fdu7$ $\"38G!)f>hH>SFdu$\"33\\g.OF;_`Fdu7$$\"3k&p\\+5`Z7%Fdu$\"3=.1.@m\"fZ&F du7$$\"3))y2vUZtKUFdu$\"3cY#))ysB=g&Fdu7$$\"3aX`P]y&*RVFdu$\"3:L()Go3/ EdFdu7$$\"3xa:urIgQWFdu$\"3\\Ju,oehReFdu7$$\"3Ib[\\6,!=b%Fdu$\"3')R4!4 !z6pfFdu7$$\"3[B2q?%pJl%Fdu$\"3MGFbKEM%3'Fdu7$$\"3)oy.9$*y8w%Fdu$\"3wx n!HVpl?'Fdu7$$\"310UGWG2l[Fdu$\"3*\\+fj)4&HK'FduF`[mF]qFinF_qFeq73F'F2 F:FB-F(6$7$$\"*++++'F^r$\"++++KTF^rFar-F(6$7$$\"++++ODFgr$\"++++#*HFgr Fjr-F(6$7$$\"++++?`Fgr$\"++++%o\"FgrFbs-F(6$7$$\"++++kkF^r$\"+++g,gF^r Fjs-F(6$7$$\"+++g>FFgr$\"+++gXEFgrFbt-F(6$7$$\"+++;+\"*F^r$\"+++W,()F^ rFjtFH-FI6'7(7$Fjil$\"3')*************f$Fdu7$$\"3))************fBFhu$ \"3))************zGFhu7$$\"36++++++?^Fhu$\"3-++++++g\"p#Fhu$\"3/+++++gXFFhu 7$$\"3C+++++;!e(Fdu$\"3t*********RQ7*FduF_oFaoFinFeo-FI6'7%Fg]nFj]nF_^ nF\\pF`pFinFeo-FI6'7$Fd^nFi^nFfpF`pFinFeo-FI6%7SFL7$$\"3Ez`g(Fau$\"3)=yiX>wU6\"Fdu7$$\"3#=H%z*3_v/*Fau$\"3a'**\\%pJT?8Fdu7$$\"3 #*H1h'3%ya5Fdu$\"3uc\\%yyUC`\"Fdu7$$\"38&yI?c1.@\"Fdu$\"3?Pq+^W@]Fdu7$$\"3U0p0$30f_\"Fdu$\"3A@7m0f*e=#Fd u7$$\"3'zh6?AKtm\"Fdu$\"3)y\"4G)4g%yBFdu7$$\"3m\"Ra3%Q*o#=Fdu$\"3g/7Q^ ^v$f#Fdu7$$\"3gnUfO'yu)>Fdu$\"3wVzNtHN3GFdu7$$\"3T\"G\"\\0ydU@Fdu$\"3> &4s3YhO,$Fdu7$$\"3akJk\"\\9PG#Fdu$\"3y=#*R*GJ))>$Fdu7$$\"39$)*e(4\")*= X#Fdu$\"3%Hl%y#*eWhs3HFdu$\"3Vl7\\#yN-+%Fdu7$$\"35( Qp8*G-rIFdu$\"3![jFT*pY.UFdu7$$\"33I(fnP&)eA$Fdu$\"3Wpn&***Gb&R%Fdu7$$ \"3/?A6Raz(Q$Fdu$\"3O@nGB;[%f%Fdu7$$\"3]XuP[7xONFdu$\"3!)=(>\\6?ex%Fdu 7$$\"3ClHP,Ax(p$Fdu$\"3#zn\"[iD)*p\\Fdu7$$\"3IC^[T\\MlQFdu$\"3KAUD=$)3 q^Fdu7$$\"37C1DX`\\6SFdu$\"3Bv*RqNxHM&Fdu7$$\"3Yi(fSgH'pTFdu$\"3*=:Zd, U$GbFdu7$$\"392se#z3LL%Fdu$\"3t*4t$Q7O=dFdu7$$\"3ouU1?*RP\\%Fdu$\"3jc* z2y/G!fFdu7$$\"3-Z4y'zZ#\\YFdu$\"3!o;J[/)*)zgFdu7$$\"3YwymE^BA[Fdu$\"3 ?#f#*y')f\\F'Fdu7$$\"3i$f9%Q%ez(\\Fdu$\"3Y'H<=?D)[kFdu7$$\"3nKh$*fE_W^ Fdu$\"3.q!Qs1(*Hj'Fdu7$$\"3)eJYPN;dH&Fdu$\"3wvb(zty&)z'Fdu7$$\"3u!pr&= HHhaFdu$\"3=*=wH\"f=ypFdu7$$\"32_<'p\"QNtFdu7$$\"3W6EY=yxSfFdu$\"3%pq060l#)[(Fdu7$$ \"3(Qi=tN'e3hFdu$\"3A]It*)fIjwFdu7$$\"3/MG2.RZqiFdu$\"3@NyN**p[IyFdu7$ $\"3'3^[^J'HOkFdu$\"3I$HR]mJ++)Fdu7$$\"3P<$G)*Q65g'Fdu$\"3!f&3/W^vm\") Fdu7$$\"3%yOP?F)f_nFdu$\"35].Ux[r=$)Fdu7$$\"3Kg&=.'GgEpFdu$\"3%o(>#Q:= 9\\)Fdu7$$\"3s!)\\$)H`Z#3(Fdu$\"3!)y_)**znXk)Fdu7$$\"3/VnT5a\"*[sFdu$ \"3&e-dh**)[1))Fdu7$$\"3S\"*yj5kX3uFdu$\"3yFVV\"H[,'*)Fdu7$F__n$\"3%3+ +++SQ7*FduF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++++9F^r$\"++++w_F^rFar-F(6 $7$$\"++++[EFgr$\"++++cHFgrFjr-F(6$7$$\"++++g`Fgr$\"++++7;FgrFbs-F(6$7 $$\"++++;!)F^r$\"+++SQyF^rFjs-F(6$7$$\"++++IHFgr$\"+++SaDFgrFbt-F(6$7$ $\"++?^g6Fgr$\"++!G84\"FgrFjtFH-FI6'7(7$$\"31+++++++KFdu$\"3#)******** *****z%Fdu7$$\"3)*************zCFhu$\"3&)************RGFhu7$$\"39+++++ +g^Fhu$\"3%*************z;Fhu7$$\"3q***********fl'Fdu$\"3G++++++w&)Fdu 7$$\"3$)**********z3HFhu$\"3))**********RaEFhu7$$\"34++++?^C5Fhu$\"35+ +++!)3X6FhuF_oFaoFinFeo-FI6'7%FhaoF]boFbboF\\pF`pFinFeo-FI6'7$FgboF\\c oFfpF`pFinFeo-FI6%7SFL7$$\"3%>bsY'H;'4#Fau$\"3*>l0VO2U7$Fau7$$\"316o,[ 5!f#RFau$\"3Zl>%Ght)=eFau7$$\"3ScL%=n,,*fFau$\"3&)yaa$ePI#))Fau7$$\"3s [34C=$[2)Fau$\"3)R1*R'p4>=\"Fdu7$$\"3G0u1$=Sc,\"Fdu$\"37A(zp\"yIx9Fdu7 $$\"3\">o')>]B#47Fdu$\"3_Icma\\q[(4oqu9)=;Fdu$\"3Il8A-Ye7BFdu7$$\"3*yzEjqGt#=Fdu$\"3wtzl*3RV f#Fdu7$$\"3n0M%*yHYU?Fdu$\"3q(fmwCc8)GFdu7$$\"3'eV'>\"4&\\KAFdu$\"3k%H !*fY8=8$Fdu7$$\"3CDZ.@m-ZCFdu$\"3:YurgY96MFdu7$$\"3'*QdF$orIm#Fdu$\"3! p_h)yK$))o$Fdu7$$\"3g#fK&pR'=(GFdu$\"3e4M6v)*z`RFdu7$$\"3S:X!=yl>1$Fdu $\"3y6'[]bx@>%Fdu7$$\"3Sb&Q8^D')G$Fdu$\"3wI,&)*4nGZ%Fdu7$$\"3/4[&)R)[6 [$Fdu$\"30FYU!4/$3ZFdu7$$\"34&Q53&Qo0PFdu$\"3%omnJ_v%z\\Fdu7$$\"3orM0$ z`\\!RFdu$\"34#)**>PO4<_Fdu7$$\"3^(*GW`w8CTFdu$\"3)z\"Hmkm>vaFdu7$$\"3 'QQ@'HRQLVFdu$\"3Zp,VQ![%=dFdu7$$\"3eZw^'3cAb%Fdu$\"3;0QTuxjpfFdu7$$\" 35o1d8Pt`ZFdu$\"3GW)zh(*fz>'Fdu7$$\"3wHJj%4o:(\\Fdu$\"3W\"zTWUA&Fdu$\"3K%y4d+o@p'Fdu7$$\"3[Z(*e\"4-jR&Fdu$\"3w>8Pm?!z!p Fdu7$$\"3uEnkO*=0h&Fdu$\"36JABNQ_QrFdu7$$\"3)4-Zj6KB$eFdu$\"3[y!\\U@$> utFdu7$$\"3)zAIa*R#)\\gFdu$\"3diRSHR@-wFdu7$$\"3#=gFbL92E'Fdu$\"3*3FF7 *[X?yFdu7$$\"3%onX:s*Q&\\'Fdu$\"3m52X0&Q+1)Fdu7$$\"3g_At'*Qr1nFdu$\"39 -+oBi'GF)Fdu7$$\"39&)f%[()=G$pFdu$\"3E5PYtOb(\\)Fdu7$$\"3O0Y(o)y6QrFdu $\"3Q#4Un?%)))p)Fdu7$$\"3_!=Oc\"z+jtFdu$\"3J:nAy\"Hl\"*)Fdu7$$\"3u\\*[ *4\"H]d(Fdu$\"3s9U2kO(*=\"*Fdu7$$\"33jf1!4yqz(Fdu$\"3v3M[b`;G$*Fdu7$$ \"3z:x&e9!)Fdu$\"3o>y4_CII&*Fdu7$$\"3R'*H'4$zsU#)Fdu$\"3o7Ld;1SR(*F du7$$\"3u!Hr6rmGY)Fdu$\"3!f*)\\JDc$Q**Fdu7$$\"3qf#e3J+%)o)Fdu$\"3$Qyph tNR,\"Fhu7$$\"3gg[be)oC\"*)Fdu$\"3#\\95hWCO.\"Fhu7$$\"3%[36c/7(=\"*Fdu $\"3)f[Ut%=]^5Fhu7$$\"3#=!*>y@#\\b$*Fdu$\"3WFjMSk'Gh*3\"Fhu7$$\"3Y\"QW;>wTz*Fdu$\"3H(yfI)QV36Fhu7$$\"38 \"zz(pd8,5Fhu$\"3N#\\xzr?i7\"Fhu7$$\"3()********>^C5Fhu$\"3))********z 3X6FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++++AF^r$\"++++?kF^rFar-F(6$7$$ \"++++gFFgr$\"++++?HFgrFjr-F(6$7$$\"+++++aFgr$\"++++S:FgrFbs-F(6$7$$\" +++++'*F^r$\"++++g&*F^rFjs-F(6$7$$\"++++MJFgr$\"++++gCFgrFbt-F(6$7$$\" ++++;9Fgr$\"++++!G\"FgrFjtFH-FI6'7(7$$\"3A+++++++SFdu$\"3w************ **fFdu7$$\"33+++++++EFhu$\"3#)*************z#Fhu7$$\"3;+++++++_Fhu$\"3 3+++++++;Fhu7$$\"3o*************R)Fdu$\"3/++++++S5Fhu7$$\"36++++++?JFh u$\"31++++++gDFhu7$$\"3/++++++'H\"Fhu$\"33++++++W8FhuF_oFaoFinFeo-FI6' 7%F]fpFbfpFgfpF\\pF`pFinFeo-FI6'7$F\\gpFagpFfpF`pFinFeo-FI6%7SFL7$$\"3 9\"yOj.N8i#Fau$\"3Gi;>Sqq+RFau7$$\"3fQnpKcI6\\Fau$\"3e-,_l\"*pdsFau7$$ \"3gxp#*>Jn'\\(Fau$\"373oGP#*>*4\"Fdu7$$\"3,rz\\z$z4,\"Fdu$\"3egZ,yUtq 9Fdu7$$\"3S(y7X(G4s7Fdu$\"3o\\LPLZ;O=Fdu7$$\"3-m*f%Rh4::Fdu$\"39s<7Q56 r@Fdu7$$\"33Q/$G7Iww\"Fdu$\"3`Y]\"R\"=Q?H#Fdu$\"3r-s9?Hq4KFdu7$$\"3g#=^,51Gc#Fdu$\"33 Zp<&)pSgNFdu7$$\"3$oZXja@@!GFdu$\"3_*yLhkic'QFdu7$$\"3)Q4\\A9VC2$Fdu$ \"3os'Q#3NMTFdu$\"3O7lz,29([&Fdu7$$\"3/D#)QW'pvP%Fdu$\"3)[RtTeC%pdFdu7$$\"3u :(*>DwNhYFdu$\"3E*[.dQ>O4'Fdu7$$\"3e8o\\.PK8\\Fdu$\"3I0Uxo$eoP'Fdu7$$ \"3E7jcgSe!>&Fdu$\"3ACl`D>g$o'Fdu7$$\"3Q^`fv*z`X&Fdu$\"3u*psm\"G!=(pFd u7$$\"3Uw968EYKdFdu$\"3!Q\"*=_$pZosFdu7$$\"3%pw)*R!\\h()fFdu$\"383*[:Q +t`(Fdu7$$\"3;HOIoOdjiFdu$\"3VNfi*o%RByFdu7$$\"3WK=/G)=5b'Fdu$\"3R7Ii( 4Ij6)Fdu7$$\"3w#yrUz$*=!oFdu$\"3O&))zMwGyO)Fdu7$$\"3g#zKr8;N2(Fdu$\"39 vIbfZ!ej)Fdu7$$\"3;SmGll%[N(Fdu$\"3qj(piP$o3*)Fdu7$$\"3)G._\"\\hwIwFdu $\"3c(QE:;a<<*Fdu7$$\"3eXL\\T?P)*yFdu$\"34TOP*GTEU*Fdu7$$\"37.GD8\\A'> )Fdu$\"3K_SIj1-(p*Fdu7$$\"3wBb-vS\\k%)Fdu$\"3C>'=c#o\")R**Fdu7$$\"3k)) )[oLy:v)Fdu$\"3$4]j)[`^>5Fhu7$$\"3*Rf:7$pG7!*Fdu$\"3\\*Hi;)fIU5Fhu7$$ \"3'zZ>Uh9zH*Fdu$\"3[H\">,MYo1\"Fhu7$$\"3o(fN)o@Bn&*Fdu$\"3Y4QYl'z&*3 \"Fhu7$$\"3!GygHK=$\\)*Fdu$\"34hAg!ftH6\"Fhu7$$\"3@065Phc75Fhu$\"3!yUB s5#[N6Fhu7$$\"3d=8NDQbT5Fhu$\"3')zuN?Hme6Fhu7$$\"3sT_;Sj_p5Fhu$\"3)R%* 40D>1=\"Fhu7$$\"3+'G&oWc=)4\"Fhu$\"3-/\"*35\"*p-7Fhu7$$\"3/]xj*efm7\"F hu$\"3.)zkzJCUA\"Fhu7$$\"3%>'3-\\'oG:\"Fhu$\"3*Q]GeXyOC\"Fhu7$$\"3%o*o Rz\"eH=\"Fhu$\"3AuaZ@Lfl7Fhu7$$\"3y!e')4`:*47Fhu$\"3ICoH2+&[G\"Fhu7$$ \"3`=#4&*4.(Q7Fhu$\"3:%QA;#\\-08Fhu7$$\"3Y@jUU\"*Hm7Fhu$\"3W$=)G$H\"*R K\"Fhu7$Fggp$\"3I++++++W8FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++++IF^r$ \"++++kvF^rFar-F(6$7$$\"++++sGFgr$\"++++%)GFgrFjr-F(6$7$$\"++++SaFgr$ \"++++o9FgrFbs-F(6$7$$\"+++g@6Fgr$\"+++k;6FgrFjs-F(6$7$$\"+++gJLFgr$\" +++SiBFgrFbt-F(6$7$$\"++!G`n\"Fgr$\"++?JP9FgrFjtFH-FI6'7(7$F[bo$\"3u** ***********>(Fdu7$$\"3>++++++?FFhu$\"3!)************fFFhu7$$\"3A++++++ S_Fhu$\"3-++++++?:Fhu7$$\"31+++++g<5Fhu$\"3,+++++g47Fhu7$$\"3?+++++![K $Fhu$\"39+++++SiCFhu7$$\"31++++!G8d\"Fhu$\"31++++?F5:FhuF_oFaoFinFeo-F I6'7%F`jqFcjqFhjqF\\pF`pFinFeo-FI6'7$F][rFb[rFfpF`pFinFeo-FI6%7SFL7$$ \"3q*oOTDdp9$Fau$\"3oqU1O#)QvYFau7$$\"3ky(G6Mm#)*eFau$\"3=oMI$=$=!p)Fa u7$$\"3MT@uDy!o+*Fau$\"3xt#)QZ!GYJ\"Fdu7$$\"3n;P$RH7^@\"Fdu$\"3)yh%)e; 6pv\"Fdu7$$\"3ql^*f+O&H:Fdu$\"3;r3'GDg3>#Fdu7$$\"3PoY2h2NA=Fdu$\"3[NI9 !y`we#Fdu7$$\"3#p[#\\>U%o7#Fdu$\"31f)Heo@H*HFdu7$$\"3dvL'HD@JW#Fdu$\"3 D9zom<41MFdu7$$\"3iJAZ?kufFFdu$\"31/NCH0$>\"QFdu7$$\"3MWFT@(Go3$Fdu$\" 3ki(fM%Q:BUFdu7$$\"3]#*H[+T2wLFdu$\"3[\\roD^9!e%Fdu7$$\"3`f#\\\"*>jHq$ Fdu$\"3WCJz*f6i(\\Fdu7$$\"3Z!4d$z3_KSFdu$\"3)[v^/,DxO&Fdu7$$\"3G_ZT;ZM ^VFdu$\"3=a(Q!>b=RdFdu7$$\"3,h^`St*=k%Fdu$\"35uV8^XcrgFdu7$$\"3i#QjuPW '))\\Fdu$\"3aj6rijpgkFdu7$$\"3GnZOt'GMG&Fdu$\"3!Q'Rwz!y^y'Fdu7$$\"3!p \"=NAN^FcFdu$\"3!*)z[GC#pcrFdu7$$\"3m80!H]DJ$fFdu$\"3!pvp^KS-[(Fdu7$$ \"3]fWGg*G&piFdu$\"3QDO?Hf^HyFdu7$$\"3U-lO#)4\"4f'Fdu$\"3G$\\**pwzl:)F du7$$\"3c\\j4'*QIFpFdu$\"3U:%*\\G-6#\\)Fdu7$$\"3CZC3SW:PsFdu$\"3q0ig5< 5&z)Fdu7$$\"3K4**y#=`Bd(Fdu$\"3Ah'Gw4Nk6*Fdu7$$\"389,?Wee@zFdu$\"3qmsD i*3UW*Fdu7$$\"3x)zDx$*[kA)Fdu$\"3:.m$R'*yXs*Fdu7$$\"3#*4t)o3#ec&)Fdu$ \"3>MSt!RAA+\"Fhu7$$\"3+KZk'pq&)*))Fdu$\"3O\\'>nJ6C.\"Fhu7$$\"3+T)Rw** HSB*Fdu$\"3o3P2'='Rh5Fhu7$$\"3CKP1L/Uf&*Fdu$\"3'GZ)3sd#*)3\"Fhu7$$\"3y -.4%>E;#**Fdu$\"3(p::\\3-*=6Fhu7$$\"3X%fi6I)yC5Fhu$\"3Qi,$pH4`9\"Fhu7$ $\"3AL@dRUqf5Fhu$\"3yZvH;b&H<\"Fhu7$$\"3A0L*o^894\"Fhu$\"3e^aa)Q>v>\"F hu7$$\"3(*oqz#Gah7\"Fhu$\"37#3Ao(e%QA\"Fhu7$$\"3%*QhpW7\"*e6Fhu$\"3)H2 MoD:\"[7Fhu7$$\"3kjy\"*o0A$>\"Fhu$\"3qWd?Hf'HF\"Fhu7$$\"3\"['e/g+$oA\" Fhu$\"3Ui9y#\\_nH\"Fhu7$$\"3?B^ztM3i7Fhu$\"3ZYbxLu6@8Fhu7$$\"3MbB=&e*4 'H\"Fhu$\"3Q7**y(QmSM\"Fhu7$$\"3,;0,Wu%4L\"Fhu$\"3)fgklJ7qO\"Fhu7$$\"3 uw=J9ecl8Fhu$\"3s#z&)fc[#*Q\"Fhu7$$\"3'))ed.PEuR\"Fhu$\"3S)\\DM5F#49Fh u7$$\"3S&et)>&)*RV\"Fhu$\"3>HEG!3#fJ9Fhu7$$\"3')=Skozvm9Fhu$\"3!\\3%G( o:6X\"Fhu7$$\"3mjl2a\\t,:Fhu$\"3=WHIV]Vr9Fhu7$$\"3#fF`3Bd_`\"Fhu$\"3]6 cA$30/\\\"Fhu7$$\"3%)********zKr:FhuFj[rF]qFinF_qFeq73F'F2F:FB-F(6$7$$ \"+++++QF^r$\"++++3()F^rFar-F(6$7$$\"++++%)HFgr$\"++++[GFgrFjr-F(6$7$$ \"++++![&Fgr$\"++++'R\"FgrFbs-F(6$7$$\"+++S'G\"Fgr$\"+++wl7FgrFjs-F(6$ 7$$\"+++!G_$Fgr$\"+++ghAFgrFbt-F(6$7$$\"++SMP>Fgr$\"++gTk:FgrFjtFH-FI6 '7(7$$\"3a+++++++cFduF]gp7$F`boFdjq7$$\"3E++++++!G&Fhu$\"3%*********** **R9Fhu7$$\"3!***********R)>\"Fhu$\"31+++++Sm8Fhu7$$\"36+++++?BNFhu$\" 39+++++ghBFhu7$$\"35++++SM\\=Fhu$\"33++++g0X;FhuF_oFaoFinFeo-FI6'7%Fa^ sFd^sFe^sF\\pF`pFinFeo-FI6'7$Fj^sF__sFfpF`pFinFeo-FI6%7SFL7$$\"3E&*3/D %GIn$Fau$\"3Ma[&\\9_#[aFau7$$\"3#)\\Qlq`x')oFau$\"3v:^)oML;,\"Fdu7$$\" 3g1I@@y/_5Fdu$\"3!\\#[PYHfG:Fdu7$$\"3uEC4/_()>9Fdu$\"3>GDj\"3Z//#Fdu7$ $\"3oKtRoi&zy\"Fdu$\"3G'\\fXo29a#Fdu7$$\"3ruh]1]'48#Fdu$\"3)=.a5ab$)*H Fdu7$$\"3In'f='>(y[#Fdu$\"33;uGm2#RY$Fdu7$$\"3],Z\"pG8)eGFdu$\"3.wn%o1 Nt$RFdu7$$\"3$pR!)eMx.B$Fdu$\"39maj%*z4,WFdu7$$\"3!ofgu6DWh$Fdu$\"3'y` #yZDqp[Fdu7$$\"3INKjm_@aRFdu$\"3)[7E;R)Rv_Fdu7$$\"3^#)pQ'e1%QVFdu$\"3# *Q!H-W\\Us&Fdu7$$\"3))fmhFdu7$$\"3\"[L$3\"*>'4 5&Fdu$\"3[Y]*o05]e'Fdu7$$\"3zCOV02)GW&Fdu$\"33w:C&=-$epFdu7$$\"3E3R$Q^ w5&eFdu$\"3iX9oYP'RR(Fdu7$$\"3If?4?$=#)>'Fdu$\"3y9())Q=sgv(Fdu7$$\"33F 2JT?a.mFdu$\"35c&fbf.$p\")Fdu7$$\"3y^(3L%)[O'pFdu$\"3M]CMa)\\z_)Fdu7$$ \"3Qo@9+#R,O(Fdu$\"3<\"R>\"G=x8*)Fdu7$$\"3W)p>wT9!RxFdu$\"3G&oD4,&)Fdu$\"3sN5$HMC E(**Fdu7$$\"3M=u\\)*3Z'*))Fdu$\"3j_W!)QzAK5Fhu7$$\"3O-s#zTg%3$*Fdu$\"3 k*e(y'*Qun5Fhu7$$\"33r3Q1g9o'*Fdu$\"3wR7L=u*z4\"Fhu7$$\"3Jizsmxw05Fhu$ \"3.oCsR?)*H6Fhu7$$\"3/&*fw1B7Y5Fhu$\"3?g!zHSwA;\"Fhu7$$\"3i!QId\"zq&3 \"Fhu$\"3/!fi5MPJ>\"Fhu7$$\"3SJC'G!f5C6Fhu$\"3@)\\PnY5BA\"Fhu7$$\"3+Af j0z%o;\"Fhu$\"33A#pqN:RD\"Fhu7$$\"37/l[ukM07Fhu$\"3/[(R;16;G\"Fhu7$$\" 3]!)R7S_aY7Fhu$\"32j3nM#Gu&RG\"Fhu$\"3$))G2JZPfL\"Fhu 7$$\"3e#*Q0PA%\\K\"Fhu$\"3+>mcY&)4j8Fhu7$$\"39,mwR>ej8Fhu$\"3oUU=&>*)z Q\"Fhu7$$\"3(z`I8'p//9Fhu$\"3$o?+*31K89Fhu7$$\"3*QCiV%*zOW\"Fhu$\"3zSQ ^.=TP9Fhu7$$\"3wY\"Fhu7$$\"3[/#**QfR`_\"Fhu$ \"3QfTU\\Q%[[\"Fhu7$$\"3A*ze2S1kc\"Fhu$\"3d0!**QQ%f2:Fhu7$$\"3+>,'Q7#> 2;Fhu$\"3qv3>E1ZH:Fhu7$$\"3JzeMy&=Zk\"Fhu$\"3yx)e.et*[:Fhu7$$\"3Ik!4Ug \"y(o\"Fhu$\"37$y+;OC1d\"Fhu7$$\"3NTs6>FME45wsn3;Fhu7$$\"3/]0\"p]Jp!=Fhu$\"3()z)RfhSli\"Fhu 7$$\"3K++++SM\\=Fhu$\"3')********f0X;FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$ \"+++++YF^r$\"++++_)*F^rFar-F(6$7$$\"++++'4$Fgr$\"++++7GFgrFjr-F(6$7$$ \"++++?bFgr$\"++++C8FgrFbs-F(6$7$$\"+++Sa9Fgr$\"+++O.9FgrFjs-F(6$7$$\" +++g2PFgr$\"+++gd@FgrFbt-F(6$7$$\"++g*3?#Fgr$\"++SYi;FgrFjtFH-FI6'7(7$ $\"39+++++++kFdu$\"3k*************f*Fdu7$Fiu$\"3;++++++!o#Fhu7$$\"3G++ ++++?`Fhu$\"35++++++g8Fhu7$$\"31+++++S#Q\"Fhu$\"3'***********R5:Fhu7$$ \"3%)**********>:PFhu$\"31+++++gdAFhu7$$\"37++++g*)G@Fhu$\"3'********* R]\\%Fau$\"3&fz%*)f**H>iFau7$$\"3MeGh=\\ #o(yFau$\"3/DA[#y:O:\"Fdu7$$\"3h'>$>qcw.7Fdu$\"3uPB'>p'4Tyr$f$e87Fhu7$$\"3K3z'fPwu:\"Fhu$\"3%Rn$yML1Z7Fhu7$$\"3H'4dNA_S?\"F hu$\"3SV8oSDl78Fhu7$$\"3_kglM([S H\"Fhu$\"3T%)e/h#)pU8Fhu7$$\"3XfgB**oOV8Fhu$\"3q&\\iNL`]P\"Fhu7$$\"35N uIJBy(Q\"Fhu$\"30mu&GdGKS\"Fhu7$$\"3dz!=.(\\IN9Fhu$\"3$yz0!p)zBV\"Fhu7 $$\"3GV_J&f^%y9Fhu$\"3gN98n!pzX\"Fhu7$$\"31Htb0$3d_\"Fhu$\"3cGy)zO^][ \"Fhu7$$\"3i9*GuC]-d\"Fhu$\"3c].=Nlo4:Fhu7$$\"34'o#*>&R)oh\"Fhu$\"3Tiq A(okX`\"Fhu7$$\"3?(G?%)QXDm\"Fhu$\"3-<00T/.e:Fhu7$$\"3A1>W%e:/r\"Fhu$ \"37=EdfWp\"e\"Fhu7$$\"35Cl/a&zlv\"Fhu$\"30^>oZ%=Og\"Fhu7$$\"3y>ui'[VQ !=Fhu$\"3n=]RSV;D;Fhu7$$\"3OGZE#3l2&=Fhu$\"3e%z(fMRmX;Fhu7$$\"3GE.w#[> R*=Fhu$\"3A^S&onVPm\"Fhu7$$\"3^V.S$z=M%>Fhu$\"3=FHV/)yNo\"Fhu7$$\"3)*Q ,p(HAx)>Fhu$\"3()y`^.E_+kZL$)3CFhu$\"3&* ********zwC=FhuF_oFaoFinFeo-FI6'7%F]fuF`fuFefuF\\pF`pFinFeo-FI6'7$Fjfu F_guFfpF`pFinFeo-FI6%7SFL7$$\"3?&oOS&G^EZFau$\"3-sa\"R(G`))pFau7$$\"3W @nM#=2%o))Fau$\"3m$e(Q0k'\\H\"Fdu7$$\"3c0*)zr:$eN\"Fdu$\"3Q#pm=/U@&>Fd u7$$\"3\\oJ3DTEJ=Fdu$\"3m$3eK)Gi*f#Fdu7$$\"3J)>#\\\\\\n2BFdu$\"3L%))e5 _6,B$Fdu7$$\"3-W6_FH=_FFdu$\"3*yoV3cDB!QFdu7$$\"3kA0sjrA:KFdu$\"3?b(z) G**Q#Q%Fdu7$$\"3cD\"e&*>4qp$Fdu$\"3kuAG?o1p\\Fdu7$$\"3#=oC\\%H5!=%Fdu$ \"3MA210obSbFdu7$$\"3#)*>XGL;*zYFdu$\"39Qzh&RGY6'Fdu7$$\"3v.*oq[8D7&Fd u$\"3d:RcIsw3mFdu7$$\"3$Q9#[IDMBcFdu$\"3S?#Q'*)[B_rFdu7$$\"3Q*\\exb%)) GhFdu$\"37.9\"=GeVo(Fdu7$$\"3E][Wsq]=mFdu$\"3[Nb#[$eO%=)Fdu7$$\"3#\\u> rMS^1(Fdu$\"342f/\"Fhu7$$\"33zbX!4BId*Fdu$\"33h<:#>k+4\"Fh u7$$\"3!*o6VW)3p+\"Fhu$\"3yzOO(*o!48\"Fhu7$$\"3+fBa0D&)e5Fhu$\"3k(o;Jh )Gs6Fhu7$$\"3)**RZE$>r16Fhu$\"3,Z+d.v=47Fhu7$$\"3)4z>vz(\\e6Fhu$\"3u(Q \\uD7yC\"Fhu7$$\"3**RM]w%fC@\"Fhu$\"3mq>^()*[mG\"Fhu7$$\"37Z5q&)zcf7Fh u$\"3A$\\5Td)R>8Fhu7$$\"3a.baC,e58Fhu$\"3c<2y(=qON\"Fhu7$$\"3WU;wv'=MO \"Fhu$\"3uKE5ua)yQ\"Fhu7$$\"3q!3!e(yQ_T\"Fhu$\"3@F(z'f`>?9Fhu7$$\"3/Fe _!\\!\\l9Fhu$\"3+x]<.#p.X\"Fhu7$$\"30Z&R%\\\"39_\"Fhu$\"35u[;lui#[\"Fh u7$$\"3')R&eaFaHT#H:)p$>Fhu$ \"39:_N*[`5o\"Fhu7$$\"3a[]N`E:*)>Fhu$\"3C_D$[*p0,gQbP$H.U9# Fhu$\"3eEk8$=jVv\"Fhu7$$\"3[IWW[9-+AFhu$\"37_?yZSMr#\\* \\AFhu$\"3-J;>4)eby\"Fhu7$$\"3\\n%>J[yJI#Fhu$\"3)\\#p/(=5(*z\"Fhu7$$\" 3/RcJ4X6aBFhu$\"3Nd$*HA5H7=Fhu7$Fegu$\"3<++++!oZ#=FhuF]qFinF_qFeq73F'F 2F:FB-F(6$7$$\"+++++iF^r$\"++++97FgrFar-F(6$7$$\"++++?LFgr$\"++++SFFgr Fjr-F(6$7$$\"+++++cFgr$\"++++!=\"FgrFbs-F(6$7$$\"+++++=Fgr$\"++++W;Fgr Fjs-F(6$7$$\"++++eSFgr$\"++++S>FgrFbt-F(6$7$$\"++++GFFgr$\"++++w(=FhuF_oFaoFinFeo-FI6'7%F^jvFbjvFejvF\\pF`pFinFe o-FI6'7$FhjvFijvFfpF`pFinFeo-FI6%7SFL7$$\"3]7b1EP#RD&Fau$\"3ak#[+3_fv( Fau7$$\"3%3N'>fV^h)*Fau$\"3h+r'e*foN9Fdu7$$\"3-cUJoFC3:Fdu$\"3)pw.QvJ< ;#Fdu7$$\"3*=TnEpwy.#Fdu$\"3G;NQ7iFvGFdu7$$\"3+#[]*\\n%*oDFdu$\"3alS4W XHoNFdu7$$\"39!QbC)=ukIFdu$\"3Yt:PS&Qc>%Fdu7$$\"3G_(>K,%[\"e$Fdu$\"32( )*35iI*H[Fdu7$$\"3*QeV8z1%>TFdu$\"3'=bl/ch'paFdu7$$\"37BO=ja/fYFdu$\"3 'p>raI+54'Fdu7$$\"3jmpk,(*f<_Fdu$\"3Z_bm*)p@8nFdu7$$\"3hKGSnbR7dFdu$\" 3IHU^xx:ZsFdu7$$\"3)Q5VmnuCF'Fdu$\"3PizI4HaKyFdu7$$\"3*ROi.K)*z$oFdu$ \"3!=a&*zJ1PS)Fdu7$$\"3K=b'4ddeQ(Fdu$\"3#[*>2$Qu%Q*)Fdu7$$\"3_slB?(>d) yFdu$\"3!*pa%Q0t3T*Fdu7$$\"3]q&Hrw2H[)Fdu$\"3K<(*zxGLc**Fdu7$$\"3%eia) **[4\"**)Fdu$\"3O&=R9Un//\"Fhu7$$\"3!Gno%>$oZe*Fdu$\"3)\\TkOOY54\"Fhu7 $$\"3ia-5jWC65Fhu$\"3Aa\"3et9W8\"Fhu7$$\"3/%4+O/j$p5Fhu$\"3?S54UA\\!= \"Fhu7$$\"3*>-7rR8\\7\"Fhu$\"3`)>XN*f\"HA\"Fhu7$$\"3')**)*f8%zI=\"Fhu$ \"3[cAB=2ol7Fhu7$$\"36.(Q\\%)omB\"Fhu$\"3um.`&Q7OI\"Fhu7$$\"3uy,q&e\\Y H\"Fhu$\"3$pkr\\H*4V8Fhu7$$\"3Mz()=!ee]N\"Fhu$\"33frK^zb#Q\"Fhu7$$\"3b 9Vygty29Fhu$\"3z_j16Xi:9Fhu7$$\"3y#o(f+^([Y\"Fhu$\"3=kmd5l+]9Fhu7$$\"3 7+N5>N*R_\"Fhu$\"3'=1>vu&3%[\"Fhu7$$\"3E:aumu&>e\"Fhu$\"3(G&f-+&>g^\"F hu7$$\"3$\\F5B7^\"Q;Fhu$\"3y>lGTLgX:Fhu7$$\"3b!\\wa?g1q\"Fhu$\"3;_ik-# \\pd\"Fhu7$$\"3IsHx52#pv\"Fhu$\"3b&o%[>sx.;Fhu7$$\"3DHOafr2<=Fhu$\"39H *\\cMN5j\"Fhu7$$\"3C(G%op`lr=Fhu$\"37[6q5\"3Xl\"Fhu7$$\"3t$oj2r'QJ>Fhu $\"3gC1SSb%)y;Fhu7$$\"33s\\'=[Rw)>Fhu$\"3F0XxQ;\\+hV*p#H=s\"Fhu7$$\"3t^:C\\8//@Fhu$\"36!ecIIF9u\"Fhu7$$\"3 5!GAa0BV;#Fhu$\"3]8Hm3shgUu%R#Fhu$\"3089V*)yl@=Fhu7$$\"3!eLQ.$4qcCFhu$\"3d\\6lI(3 [$=Fhu7$$\"3#4_q-+w?^#Fhu$\"33+J'G\\8a%=Fhu7$$\"3E_Deab1rDFhu$\"3'y&Q> @p_b=Fhu7$$\"3yWKOzhYFEFhu$\"3#\\xWA&H1k=Fhu7$$\"3s***********zo#FhuFa [wF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++++qF^r$\"+++SG8FgrFar-F(6$7$$\"++ ++KMFgr$\"++++/FFgrFjr-F(6$7$$\"++++ScFgr$\"++++36FgrFbs-F(6$7$$\"+++g x>Fgr$\"+++/ZLFhuFdgp7$$\"3R++++++SaFhu$\"36++++++?6Fhu7$$\"3++++ ++g`>Fhu$\"3#***********fl=Fhu7$$\"3q**********z_UFhu$\"36+++++SE>Fhu7 $$\"3))********zClHFhu$\"3-++++?N#*=FhuF_oFaoFinFeo-FI6'7%Fh]xF]^xF`^x F\\pF`pFinFeo-FI6'7$Fe^xFj^xFfpF`pFinFeo-FI6%7SFL7$$\"3%)>J2z(y5dyLFdu$\"3%)[w3!)**>$e%Fdu7$$\" 3A;_V)G8%\\RFdu$\"3WelNgCsp_Fdu7$$\"3X7r;#GLRa%Fdu$\"3-Wp*3SY,'fFdu7$$ \"3$ok:qN119&Fdu$\"35'oJH(zoGmFdu7$$\"31]];WRVedFdu$\"3g&4B3LIgH(Fdu7$ $\"3:\"\\Y=J;fI'Fdu$\"3CaJve!po'yFdu7$$\"3kk'[M@r)Fdu$\"3#[AfrTMg,\"Fhu7$$\"3_u#)*fe(ft$*Fd u$\"3U;m[zVFt5Fhu7$$\"3D86m]4`O**Fdu$\"3`7x&>o?,7\"Fhu7$$\"3c=&4;B>%f5 Fhu$\"3uKw6)RKF<\"Fhu7$$\"3Fhu$\"3'3)yAZ:R$o\"Fhu7$$\"3ww.` ))yL4?Fhu$\"3`GQ+=W^35]SN@Fhu$\"3!)*Qi'G4e^#Fhu$\"3'Qh/<^p0x\"Fhu7 $$\"3]D=1Qz=iAFhu$\"3?OhT*)Q!*)y\"Fhu7$$\"3Kj[ui5`DBFhu$\"3eweyIjM0=Fh u7$$\"3%[2$G/>#=R#Fhu$\"3D5v>wQ+@=Fhu7$$\"35^'y.-JcX#Fhu$\"3QZ#*)[=$fM =Fhu7$$\"3,Y\"GfBL3_#Fhu$\"3!=!*zy#=*p%=Fhu7$$\"3#\\HR/\\Hae#Fhu$\"3qP q?4d!y&=Fhu7$$\"3k5lIskrWEFhu$\"3Y?W'fe`k'=Fhu7$$\"3Un!z+goDr#Fhu$\"3S 4K/#\\h[(=Fhu7$$\"3c&yX[:bJx#Fhu$\"3M&*eCRT.\")=Fhu7$$\"3)=%4FgQjPGFhu $\"3e8gXyyA')=Fhu7$$\"3#GvSW#*>#**GFhu$\"3%>Xnvtq)*)=Fhu7$F`_x$\"3C+++ +?N#*=FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++++yF^r$\"+++!GW\"FgrFar-F( 6$7$$\"++++WNFgr$\"++++oEFgrFjr-F(6$7$$\"++++!o&Fgr$\"++++O5FgrFbs-F(6 $7$$\"+++Se@Fgr$\"+++cQ=FgrFjs-F(6$7$$\"+++!GQ%Fgr$\"+++g4F hu7$$\"3-+++++?>WFhu$\"35+++++g4=Fhu7$$\"33++++SURKFhu$\"3)*********f( p)=FhuF_oFaoFinFeo-FI6'7%FiayFjayF_byF\\pF`pFinFeo-FI6'7$FbbyFebyFfpF` pFinFeo-FI6%7SFL7$$\"3JH\"G+#o25jFau$\"3Q5dxSTN&G*Fau7$$\"3%flgTAF_=\" Fdu$\"3WVAh+_B:>v+=q&Fdu7$$\"3 ]!Rv&Gb`q\\Fdu$\"3Pm(H][u0W'Fdu7$$\"36k@t[&4Zi&Fdu$\"3ey28&)ep`rFdu7$$ \"3Ce>8NLJ-jFdu$\"3%\\1eV9uJ'yFdu7$$\"3rITUL#QH!pFdu$\"3%oVc7cQ!o%)Fdu 7$$\"3)pe]/HmJe(Fdu$\"3'*[*=`c!pE\"*Fdu7$$\"3+C&o7]F.F)Fdu$\"3]go'z&\\ \\k(*Fdu7$$\"3P\"Fhu7$$\"3op[(H9W5;\"Fhu$\"3+x`gKqn\\7Fhu7$$\"3WZk>[EAD7Fhu $\"3Qh;.Fti&H\"Fhu7$$\"3W6$fl,,fH\"Fhu$\"3%>#ft3S$RM\"Fhu7$$\"3In'3!RH Wj8Fhu$\"3qIxB*)3!zQ\"Fhu7$$\"3-9l:ge9M9Fhu$\"3=\"z\"ok4oJ9Fhu7$$\"3)* 46*ecj#*\\\"Fhu$\"3-^V)Q.<+Z\"Fhu7$$\"3y.W&4;)op:Fhu$\"3oCI@'*4Q4:Fhu7 $$\"3@Tx3wV-V;Fhu$\"3U5_****e5[:Fhu7$$\"3\\f*o#=2+2J.=>Fhu$\"3CT')eMi_t;Fhu7$$\"3eZu(4#)3g)>Fhu$\"3 ;P>5)R\"))*p\"Fhu7$$\"3Ye6(>$4bh?Fhu$\"3a!fD$)H\"3Fx\"Fhu7$$\"3s]9t &=axE#Fhu$\"3#pdBN<2/z\"Fhu7$$\"3?>aL\\LhRBFhu$\"3A&=TJa/)3=Fhu7$$\"3h hgn5&)>2CFhu$\"3q&yW\\+6W#=Fhu7$$\"33@xIYnzxCFhu$\"3k7noPh(*Q=Fhu7$$\" 37A+ZHZvYDFhu$\"3%zGR=H1:&=Fhu7$$\"3az=7<)f)=EFhu$\"3a+3m]<$G'=Fhu7$$ \"3)y>bNm-#)o#Fhu$\"3/sQL_W-s=Fhu7$$\"3+c4DbD**eFFhu$\"3iI&\\Si.(z=Fhu 7$$\"3k/>4%oa!HGFhu$\"3or()fq$=c)=Fhu7$$\"3[\"\\r,E%H$*GFhu$\"3+c3'H1w &*)=Fhu7$$\"3cMOm=etmHFhu$\"3mB7'4(4R#*=Fhu7$$\"3b(Gc')>RA.$Fhu$\"3+Fh 0L#oL*=Fhu7$$\"3C&=&=s.(=5$Fhu$\"3![%G$o3EG*=Fhu7$$\"3;d!)R;sHoJFhu$\" 3)>\\FgrFjs-F(6$7$$\" +++gNXFgr$\"+++g*e\"FgrFbt-F(6$7$$\"++gP,NFgr$\"++S=dbFhuFcb t7$$\"3/+++++S]BFhu$\"3@+++++SQ?Fhu7$$\"3;+++++?zXFhu$\"3)***********f *o\"Fhu7$$\"30++++gP4NFhu$\"3+++++S-d=FhuF_oFaoFinFeo-FI6'7%FdezFgezFj ezF\\pF`pFinFeo-FI6'7$F]fzFbfzFfpF`pFinFeo-FI6%7SFL7$$\"3Lg\"**el;)QoF au$\"3\\7L9QRt/5Fdu7$$\"3]'*[=M2*\\G\"Fdu$\"3W'o4aPmS&=Fdu7$$\"3Am$f,b ?v'>Fdu$\"3;$4q*QYz\"y#Fdu7$$\"3K6*)GkhIhEFdu$\"3!y:(\\s&ylo$Fdu7$$\"3 KPAc\"G_#eLFdu$\"3M[![,5mwFdu7$$\"3I*>!G\\@8\\oFdu$\"3eaz`b Tv9%)Fdu7$$\"3s.+;XQK.vFdu$\"37H)**>x.30*Fdu7$$\"3lI1SZ`OW#)Fdu$\"3'=n cBh!*3u*Fdu7$$\"3%Ggol`!3$**)Fdu$\"3FEcZ%zR1/\"Fhu7$$\"3\\$e`)[qv=(*Fd u$\"3[\")pR^')>-6Fhu7$$\"3T6dmiu5Q5Fhu$\"3o`;/O)Hf:\"Fhu7$$\"3&>n'3M=D <6Fhu$\"3IyjNUv:<7Fhu7$$\"3OE[%)>tg%=\"Fhu$\"3=GGEc:ym7Fhu7$$\"31Ct%fh !Hj7Fhu$\"3)HCBs@T>K\"Fhu7$$\"3w!yKh<;KL\"Fhu$\"3/ICKal[o8Fhu7$$\"3MeH ZC;@59Fhu$\"3ueD$fN9rT\"Fhu7$$\"3_K9w@ax$[\"Fhu$\"3[q<@&>p5Y\"Fhu7$$\" 3K9h4uJwg:Fhu$\"3dH_dI8^/:Fhu7$$\"3x`jj)H\\;j\"Fhu$\"3&[(Q>w)[Aa\"Fhu7 $$\"3oI$4,!yG3k!)y\"Fhu$\"3.W2]e<2= ;Fhu7$$\"37_hb(RIw&=Fhu$\"3cvR)od*e[;Fhu7$$\"3%R2r\\FoG$>Fhu$\"3is&)Q; 9Vz;Fhu7$$\"3i.;>`po5?Fhu$\"3rE0]]s+43I!=Fhu7$$\"3'Q1S,HHZR#Fhu$ \"3%*[q)eba;#=Fhu7$$\"3)zED\\:neY#Fhu$\"3)zh8;zQm$=Fhu7$$\"3M\"eX&4ZcV DFhu$\"3A+@Z-M'4&=Fhu7$$\"3ylrIh!zlh#Fhu$\"3-^5;t6]i=Fhu7$$\"3a;g4l3y# p#Fhu$\"3ScnHsIds=Fhu7$$\"3!H2(yZ>9nFFhu$\"3Wpn%yex/)=Fhu7$$\"3)z*oBN& =[%GFhu$\"3a!ea24>n)=Fhu7$$\"3&p)e\"o>S%>HFhu$\"3#yO[x\"py!*=Fhu7$$\"3 cB0=i[`&*HFhu$\"3O`V*3X)*H*=Fhu7$$\"3yY@^Y$f22$Fhu$\"3yx5Gg%pK*=Fhu7$$ \"3k$eNLz^'RJFhu$\"3aHhk56&=*=Fhu7$$\"3%p/*3ZRJ=KFhu$\"35$=3k!eG))=Fhu 7$$\"3[;f)pk!Q)G$Fhu$\"3U.*4!fKO$)=Fhu7$$\"3uIeKi?wiLFhu$\"3Y-QKuXLw=F hu7$$\"3\"\\0&3F&>OV$Fhu$\"3[**\\%p)z!z'=Fhu7$$\"3h********fP4NFhuFjfz F]qFinF_qFeq73F'F2F:FB-F(6$7$$\"*+++S*Fgr$\"+++gr;FgrFar-F(6$7$$\"++++ oPFgr$\"++++'f#FgrFjr-F(6$7$$\"++++gdFgr$\"++++?*)F^rFbs-F(6$7$$\"+++g HDFgr$\"+++/()>FgrFjs-F(6$7$$\"++++#o%Fgr$\"+++Sm9FgrFbt-F(6$7$$\"++?& *\\PFgr$\"++!)30 &Rx$Fhu$\"3#*********zk.=FhuF_oFaoFinFeo-FI6'7%Fai[lFbi[lFgi[lF\\pF`pF inFeo-FI6'7$Fji[lF_j[lFfpF`pFinFeo-FI6%7SFL7$$\"3CK[l$4(*zO(Fau$\"3-Fdu7$$\"3u%\\a\"\\`G@@ Fdu$\"3]Io>w$*e&)HFdu7$$\"3Wa$R!R3HqGFdu$\"3'[Yg)))e\"=&RFdu7$$\"3M)4@ ^!)RJi$Fdu$\"3'f@1U@6-)[Fdu7$$\"3&RnHT#fXFVFdu$\"3([:6Om)f6dFdu7$$\"3m xnS=E)G1&Fdu$\"3$Gb\"o6z&Ha'Fdu7$$\"3[;HL9ZvHeFdu$\"3K#*H@eHZrtFdu7$$ \"3!GKk]^Q-g'Fdu$\"3uD\"*\\an(f;)Fdu7$$\"33yAMhYy)R(Fdu$\"3Gb-j*3w3&*) Fdu7$$\"3ki$y+mNp5)Fdu$\"3\\x!fz<-`h*Fdu7$$\"3[c0D(HE#4*)Fdu$\"3\\`8#z ]TL.\"Fhu7$$\"3')H&)48w\"*>(*Fdu$\"32(H3)H[H-6Fhu7$$\"3/&yj)o5d]5Fhu$ \"3QnvKB[yl6Fhu7$$\"3)Hf.&o.HA6Fhu$\"3<([;N$y%4A\"Fhu7$$\"3Lt\"p[**))z ?\"Fhu$\"3AnpLYb[$G\"Fhu7$$\"3/ofb5D:Fhu$\"3!>os$*47Z[\"Fhu7$$\"3y3=uu8p/;Fhu$\"3kVjCW( \\\"G:Fhu7$$\"3SGpVl]&zo\"Fhu$\"3W(4az+52d\"Fhu7$$\"3?=PGiDfkq;C;u%=Fhu$\"3\"QeYa-IUk\"Fhu7$$\"3Sw@=62fL>Fhu$\" 3Qj!*)))y;(z;Fhu7$$\"3Y+QY3#*p3?Fhu$\"3&y]jc&3G3s?\"ou#Fhu$\"3sB-HDX]y=Fhu7$$\"3rLR`'*o/DGFhu$\"3!=W>52D`)=Fhu7 $$\"3$oH&)oB8m!HFhu$\"31$o(y]X5u=Fhu7$$\"317BNYVTmMFhu$\"3 #*yqQPYVj=Fhu7$$\"3'>c=^,K1a$Fhu$\"3WNL#=qm>&=Fhu7$$\"3%fU$p-fH>OFhu$ \"3bQ$G*ojwP=Fhu7$$\"3[S;NG$3Tp$Fhu$\"39z+I2XHA=Fhu7$$\"3U++++?&Rx$Fhu Fgj[lF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"++++?5Fgr$\"++++'y\"FgrFar-F(6$7$ $\"++++!)QFgr$\"++++gDFgrFjr-F(6$7$$\"+++++eFgr$\"+++++#)F^rFbs-F(6$7$ $\"++++?FFgr$\"++++W?FgrFjs-F(6$7$$\"++++A[Fgr$\"++++S8FgrFbt-F(6$7$$ \"++++#*RFgr$\"++++K;FgrFjtFH-FI6'7(7$Fajv$\"3/+++++++=Fhu7$$\"3#)**** *********z$Fhu$F[jlFhu7$$\"3k*************f&FhuF_jv7$Ffjq$\"39++++++g@ Fhu7$$\"3))************z[FhuFh^s7$$\"3.++++++KSFhu$\"3)************zs \"FhuF_oFaoFinFeo-FI6'7%F^]]lFa]]lFe]]lF\\pF`pFinFeo-FI6'7$Fh]]lF[^]lF fpF`pFinFeo-FI6%7SFL7$$\"3QrPESph(*yFau$\"3aA#\\[U)ec6Fdu7$$\"3'QH6Z3l \\[\"Fdu$\"3(3'Q=rd%)H@Fdu7$$\"3ir)zWl\"QvAFdu$\"3szc.kA%z=$Fdu7$$\"3E 8;_O<&)zIFdu$\"31vH41gY9UFdu7$$\"3)43im)z*)))QFdu$\"37]J\"=2`!)>&Fdu7$ $\"3U9&4%)=Uhk%Fdu$\"3WHhKZvPwgFdu7$$\"3^S\\^Y&[rV&Fdu$\"35AZ()*Q2@&pF du7$$\"3OAbxm`EiiFdu$\"3=N+nX;Z*Fdu7$$\"3MaM?\">OOr)Fdu$\"3H$*4hE r;;5Fhu7$$\"3AC@lM*ovd*Fdu$\"3/xP7d$Rvg]/\"Fhu$\"3Y4 [%4sP9;\"Fhu7$$\"37pA&yM%oH6Fhu$\"3^rh0))p[E7Fhu7$$\"3lpT^.u\"p?\"Fhu$ \"3U*y\"f;es#G\"Fhu7$$\"3wygY#G'>*H\"Fhu$\"3;6I*QXGhM\"Fhu7$$\"3Gy^)H \\7xP\"Fhu$\"3cy-ss*>pR\"Fhu7$$\"3ud6LYWSp9Fhu$\"3AFFhu$ \"3np!fHAE-q\"Fhu7$$\"3id='))[S%z?Fhu$\"3w.l)zU/Kt\"Fhu7$$\"3u6[V**\\- g@Fhu$\"3#)*o\"z[QHfk-<3=Fhu7$$\"3SSvhgx$\\U#Fhu$\"3()QI\\YqBG=Fhu7$$\"3 sDAIMd$*4DFhu$\"30i!GslB]%=Fhu7$$\"3L$o;iz\"=/EFhu$\"3g'[%=bFng=Fhu7$$ \"3#RSb*)Q9()o#Fhu$\"3YEViJh3s=Fhu7$$\"3[JBy\\[xyFFhu$\"3mtJ:=Ja\")=Fh u7$$\"3)yDNn0w,'GFhu$\"3Mr%p+O5x)=Fhu7$$\"3=[!R'=%3*[HFhu$\"3WX1BI\\(= *=Fhu7$$\"3tM.php6KIFhu$\"3q))f=`xO$*=Fhu7$$\"3vYT8/ow=JFhu$\"3)f!4qI# \\C*=Fhu7$$\"3`NqnKa7.KFhu$\"3IgPC\"zN\"*)=Fhu7$$\"3M%R)[_`-\"H$Fhu$\" 3)GzSE?XJ)=Fhu7$$\"3uLE^%oX_P$Fhu$\"3yOSc\"ps\\(=Fhu7$$\"3!e1\\cK#*3Y$ Fhu$\"3'QW?)4L@k=Fhu7$$\"3%z`%eLLJXNFhu$\"3[%R)eo1=^=Fhu7$$\"3FgrFar-F(6$7$Fg\\]l$\"++++CDF grFjr-F(6$7$$\"++++SeFgr$\"++++![(F^rFbs-F(6$7$$\"+++g8HFgr$\"+++W*3#F grFjs-F(6$7$$\"+++gb\\Fgr$\"+++S57FgrFbt-F(6$7$$\"++!ojA%Fgr$\"++?2R:F grFjtFH-FI6'7(7$FhfuF^v7$$\"3$*************>RFhuF[^n7$$\"3o*********** *RcFhuFajq7$$\"3!)**********fpHFhu$\"3++++++g,AFhu7$$\"3P+++++!3-&Fhu$ \"3,+++++S58Fhu7$$\"3'*********zO#G%Fhu$\"3-++++?BJ;FhuF_oFaoFinFeo-FI 6'7%Ff`^lFg`^lFj`^lF\\pF`pFinFeo-FI6'7$F]a^lFba^lFfpF`pFinFeo-FI6%7SFL 7$$\"3I%f%p-]nF%)Fau$\"3lnc4EYCK7Fdu7$$\"3aL;okV<&e\"Fdu$\"3S4Cp_bzmAF du7$$\"3**='>%3n!)HCFdu$\"3i=D?gg&))Q$Fdu7$$\"3G&z6^8#)**G$Fdu$\"3W\"e =ekNXZ%Fdu7$$\"3W#)z1\\=6Fhu$\"3([FNqe\"4=7Fhu7$ $\"3z'>Ukn'=47Fhu$\"3a19*4,MVG\"Fhu7$$\"3i+R^AP&>H\"Fhu$\"34-6XI')HT8F hu7$$\"3-_'o/tI3R\"Fhu$\"3vXKVJ#H^S\"Fhu7$$\"3'R(y^$yW\\Z\"Fhu$\"3]C#H 4;?fX\"Fhu7$$\"3aW8b.*\\Jd\"Fhu$\"3)z`8Gdg7^\"Fhu7$$\"3))[x%R3i.m\"Fhu $\"3#yxd/QGpb\"Fhu7$$\"3YXQ+^,JcFhu$\"3SmqGYnl$o\"Fhu7$$\"35#4` /`Z;.#Fhu$\"3'ysZ6$\\d;E6)py,iy:mx$Fh u$\"3\")**)[#fw)H!=Fhu7$$\"3iU%oB'\\9dQFhu$\"3qj!*3!e\\=y\"Fhu7$$\"3AR /Q,ui[RFhu$\"3AUE&>%)[]v\"Fhu7$$\"3Bn%G>du'HSFhu$\"3+.O*ox)yGX)p\"Fhu7$$\"3oeJ-!Hri>%Fhu$\"35tL4!)f>n;Fhu Fga^lF]qFinF_qFeq73F'F2F:FB-F(6$7$Fdhv$\"+++![,#FgrFar-F(6$7$$\"++++/T Fgr$\"++++)[#FgrFjr-F(6$7$$\"++++!)eFgr$\"++++gnF^rFbs-F(6$7$$\"+++S5J Fgr$\"+++OB@FgrFjs-F(6$7$$\"+++!G3&Fgr$\"+++gx5FgrFbt-F(6$7$$\"++S!>X% Fgr$\"++gXF9FgrFjtFH-FI6'7(7$F[ctF\\[w7$$\"3/++++++SSFhu$\"3%)******** ****>BFhu7$$\"3q************zcFhuFabt7$$\"3))**********R#=$Fhu$\"3%*** ********RIAFhu7$$\"3y**********>b^Fhu$\"3)***********fx6Fhu7$$\"3$**** *****R!R_%Fhu$\"31++++g\\9:FhuF_oFaoFinFeo-FI6'7%F\\d_lF]d_lFbd_lF\\pF `pFinFeo-FI6'7$Fed_lFjd_lFfpF`pFinFeo-FI6%7SFL7$$\"3S?f\"z3q\"e*)Fau$ \"3Z*Gv;H?xI\"Fdu7$$\"3g)=-TnIbo\"Fdu$\"31xkjj(>JS#Fdu7$$\"3'*eyD`xb%e #Fdu$\"3MFKTANL)e$Fdu7$$\"3n/g=8`n+NFdu$\"38y6mH:.KZFdu7$$\"3m\"f@s3tH U%Fdu$\"3__@%*y^m@eFdu7$$\"3)\\UI$*\\npG&Fdu$\"3OpW^g%Q!*y'Fdu7$$\"3C) QC**zK,>'Fdu$\"3Op;(ymCxu(Fdu7$$\"3aC=_tI#G8(Fdu$\"3y#e\"3#eBPp)Fdu7$$ \"3eT:[csp!3)Fdu$\"3=))eIO=?\"f*Fdu7$$\"3]\"Fhu7$$\"3OJc]<@E#>\"Fhu$\"3'Q!)HsazAF\"Fhu7$$\"3Qb\\A*=\\!*G\"Fhu $\"3/')=adZNR8Fhu7$$\"3'RC>.[ktP\"Fhu$\"3+mzF?6q'R\"Fhu7$$\"3Id\"oCP\\ G[\"Fhu$\"3'ySm`%3`g9Fhu7$$\"3MkK8HDcs:Fhu$\"35nRo;?%4^\"Fhu7$$\"3+]i* >7tsn\"Fhu$\"3_by!*eCTl:Fhu7$$\"3!=7O_/H-x\"Fhu$\"3sz\"4*>Q$*4;Fhu7$$ \"3WLdiwTXs=Fhu$\"3oW\"4'fG%[l\"Fhu7$$\"3QmMtAv(*p>Fhu$\"35,%G?QFQp\"F hu7$$\"3%=Hp53n=2#Fhu$\"3\")R1!=.F1t\"Fhu7$$\"3=T#f!fr]l@Fhu$\"39rQL&e !)4w\"Fhu7$$\"3IXQ>T'QlE#Fhu$\"3CIFd`)z+z\"Fhu7$$\"3urWN!3c9P#Fhu$\"3* 349'z7O;=Fhu7$$\"3[a&Q`\"*=FY#Fhu$\"3QuBK%e8g$=Fhu7$$\"3_'H#*=Fhu7$$\"3+H?`Ku$y0$Fhu$\"3M.E@M\"eL*=Fhu7$$\"3 Er$yskn%eJFhu$\"3U65)\\-(=\"*=Fhu7$$\"3w**3bkL?\\KFhu$\"3=40X$e=j)=Fhu 7$$\"3U5hR))*fyM$Fhu$\"31T(=5m\"*y(=Fhu7$$\"377z7jN7SMFhu$\"3)on2\"*o \\q'=Fhu7$$\"3[**[%[HHf`$Fhu$\"3A-*f\\#)[F&=Fhu7$$\"3w;^P/$=*GOFhu$\"3 3%[;S:%)e$=Fhu7$$\"3I(>BkG'\\DPFhu$\"3u90Z\"GG_\"=Fhu7$$\"3xnZJ&)Hr$R\"zF'4=%Fhu$\"3E#HXN\\\"Ht;Fhu7$$\"372N<\"zqXE%Fhu$\"3>eEeyB!*Q;Fh u7$$\"3i(Qj^/?EN%Fhu$\"3/PzTv=t*f\"Fhu7$$\"3%G)y7%R)yNWFhu$\"3c'zJ)))z ')f:Fhu7$F`e_l$\"3%)********f\\9:FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"++ ++g7Fgr$\"+++?H@FgrFar-F(6$7$$\"++++;UFgr$\"++++_CFgrFjr-F(6$7$$\"++++ ?fFgr$\"++++SgF^rFbs-F(6$7$$\"+++S5LFgr$\"+++wX@FgrFjs-F(6$7$$\"+++g._ Fgr$\"*++gT*FgrFbt-F(6$7$$\"++gXnYFgr$\"++SI)H\"FgrFjtFH-FI6'7(7$Fh^sF i]]l7$$\"39++++++gTFhu$\"3!)************zAFhu7$$\"3v************>dFhuF b^s7$$\"33+++++S)R$Fhu$\"3'***********RYAFhu7$$\"3!***********>$G&Fhu$ \"33+++++gT5Fhu7$$\"3o********fXbZFhu$\"3%*********R%*y8FhuF_oFaoFinFe o-FI6'7%Fig`lFjg`lF_h`lF\\pF`pFinFeo-FI6'7$Fbh`lFgh`lFfpF`pFinFeo-FI6% 7SFL7$$\"3Atj*G+,\"*[*Fau$\"3[E7*3a:IQ\"Fdu7$$\"3y]qqKK.'y\"Fdu$\"3%=( >G%=>)QDFdu7$$\"3\"4ry7.K'RFFdu$\"3k#o$Q3uP'y$Fdu7$$\"3(*[.7\\X#>r$Fdu $\"3=eYCz.'p)\\Fdu7$$\"3I+d+)Qj7p%Fdu$\"3.M')pY?YFhFdu7$$\"3hkAKD=14cF du$\"38lqj5_'p8(Fdu7$$\"33D7B:0yolFdu$\"3()*))pw2jU8)Fdu7$$\"3Ay)=4%Qw qvFdu$\"3rDF%49F\\6*Fdu7$$\"3[dY//mXy&)Fdu$\"3-'pR[)Gz?ICp8Fhu$\"3#HA;J4y:R\"Fhu7$$\"3OdmuJ[6j9Fhu$\"3%R#fDJ\"o* [9Fhu7$$\"3+de0U#4_d\"Fhu$\"3mK75iiP7:Fhu7$$\"3Q7@;p\\^q;Fhu$\"3)eu`1I O?c\"Fhu7$$\"3wx-d^Jr\"y\"Fhu$\"3'oF5ck)=:;Fhu7$$\"3e]N(yx'Q!)=Fhu$\"3 )Rp*Q*R`\"e;Fhu7$$\"3Q))3co#[)))>Fhu$\"3p'Qy/MV4q\"Fhu7$$\"3%3:4V#[E#4 #Fhu$\"39W-iN%>vt\"Fhu7$$\"39)f<#fhC+AFhu$\"3_(e=z259x\"Fhu7$$\"3=$zNn ;?%*H#Fhu$\"3o$*RQ(yW()z\"Fhu7$$\"3v)*Qy_lM1CFhu$\"3O:GfMWAC=Fhu7$$\"3 'Gt90-&Hcs#HuUv=Fhu7$$\"3I3bkLqrCGFhu$\"3?$f6yo+`)=Fhu7$$ \"3!\\s`WZ/\"HHFhu$\"3M3b3Jg<\"*=Fhu7$$\"3va69I*4(HIFhu$\"3egC\"4GdL*= Fhu7$$\"3PT*)Qr:*39$Fhu$\"3)QuGl+6=*=Fhu7$$\"3o#fY^Li-C$Fhu$\"3S>#)o** 3#p)=Fhu7$$\"3wxcm(4UdM$Fhu$\"37]9;&*o5y=Fhu7$$\"3)3y^7M92W$Fhu$\"3?Iq >?5(p'=Fhu7$$\"3sldVu6#Qa$Fhu$\"3e$fOT-K9&=Fhu7$$\"3QDq2!***3SOFhu$\"3 M!)[>_!fO$=Fhu7$$\"3pqRA.T))RPFhu$\"3;0&)Qa](=\"=Fhu7$$\"3)\\J-iDnl$QF hu$\"3uJOC\\\"ouy\"Fhu7$$\"3+ZBR]^yORFhu$\"3+#oV-$ooed\\ yxA.%Fhu$\"359KzVc!zs\"Fhu7$$\"3$*eYL?_#)GTFhu$\"3eN9%e.%Q$p\"Fhu7$$\" 3/$o1&HWSBUFhu$\"3-SBn$frhl\"Fhu7$$\"3K8Mv!)HB4VFhu$\"3qoVHnvT>;Fhu7$$ \"3!z3C<(eE1WFhu$\"3cU=o'pUVd\"Fhu7$$\"3cDrN\"*py\"\\%Fhu$\"3c1e*fL99` \"Fhu7$$\"37jUW1Af\"e%Fhu$\"3!eIr%yr)H[\"Fhu7$$\"3Cv<@wy:mYFhu$\"3=5#) **\\b7M9Fhu7$F]i`l$\"3;++++S%*y8FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$Fb\\]l $\"+++gVAFgrFar-F(6$7$$\"++++GVFgr$\"++++;CFgrFjr-F(6$7$$\"++++gfFgr$F \\\\nF^rFbs-F(6$7$$\"+++g8NFgr$\"+++kc@FgrFjs-F(6$7$$\"++++=`Fgr$\"+++ +C!)F^rFbt-F(6$7$$\"++?(=([Fgr$\"++!oF:\"FgrFjtFH-FI6'7(7$F[[rF]h`l7$$ \"3E++++++!G%FhuF]jl7$$\"3!)************fdFhuF[bo7$$\"3$***********f(e(\\Fhu$\"3$*********zsD7FhuF_oFaoFinFeo-FI6'7%Fc[blFd[ blFg[blF\\pF`pFinFeo-FI6'7$Fj[blF_\\blFfpF`pFinFeo-FI6%7SFL7$$\"3Nd/Yb m/-5Fdu$\"3c;m/$\\I\"e9Fdu7$$\"3C6.Bg7o')=Fdu$\"3_/[*[f%*Qn#Fdu7$$\"3e )HmZyE]*GFdu$\"3_j(HeZ!*H)RFdu7$$\"3:N4H@JsBRFdu$\"3&o\"H>;*G$R_Fdu7$$ \"3I-JI56Pg\\Fdu$\"3f$zuFdu7$$ \"3U/lF;#=)[pFdu$\"3mYUmu]L8&)Fdu7$$\"3.Susf$3/,)Fdu$\"3pX&yw%fWE&*Fdu 7$$\"3sU@n0dAy!*Fdu$\"3wvY!\\'3,[5Fhu7$$\"3L*z*fI_e=5Fhu$\"3>>S\"))\\J .9\"Fhu7$$\"3!*>c#Qddo6\"Fhu$\"37RW,=/'o@\"Fhu7$$\"3Y05GD_@G7Fhu$\"3CH tz*3@xH\"Fhu7$$\"3[2d;\">F2M\"Fhu$\"3S(\\JlA]LP\"Fhu7$$\"3g6Xt0$R(\\9F hu$\"3zII7$)G.T9Fhu7$$\"3)**f7;$*p\"\\:Fhu$\"3[;&o&3X8)\\\"Fhu7$$\"3M: I#GPnym\"Fhu$\"39ek/[%32c\"Fhu7$$\"3W!=NHM^(oFhu$\"3.-\\p`\"e;q\" Fhu7$$\"3=$zj:64a5#Fhu$\"3c,(=Qzx=u\"Fhu7$$\"3Kz[&*HVl9AFhu$\"3#y=I%)* \\gv$f`=Fhu7$$\"3u/K5C%y'HOFhu$\"3!zrDg6Md$=Fh u7$$\"3(H0O(pK%ot$Fhu$\"3P+r\"Ru*e7=Fhu7$$\"3m[;HG%=n$QFhu$\"3-lVD=uU( y\"Fhu7$$\"3Ybp*o(=0SRFhu$\"3)R10aJ)odZ5H\"Fhu7$$ \"3u++++?(e(\\Fhu$\"3:++++!GdA\"FhuF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"+++ +?9Fgr$\"++++eBFgrFar-F(6$7$$\"++++SWFgr$\"++++!Q#FgrFjr-F(6$7$$\"++++ +gFgrFf_tFbs-F(6$7$$\"++++?PFgr$\"++++c@FgrFjs-F(6$7$$\"++++EaFgr$\"++ +++mF^rFbt-F(6$7$$\"++++k]Fgr$\"++++?**F^rFjtFH-FI6'7(7$FjfpFd]]l7$$\" 3M+++++++WFhu$\"3;+++++++AFhu7$$\"3#)*************z&FhuF^fp7$$\"3&)*** *********RQFhuF]jl7$F[fz$\"33+++++++wFdu7$$\"3;++++++%=&Fhu$\"31++++++ c5FhuF_oFaoFinFeo-FI6'7%F^_clF__clFd_clF\\pF`pFinFeo-FI6'7$Fg_clFj_clF fpF`pFinFeo-FI6%7SFL7$$\"3k84+blAb5Fdu$\"3:(fWuElI`\"Fdu7$$\"3gggSwRZ( )>Fdu$\"3V\")3uvnM3GFdu7$$\"3/VZ+c#R20$Fdu$\"3Q[tY#[v\"yTFdu7$$\"3$o'Q 23V1OTFdu$\"3%\\%)H@'Q9*[&Fdu7$$\"33(e'*\\%HGI_Fdu$\"3&fg\"=cR5FnFdu7$ $\"3WYcu;?]ciFdu$\"3*)o@c*4Bh\"yFdu7$$\"3z-I1)f5-L(Fdu$\"3Wb>&Q'f(\\)) )Fdu7$$\"3!4>%\\'[.mMW8Fhu7$$\"3JzUS@eP: 9Fhu$\"3`$))QR=z-U\"Fhu7$$\"3w#3W'y\"40`\"Fhu$\"33A4(H*zu([\"Fhu7$$\"3 9INtM\\\\N;Fhu$\"3W'G*R(4NUa\"Fhu7$$\"3'e*3O)z!yg3hp.d0;Fhu7 $$\"3')HKy*)3An=Fhu$\"3ug`\"f>XEl\"Fhu7$$\"3E9LsV)*H\"*>Fhu$\"36:=A@x& =q\"Fhu7$$\"3X\\K,=/H,@Fhu$\"3e&Q?w6>0u\"Fhu7$$\"3zI*)Q!R`?A#Fhu$\"3#f gv[`Hxx\"Fhu7$$\"3&o8Rky\\qL#Fhu$\"3!eu*oB.=3=Fhu7$$\"3Mt?C^-&pX#Fhu$ \"3p^l3,*e[$=Fhu7$$\"3u%Qou;-pc#Fhu$\"37&*y!p)H&[&=Fhu7$$\"3uH0$)*f^_o #Fhu$\"39sn&y*pmr=Fhu7$$\"3U([Jb+=y!GFhu$\"3/nA.V?+%)=Fhu7$$\"3Dt?u]L8 9HFhu$\"3!efgT$)f0*=Fhu7$$\"3TesMi<[GIFhu$\"3=H,N(=^L*=Fhu7$$\"3u_ZKC< ,YJFhu$\"3e&\\l@#*R;*=Fhu7$$\"3)4[e;q7.E$Fhu$\"39#*pU]G`&)=Fhu7$$\"3%G fW*>]=qLFhu$\"3e&eUp4Jb(=Fhu7$$\"3%R\"*3t\"fD\"\\$Fhu$\"3ic?=qS!)f=Fhu 7$$\"3-\"*R>@48*f$Fhu$\"3]Sm$)RzgT=Fhu7$$\"3\\3rgwxE8PFhu$\"3GBr;%z@!= =Fhu7$$\"3$pa@8y'o:QFhu$\"33&>?Fq;Iz\"Fhu7$$\"3h!)=mb#zk#RFhu$\"3a_`** Q=\"=w\"Fhu7$$\"3UYd5BQ_HSFhu$\"3+i@&>%)R)G\\Fhu$\"3uB)eoDBuE\"Fhu7$$\"3HJcpJXD5]Fhu$\"3Q4+\"R:w%*>\" Fhu7$$\"3+bmrh)R^4&Fhu$\"3>Ea@U9rJ6Fhu7$$\"3/,+++++%=&FhuF``clF]qFinF_ qFeq73F'F2F:FB-F(6$7$$\"+++++:Fgr$\"+++SsCFgrFar-F(6$7$$\"++++_XFgr$\" ++++WBFgrFjr-F(6$7$$F`f`lFgr$F\\[]lF^rFbs-F(6$7$$\"+++gHRFgr$\"+++%Q9# FgrFjs-F(6$7$$\"+++gFbFgr$\"++++W^F^rFbt-F(6$7$$\"++!)oU_Fgr$\"+++_r\" )F^rFjtFH-FI6'7(7$FeboF]by7$$\"3d************>XFhuFi]]l7$$\"3&)******* *****ReFhuFiao7$$\"3))**********flSFhu$\"3,+++++g+++++74()FduF_oFaoFin Feo-FI6'7%FebdlFfbdlFibdlF\\pF`pFinFeo-FI6'7$F\\cdlFacdlFfpF`pFinFeo-F I6%7SFL7$$\"36pygz'\\%36Fdu$\"3S1$)Q$)*>yg\"Fdu7$$\"39\"Ro4g5%)3#Fdu$ \"3n6h32lP%Fdu7$$\"3k__%yQT*[VFd u$\"3&)Q$z'Q>TOdFdu7$$\"3\"p%*oHe&)4]&Fdu$\"3a3D9;c(4-(Fdu7$$\"33dz_hJ !=e'Fdu$\"3)R!R,f*)RZ\")Fdu7$$\"3[%\\$fbB#Hr(Fdu$\"35W-B]5A\\#*Fdu7$$ \"3S3ewxgf%*))Fdu$\"3l[(ey=k?.\"Fhu7$$\"3!zeOg))[$35Fhu$\"3T(oxE4t?8\" Fhu7$$\"3j)[*['[\"oJ6Fhu$\"3#4LH#R%yzA\"Fhu7$$\"3*4NlZJ+6C\"Fhu$\"3)G2 o\"Rpo18Fhu7$$\"3SlW\")p[1l8Fhu$\"3-a*R^tI*)Q\"Fhu7$$\"3mf]r@5F!\\\"Fh u$\"3Ux207Z$[Y\"Fhu7$$\"32K?6uP_6;Fhu$\"3F6&o!)G_<`\"Fhu7$$\"3]0f#f*\\ 0AFhu$\"37tcaG8E#p\"Fhu7$$\"3oI661YK'4#Fhu$\"3lB@K5D()Ql&3Y3>-q#Fhu$\"3KLvHg!\\M(=Fhu7$$\"3C'>otehS#GFhu$\"3+a&>!G@D& )=Fhu7$$\"3%*4`tw^<_HFhu$\"3izut2v(>*=Fhu7$$\"3/;VRRe)=Fhu7$$\"3=%4yb-&oBMFhu$\"3-S'eSU\">p=Fhu7$$\"3(RAN\"3,xPNFhu$\"3 -TT^pNW_=Fhu7$$\"3I_2q'3rKm$Fhu$\"3FUb9B_!*G=Fhu7$$\"3=LJH79*[x$Fhu$\" 3$)RJ%oS:M!=Fhu7$$\"3_Il6vsw#*QFhu$\"3Qgq.`&o 6-=UFhu$\"3'GIazP\"Qe;Fhu7$$\"3>$**e**3=pK%Fhu$\"3I5xOiT[6;Fhu7$$\"3i) 3PgNz;V%Fhu$\"3Isw%)4U!>c\"Fhu7$$\"3:[_)eAY%RXFhu$\"3\"z&zT='\\h]\"Fhu 7$$\"3x.(=O.&HTYFhu$\"3Gq:HSs#)[9Fhu7$$\"3y,W8vuPVZFhu$\"3mPeGm&HmQ\"F hu7$$\"3^j\"z$*>XC%[Fhu$\"3n$yH$=RZ@8Fhu7$$\"3)*)GnB1)[J\\Fhu$\"3GDb(R ')Q'e7Fhu7$$\"3#G#QS[]4J]Fhu$\"3g0[Q0SB$=\"Fhu7$$\"31d$y.j!)y6&Fhu$\"3 a64u*=:G6\"Fhu7$$\"3\"*>smR'=z?&Fhu$\"3%yC%H#)etM5Fhu7$$\"3AJu$)3`f\"H &Fhu$\"3esVm&Ho>d*Fdu7$Fgcdl$\"3I,++++74()FduF]qFinF_qFeq73F'F2F:FB-F( 6$7$$\"++++!e\"Fgr$\"+++!oe#FgrFar-F(6$7$$\"++++kYFgr$\"++++3BFgrFjr-F (6$7$$\"++++!3'Fgr$\"++++gJF^rFbs-F(6$7$$\"+++SUTFgr$\"+++;?@FgrFjs-F( 6$7$$\"+++!Gi&Fgr$FbczF^rFbt-F(6$7$$\"++Sy1aFgr$\"+++w$H'F^rFjtFH-FI6' 7(7$Fb^nFcfu7$$\"3o************RYFhuFfu7$$\"3))************zeFhuFjil7$ $\"3a**********R%H%Fhu$\"3))**********R#=#Fhu7$$\"3H+++++?JdFhu$\"39++ ++++cYFdu7$$\"3=++++SyebFhu$\"3\"**********fhr'FduF_oFaoFinFeo-FI6'7%F _felF`felFcfelF\\pF`pFinFeo-FI6'7$FffelF[gelFfpF`pFinFeo-FI6%7SFL7$$\" 39&=y*4frh6Fdu$\"33$)3=gZR#o\"Fdu7$$\"3A$R^ON!\\*=#Fdu$\"3c1k>pXQvIFdu 7$$\"3%ooq3K1JO$Fdu$\"3GV0=WBFkXFdu7$$\"3m+7)*QwMiXFdu$\"3o#Hlu')R6)fF du7$$\"3!o(H5:dYsdFdu$\"3nLE2D))*3J(Fdu7$$\"3DnyDD&G\"3pFdu$\"3A&eCACp JZ)Fdu7$$\"3Mf2(Q==p4)Fdu$\"3RQjzQc51'*Fdu7$$\"3qv*)3!*H.R$*Fdu$\"3ADn '*GDMq5Fhu7$$\"3Y&\\0e2$))e5Fhu$\"3_PM#fb7B<\"Fhu7$$\"3G#zj]3L&)=\"Fhu $\"3wl9$>y/'p7Fhu7$$\"3Fhu$\"3S(*=B&Gb]o\"Fhu7$$\"3%=KF+YcY1#Fhu$\"3%H(z1KH:G-gs/DK#Fhu$\"3@2X^8Qf/= Fhu7$$\"3yOQ_%**e_X#Fhu$\"3+`77T\"=X$=Fhu7$$\"3+*=(**QNZ\"e#Fhu$\"3#*e P[u_=d=Fhu7$$\"3u'yQ!4:%Gr#Fhu$\"39[)GS6#*[(=Fhu7$$\"3UIu)*G_3LGFhu$\" 3[oGQLy!f)=Fhu7$$\"3Kchj&y^A'HFhu$\"3/za\"eDrA*=Fhu7$$\"3S\")4I^:r&4$F hu$\"3Yhh$*Q'RH*=Fhu7$$\"3Iu-@Z-?6KFhu$\"3KcpCLNp))=Fhu7$$\"3@ic3Qm5NL Fhu$\"3cw\")fY\\;z=Fhu7$$\"3!GN.9j,@Y$Fhu$\"3;[)\\YmVS'=Fhu7$$\"3m_0B0 8B&e$Fhu$\"38qG9e\\b->0$[F hu$\"3Zc=bRqdH8Fhu7$$\"3\\kD!pR5I$\\Fhu$\"3AE'HG%z_d7Fhu7$$\"3\\m%4*f/ 1K]Fhu$\"3QieU$[vC=\"Fhu7$$\"3'4**zIu-27&Fhu$\"3*>t.!3wW56Fhu7$$\"3ESA E5EQ>_Fhu$\"3/mos/eSC5Fhu7$$\"3h%>br!>!\\I&Fhu$\"3_2`6sT&RW*Fdu7$$\"3E W,(Qn?JR&Fhu$\"3k%zwyTJ-c)Fdu7$$\"3OipL@!*euaFhu$\"3#3ce%>)=eo(FduF`ge lF]qFinF_qFeq73F'F2F:FB-F(6$7$$\"++++g;Fgr$\"+++?,FFgrFar-F(6$7$$\"+++ +wZFgr$\"++++sAFgrFjr-F(6$7$$\"++++?hFgr$\"++++SCF^rFbs-F(6$7$$\"+++Se VFgr$\"+++'\\3#FgrFjs-F(6$7$$\"+++g6dFgr$\"++++O@F^rFbt-F(6$7$$\"++g8b bFgr$\"+++C)H%F^rFjtFH-FI6'7(7$FdjlFfjq7$$\"3!)************fZFhu$\"33+ +++++!3#Fhu7$$\"3$*************>fFhuFhil7$$\"3w**********REXFhu$\"3&)* *********RM@Fhu7$$\"3Q+++++?FeFhu$\"3!************f8$Fdu7$$\"3A++++g8B dFhu$\"3/+++++k#f%FduF_oFaoFinFeo-FI6'7%FgiflFhiflF]jflF\\pF`pFinFeo-F I6'7$F`jflFejflFfpF`pFinFeo-FI6%7SFL7$$\"3NC(3o7D]@\"Fdu$\"3Gka7<(*yc< Fdu7$$\"3Of\"*=aCr!H#Fdu$\"3\">nPBurz?$Fdu7$$\"3UIk1*Rb(>NFdu$\"3u5zo9 (*=bZFdu7$$\"3U8y&)RjFwZFdu$\"3H(f6,PMLA'Fdu7$$\"3>m9GK+rWgFdu$\"3)=>* 3#*o)of(Fdu7$$\"3rh2hZdXNsFdu$\"3;@)=&4jX$z)Fdu7$$\"3cqv*yxi@[)Fdu$\"3 +kuaM]mb**Fdu7$$\"3Oq.,!3h\\y*Fdu$\"3o'\\VAltw5\"Fhu7$$\"3i/a!*4ye46Fh u$\"3i&\\+\"HxO67Fhu7$$\"3sN()o+HdX7Fhu$\"3#fel'p/y48Fhu7$$\"3ezclyN>m 8Fhu$\"3;Y/t'pY'*Q\"Fhu7$$\"3\"Q^$R9E!G]\"Fhu$\"3))*fCb!o/s9Fhu7$$\"3O +G&fl32k\"Fhu$\"3'H`))p&z\"pa\"Fhu7$$\"3=7')3r;spPo(=Fhu7$$\"3g,&o$zDS()Q\"Fhu7$$\"3\\B.DBg$Q6&F hu$\"3')od2L'*>;6Fhu7$$\"3\"*H*H!H3-7_Fhu$\"3?&)**)p$)\\5.\"Fhu7$$\"3q p%>'R&Fhu$\"3XEyF1NHG&)Fdu 7$$\"3ui+HTd]zaFhu$\"3.@z')RU6JwFdu7$$\"3M_\\I1w%[c&Fhu$\"3Hev*Qg4Nk'F du7$$\"37S^1rC/VcFhu$\"3gvy.v7XpcFdu7$F[[gl$\"3Q)********REf%FduF]qFin F_qFeq73F'F2F:FB-F(6$7$$\"++++S#F^rFjtFH-FI6'7(7$F^vF]^n7$F\\^]lF\\[w7$$\"3'*************ffFh uF_u7$$\"3i**********fhZFhu$\"3))**********ft?Fhu7$$\"3G+++++!o\"fFhu$ \"3')***********Re\"Fdu7$$\"3E++++?fqeFhu$\"3'**********z+N#FduF_oFaoF inFeo-FI6'7%Fd]hlFe]hlFf]hlF\\pF`pFinFeo-FI6'7$Fi]hlF^^hlFfpF`pFinFeo- FI6%7SFL7$$\"39[jz5sPo7Fdu$\"3l(=DN(\\+J=Fdu7$$\"3QzdJAh2#R#Fdu$\"3y;e x1)Q*RLFdu7$$\"3%f^\\T;6nn$Fdu$\"3\"[H]_0!pW\\Fdu7$$\"3'*)>^)o2s!*\\Fd u$\"3!z9U#o@+jkFdu7$$\"3\"z?(QD_qx(*>iC\\7Fhu7$$\"3sp v$3O!z-8Fhu$\"3%*R%e\\2;&[8Fhu7$$\"3P!Qj\\yC!H9Fhu$\"3zY)34\"feG9Fhu7$ $\"3-tI'GZf>d\"Fhu$\"3qSZW,[h5:Fhu7$$\"3tQ&z:&[?;\\%e \"Fhu7$$\"3/;+yTsub=Fhu$\"3w\"fX*Q*)yZ;Fhu7$$\"3I!H'4wqz#)>Fhu$\"39l\\ drkn)p\"Fhu7$$\"3:8\\/?\"GT8#Fhu$\"3#yNM-uu6v\"Fhu7$$\"3e&[SXf:CE#Fhu$ \"3)\\[lWfl*)y\"Fhu7$$\"3q>EPdJf6CFhu$\"3Sm$[RGp`#=Fhu7$$\"3YCdZB9YVDF hu$\"3u5nVld%4&=Fhu7$$\"3-p9+f&)z(o#Fhu$\"3+l%z6ov>(=Fhu7$$\"3\"4?Jh0m Z#GFhu$\"3!**QaQJ/`)=Fhu7$$\"3]])pV.Jq'HFhu$\"39pL)\\h)R#*=Fhu7$$\"3?# \\Hw/hp4$Fhu$\"3%4Ns/jOKFhu$\"3EyX9+e=()=Fhu7$$\" 3/KD@j\"o'zLFhu$\"33v#>\"[wZu=Fhu7$$\"31f\\59iY.NFhu$\"3U&)[bRH%z&=Fhu 7$$\"3w6I%o+$*ej$Fhu$\"3JE7#>++X$=Fhu7$$\"3]PAn+;;rPFhu$\"3'3st%pqL/=F hu7$$\"3Pg!)Hy6$=!RFhu$\"3%3-06YJ\"p=j)>_%Fhu$ \"3(yYGQf=b^\"Fhu7$$\"3MD)=Ht@Ak%Fhu$\"3]\"f()\\M%G[9Fhu7$$\"3G!HO](Q \"Gv%Fhu$\"3!*)4Zk?J1Q\"Fhu7$$\"3m85s5#Qc'[Fhu$\"3y@nD3r]08Fhu7$$\"33* es\"))31t\\Fhu$\"3()\\Y1<[%yA\"Fhu7$$\"3+3'=Sf:B3&Fhu$\"3L6@6_lAU6Fhu7 $$\"35.qA*\\$H%=&Fhu$\"3ws#HBTTd0\"Fhu7$$\"3#Q'\\(eKN^G&Fhu$\"3mB)pKlO Oj*Fdu7$$\"330GsqGb\"Q&Fhu$\"3M,!*R]Tqz')Fdu7$$\"31e]TV=(oY&Fhu$\"3C-L 9^IArxFdu7$$\"3g())3WlU1c&Fhu$\"3;da!4#G$Rp'Fdu7$$\"3]^o1[YuScFhu$\"3E a!Gy%p5*p&Fdu7$$\"3M!>s*3k3AdFhu$\"3N2$**4mBtg%Fdu7$$\"3))f=()Hr(ez&Fh u$\"3)>P.R%4lLNFdu7$Fd^hl$\"3i,++++3]BFduF]qFinF_qFeq7-F'F2F:FBFH-FI6' 7(FMFPFSFPFSFSF_oFaoFinFeo-FI6'7%FMFPFSF\\pF`pFinFeo-FI6'7$FPFSFfpF`pF inFeo-FI6%7UFL7$$\"3!Ggj\")e\"RulFau$\"3%p>$o[1/0>Fdu7$$\"3SFdu$\"3KQc!*[7q)p#Fdu7$$\"3CX`wx0e$ \\#Fdu$\"3U^nxUlGrMFdu7$$\"3fmSSe3(R$QFdu$\"3gvNeBhxK^Fdu7$$\"3kiuL/Un 0_Fdu$\"3oQ3[$)*\\,q'Fdu7$$\"3u%*HI&)zV\"f'Fdu$\"3qH/&o$36d\")Fdu7$$\" 3kUq)oKEI*yFdu$\"3k8klP-n<%*Fdu7$$\"3!f$4@VqDc#*Fdu$\"3ko$fn%\\Hj5Fhu7 $$\"3)eX$)=#y5o5Fhu$\"3-bZo-b^z6Fhu7$$\"3!HzA+[y9@\"Fhu$\"3qpRKOc&fG\" Fhu7$$\"35XN)H*[;Fhu7$$\"3UW`Rt?XP>Fhu$\"3u+\"G\"H'Q7o\"Fhu7$$\"3WK!Rx\"*[ *p?Fhu$\"3AV(\\dzq*HNFhu$\"3[yjv*R!Qb=Fhu7$$\"3)3]p+ZVtk$Fhu$\"3Eq'4=T \">K=Fhu7$$\"3AFhWrpe$y$Fhu$\"3tR&o_zZ7!=Fhu7$$\"3A,x$z*\\[ARFhu$\"3k7 L&fv7Iw\"Fhu7$$\"3AT^*\\\\xj0%Fhu$\"3o54q1\\Fhu$\"3YEpKQ(RvE\"Fhu7$$\"3]]xynb,K]Fhu$\"3yH2*Q^Fhu$\"3c;ulfM/&4\"Fhu7$$\"3)e63nz*4Z_Fhu$\"3#=\\=^e;0* **Fdu7$$\"3_0Z\\c.^Z`Fhu$\"3;R$\\Q5'*[-*Fdu7$$\"3soPZw#*=YaFhu$\"37=)f )y1R(*zFdu7$$\"3%oMq*[J))RbFhu$\"3xON$f`1,%pFdu7$$\"381me#ou@i&Fhu$\"3 I.aZ'Q)ROfFdu7$$\"3DM6pey$=r&Fhu$\"3'QY\"=\"Qi)\\ZFdu7$$\"3#=XpF9rwy&F hu$\"3]*zcJ92ul$Fdu7$$\"3-3C(Q0CQ'eFhu$\"3#)Rl " 0 "" {MPLTEXT 1 0 176 "x01 := (1-t)*x0+t *x1;\nx12 := (1-t)*x1+t*x2;\nx23 := (1-t)*x2+t*x3;\nx012 := (1-t)*x01+ t*x12;\nx123 := (1-t)*x12+t*x23;\nx0123 := (1-t)*x012+t*x123;\ncollect (%,[x0,x1,x2,x3],factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x01G,& *&,&\"\"\"F(%\"tG!\"\"F(%#x0GF(F(*&F)F(%#x1GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x12G,&*&,&\"\"\"F(%\"tG!\"\"F(%#x1GF(F(*&F)F(%#x2GF( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x23G,&*&,&\"\"\"F(%\"tG!\"\"F (%#x2GF(F(*&F)F(%#x3GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%x012G, &*&,&\"\"\"F(%\"tG!\"\"F(,&*&F'F(%#x0GF(F(*&F)F(%#x1GF(F(F(F(*&F)F(,&* &F'F(F/F(F(*&F)F(%#x2GF(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%x 123G,&*&,&\"\"\"F(%\"tG!\"\"F(,&*&F'F(%#x1GF(F(*&F)F(%#x2GF(F(F(F(*&F) F(,&*&F'F(F/F(F(*&F)F(%#x3GF(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&x0123G,&*&,&\"\"\"F(%\"tG!\"\"F(,&*&F'F(,&*&F'F(%#x0GF(F(*&F)F(%#x 1GF(F(F(F(*&F)F(,&*&F'F(F1F(F(*&F)F(%#x2GF(F(F(F(F(F(*&F)F(,&*&F'F(F3F (F(*&F)F(,&*&F'F(F6F(F(*&F)F(%#x3GF(F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&),&!\"\"\"\"\"%\"tGF(\"\"$F(%#x0GF(F'**F*F()F&\"\"# F(F)F(%#x1GF(F(**F*F()F)F.F(F&F(%#x2GF(F'*&)F)F*F(%#x3GF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 283 1 "y" }{TEXT -1 134 " coordinate is obtained in a similar way, so that we have the \+ following parametric equations for the curve traced out by the point U : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = \+ (1-t)^3*x[0]+3*(1-t)^2*t*x[1]+3*(1-t)*t^2*x[2]+t^3*x[3], ``],[y = (1-t )^3*y[0]+3*(1-t)^2*t*y[1]+3*(1-t)*t^2*y[2]+t^3*y[3], ``]);" "6#-%*PIEC EWISEG6$7$/%\"xG,**&,&\"\"\"F,%\"tG!\"\"\"\"$&F(6#\"\"!F,F,**F/F,*$,&F ,F,F-F.\"\"#F,F-F,&F(6#F,F,F,**F/F,,&F,F,F-F.F,F-F6&F(6#F6F,F,*&F-F/&F (6#F/F,F,%!G7$/%\"yG,**&,&F,F,F-F.F/&FC6#F2F,F,**F/F,*$,&F,F,F-F.F6F,F -F,&FC6#F,F,F,**F/F,,&F,F,F-F.F,F-F6&FC6#F6F,F,*&F-F/&FC6#F/F,F,F@" } {TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 267 31 "__ _____________________________" }{TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "From the way t hat the point U sets out from the point A and arrives at the point D, \+ it is intuitively clear that the line segments AB and CD are " }{TEXT 260 10 "tangential" }{TEXT -1 14 " to the curve." }}{PARA 0 "" 0 "" {TEXT -1 29 "We can check this as follows." }}{PARA 0 "" 0 "" {TEXT -1 8 "Firstly," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/ dt = -3*(1-t)^2*x[0]+3*(3*t-1)*(t-1)*x[1]+3*t*(2-3*t)*x[2]+3*t^2*x[3]; " "6#/*&%#dxG\"\"\"%#dtG!\"\",**(\"\"$F&*$,&F&F&%\"tGF(\"\"#F&&%\"xG6# \"\"!F&F(**F+F&,&*&F+F&F.F&F&F&F(F&,&F.F&F&F(F&&F16#F&F&F&**F+F&F.F&,& F/F&*&F+F&F.F&F(F&&F16#F/F&F&*(F+F&*$F.F/F&&F16#F+F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Similarly, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dt = -3*(1-t)^2*y[0]+3*(3*t-1)*(t-1) *y[1]+3*t*(2-3*t)*y[2]+3*t^2*y[3];" "6#/*&%#dyG\"\"\"%#dtG!\"\",**(\" \"$F&*$,&F&F&%\"tGF(\"\"#F&&%\"yG6#\"\"!F&F(**F+F&,&*&F+F&F.F&F&F&F(F& ,&F.F&F&F(F&&F16#F&F&F&**F+F&F.F&,&F/F&*&F+F&F.F&F(F&&F16#F/F&F&*(F+F& *$F.F/F&&F16#F+F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "x01 := (1-t)*x0+t*x1:\nx1 2 := (1-t)*x1+t*x2:\nx23 := (1-t)*x2+t*x3:\nx012 := (1-t)*x01+t*x12:\n x123 := (1-t)*x12+t*x23:\nx0123 := (1-t)*x012+t*x123:\ncollect(%,[x0,x 1,x2,x3],factor);\ndiff(%,t);\ncollect(%,[x0,x1,x2,x3],factor);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&),&!\"\"\"\"\"%\"tGF(\"\"$F(%#x0GF (F'**F*F()F&\"\"#F(F)F(%#x1GF(F(**F*F()F)F.F(F&F(%#x2GF(F'*&)F)F*F(%#x 3GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&),&!\"\"\"\"\"%\"tGF(\" \"#F(%#x0GF(!\"$**\"\"'F(F&F(F)F(%#x1GF(F(*(\"\"$F(F%F(F/F(F(**F.F(F)F (F&F(%#x2GF(F'*(F1F()F)F*F(F3F(F'*(F1F(F5F(%#x3GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&),&!\"\"\"\"\"%\"tGF(\"\"#F(%#x0GF(!\"$**\"\"$F( ,&F)F.F(F'F(F&F(%#x1GF(F(**F.F(F)F(,&F)F.F*F'F(%#x2GF(F'*(F.F()F)F*F(% #x3GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (-3*(1-t)^2*y [0]+3*(3*t-1)*(t-1)*y[1]+3*t*(2-3*t)*y[2]+3*t^2*y[3])/(-3*(1-t)^2*x[0] +3*(3*t-1)*(t-1)*x[1]+3*t*(2-3*t)*x[2]+3*t^2*x[3]);" "6#/*&%#dyG\"\"\" %#dxG!\"\"*&,**(\"\"$F&*$,&F&F&%\"tGF(\"\"#F&&%\"yG6#\"\"!F&F(**F,F&,& *&F,F&F/F&F&F&F(F&,&F/F&F&F(F&&F26#F&F&F&**F,F&F/F&,&F0F&*&F,F&F/F&F(F &&F26#F0F&F&*(F,F&*$F/F0F&&F26#F,F&F&F&,**(F,F&*$,&F&F&F/F(F0F&&%\"xG6 #F4F&F(**F,F&,&*&F,F&F/F&F&F&F(F&,&F/F&F&F(F&&FI6#F&F&F&**F,F&F/F&,&F0 F&*&F,F&F/F&F(F&&FI6#F0F&F&*(F,F&*$F/F0F&&FI6#F,F&F&F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "t=0" "6#/%\"t G\"\"!" }{TEXT -1 9 " we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx= (y[1]-y[0])/(x[1]-x[0])" "6#/*&%#dyG\"\"\"%#dxG! \"\"*&,&&%\"yG6#F&F&&F,6#\"\"!F(F&,&&%\"xG6#F&F&&F36#F0F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 28 "which is the gradient of AB. " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "t=1" "6#/%\"tG \"\"\"" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = (y[3]-y[2])/(x[3]-x[2]);" "6#/*&%#dyG\"\"\"%#dx G!\"\"*&,&&%\"yG6#\"\"$F&&F,6#\"\"#F(F&,&&%\"xG6#F.F&&F46#F1F(F(" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 28 "which is the gradient of CD." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Example " }} {PARA 0 "" 0 "" {TEXT -1 31 "If A, B, C and D are the points" } {XPPEDIT 18 0 " ``(0,0), ``(2,3), ``(5,2)" "6%-%!G6$\"\"!F&-F$6$\"\"# \"\"$-F$6$\"\"&F)" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(6,0)" "6#-%! G6$\"\"'\"\"!" }{TEXT -1 45 " respectively, then the parametric equati ons " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x \+ = (1-t)^3*x[0]+3*(1-t)^2*t*x[1]+3*(1-t)*t^2*x[2]+t^3*x[3], ``],[y = (1 -t)^3*y[0]+3*(1-t)^2*t*y[1]+3*(1-t)*t^2*y[2]+t^3*y[3], ``]);" "6#-%*PI ECEWISEG6$7$/%\"xG,**&,&\"\"\"F,%\"tG!\"\"\"\"$&F(6#\"\"!F,F,**F/F,*$, &F,F,F-F.\"\"#F,F-F,&F(6#F,F,F,**F/F,,&F,F,F-F.F,F-F6&F(6#F6F,F,*&F-F/ &F(6#F/F,F,%!G7$/%\"yG,**&,&F,F,F-F.F/&FC6#F2F,F,**F/F,*$,&F,F,F-F.F6F ,F-F,&FC6#F,F,F,**F/F,,&F,F,F-F.F,F-F6&FC6#F6F,F,*&F-F/&FC6#F/F,F,F@" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "become " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 3*t*(t+1)*(2-t), \+ ``],[y = 3*t*(t-1)*(t-3), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG**\"\"$\"\" \"%\"tGF+,&F,F+F+F+F+,&\"\"#F+F,!\"\"F+%!G7$/%\"yG**F*F+F,F+,&F,F+F+F0 F+,&F,F+F*F0F+F1" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "x01 := (1-t)*x0+t*x1:\nx12 \+ := (1-t)*x1+t*x2:\nx23 := (1-t)*x2+t*x3:\nx012 := (1-t)*x01+t*x12:\nx1 23 := (1-t)*x12+t*x23:\nx0123 := (1-t)*x012+t*x123:\nsubs(\{x0=0,x1=2, x2=5,x3=6\},%);\nx=factor(%);\ny01 := (1-t)*y0+t*y1:\ny12 := (1-t)*y1+ t*y2:\ny23 := (1-t)*y2+t*y3:\ny012 := (1-t)*y01+t*y12:\ny123 := (1-t)* y12+t*y23:\ny0123 := (1-t)*y012+t*y123:\nsubs(\{y0=0,y1=3,y2=2,y3=0\}, %);\ny=factor(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"\"F&% \"tG!\"\"F&,&*&F'F&F%F&\"\"#*&F'F&,&F+F&*&\"\"$F&F'F&F&F&F&F&F&*&F'F&, &*&F%F&F-F&F&*&F'F&,&\"\"&F&F'F&F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,$*(%\"tG\"\"\",&F'F(F(F(F(,&F'F(\"\"#!\"\"F(!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"\"\"F&%\"tG!\"\"F&,&*&F'F&F%F& \"\"$*&F'F&,&F+F&F'F(F&F&F&F&*&F'F&,&*&F%F&F-F&F&*&F'F&,&\"\"#F&*&F3F& F'F&F(F&F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,$*(%\"tG\"\" \",&!\"\"F(F'F(F(,&F'F(\"\"$F*F(F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 29 "The picture shows the points " } {XPPEDIT 18 0 "A(0,0), B(2,3), C(5,2)" "6%-%\"AG6$\"\"!F&-%\"BG6$\"\"# \"\"$-%\"CG6$\"\"&F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D(6,0)" "6#- %\"DG6$\"\"'\"\"!" }{TEXT -1 60 ", together with the curve given by th e parametric equations:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = 3*t*(t+1)*(2-t), ``],[y = 3*t*(t-1)*(t-3), ``]); " "6#-%*PIECEWISEG6$7$/%\"xG**\"\"$\"\"\"%\"tGF+,&F,F+F+F+F+,&\"\"#F+F ,!\"\"F+%!G7$/%\"yG**F*F+F,F+,&F,F+F+F0F+,&F,F+F*F0F+F1" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plot([[[0,0],[2,3],[5,2],[6,0]],\n [3*t*(t+1)*(2-t ),3*t*(t-1)*(t-3),t=0..1]],\n color=[brown,green],thickness =2);" }}{PARA 13 "" 1 "" {GLPLOT2D 334 237 237 {PLOTDATA 2 "6'-%'CURVE SG6$7&7$$\"\"!F)F(7$$\"\"#F)$\"\"$F)7$$\"\"&F)F+7$$\"\"'F)F(-%'COLOURG 6&%$RGBG$\")#)eqk!\")$\"))eqk\"F;F<-F$6$7UF'7$$\"3!Ggj\")e\"Rul!#>$\"3 %p>$o[1/0>FJ7$$\"3SFJ$\"3K Qc!*[7q)p#FJ7$$\"3CX`wx0e$\\#FJ$\"3U^nxUlGrMFJ7$$\"3fmSSe3(R$QFJ$\"3gv NeBhxK^FJ7$$\"3kiuL/Un0_FJ$\"3oQ3[$)*\\,q'FJ7$$\"3u%*HI&)zV\"f'FJ$\"3q H/&o$36d\")FJ7$$\"3kUq)oKEI*yFJ$\"3k8klP-n<%*FJ7$$\"3!f$4@VqDc#*FJ$\"3 ko$fn%\\Hj5!#<7$$\"3)eX$)=#y5o5Fjo$\"3-bZo-b^z6Fjo7$$\"3!HzA+[y9@\"Fjo $\"3qpRKOc&fG\"Fjo7$$\"35XN)H*[;Fjo7$$\"3UW`Rt?XP>Fjo$\"3u+\"G\"H'Q7o\"Fjo7 $$\"3WK!Rx\"*[*p?Fjo$\"3AV(\\dzq*HNFjo$\"3[yjv*R!Qb=Fjo7$$\"3)3]p+ZVtk $Fjo$\"3Eq'4=T\">K=Fjo7$$\"3AFhWrpe$y$Fjo$\"3tR&o_zZ7!=Fjo7$$\"3A,x$z* \\[ARFjo$\"3k7L&fv7Iw\"Fjo7$$\"3AT^*\\\\xj0%Fjo$\"3o54q1\\Fjo$\"3YEpKQ(RvE\"Fjo7$$\"3]]xynb,K]Fjo$\"3yH 2*Q^Fjo$\"3c;ulfM/&4\"Fjo7$$\"3)e63nz*4Z_Fjo$ \"3#=\\=^e;0***FJ7$$\"3_0Z\\c.^Z`Fjo$\"3;R$\\Q5'*[-*FJ7$$\"3soPZw#*=Ya Fjo$\"37=)f)y1R(*zFJ7$$\"3%oMq*[J))RbFjo$\"3xON$f`1,%pFJ7$$\"381me#ou@ i&Fjo$\"3I.aZ'Q)ROfFJ7$$\"3DM6pey$=r&Fjo$\"3'QY\"=\"Qi)\\ZFJ7$$\"3#=Xp F9rwy&Fjo$\"3]*zcJ92ul$FJ7$$\"3-3C(Q0CQ'eFjo$\"3#)Rl " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "General Bezier curves and Bernstein polynomials " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Given a sequence of points" }{XPPEDIT 18 0 " ``(x[0], y[0]),``( x[1], y[1]),` . . . `, ``(x[n], y[n])" "6&-%!G6$&%\"xG6#\"\"!&%\"yG6#F )-F$6$&F'6#\"\"\"&F+6#F1%(~.~.~.~G-F$6$&F'6#%\"nG&F+6#F9" }{TEXT -1 6 ", the " }{TEXT 260 24 "Bezier curve of degree n" }{TEXT -1 49 " is th e curve given by the parametric equations: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = Sum(matrix([[n], [k]])*( 1-t)^(n-k)*t^k*x[k],k = 0 .. n), ``],[y = Sum(matrix([[n], [k]])*(1-t) ^(n-k)*t^k*y[k],k = 0 .. n), ``]);" "6#-%*PIECEWISEG6$7$/%\"xG-%$SumG6 $**-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\"),&F5F5%\"tG!\"\",&F2F5F4F9F5)F8F 4F5&F(6#F4F5/F4;\"\"!F2%!G7$/%\"yG-F*6$**-F.6#7$7#F27#F4F5),&F5F5F8F9, &F2F5F4F9F5)F8F4F5&FD6#F4F5/F4;F@F2FA" }{TEXT -1 13 "------- (i), " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "matrix([[n], [k]]) = n!/(k!*(n-k)!);" "6#/-%'matrixG6#7 $7#%\"nG7#%\"kG*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F.6#,&F)F0F+!\"\"F 0F7" }{TEXT -1 29 " is the binomial coefficient." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In particular, if the " } {TEXT 282 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0],x[1],` . . . `, x[n]" "6&&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 46 " are equally spaced between 0 and 1, that is, " }{XPPEDIT 18 0 "x[k]=k/n" "6#/&%\"xG6#%\"kG*&F'\"\"\"%\"nG!\"\"" }{TEXT -1 6 " for \+ " }{XPPEDIT 18 0 "k=0, ` . . . `, n" "6%/%\"kG\"\"!%(~.~.~.~G%\"nG" } {TEXT -1 6 ", then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(matrix([[n], [k]])*(1-t)^(n-k)*t^k*x[k],k = 0 .. n) = t;" "6#/-%$ SumG6$**-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\"),&F0F0%\"tG!\"\",&F-F0F/F4F 0)F3F/F0&%\"xG6#F/F0/F/;\"\"!F-F3" }{TEXT -1 14 " ------- (ii)." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 19 "___________________" } {TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "(ii) r esembles the basic formula" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(matrix([[n], [k]])*(1-t)^(n-k)*t^k,k = 0 .. n) = 1; " "6#/-%$SumG6$*(-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\"),&F0F0%\"tG!\"\",& F-F0F/F4F0)F3F/F0/F/;\"\"!F-F0" }{TEXT -1 16 " ------- (iii), " }} {PARA 0 "" 0 "" {TEXT -1 50 "which follows directly from the binomial \+ theorem: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^n= Sum(matrix([[n], [k]])*a^(n-k)*b^k,k = 0 .. n)" "6#/),&%\"aG\"\"\"%\"b GF'%\"nG-%$SumG6$*(-%'matrixG6#7$7#F)7#%\"kGF')F&,&F)F'F4!\"\"F')F(F4F '/F4;\"\"!F)" }{TEXT -1 1 "," }}{PARA 258 "" 0 "" {TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "by taking " }{XPPEDIT 18 0 "a=1-x" "6#/% \"aG,&\"\"\"F&%\"xG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b=x" "6# /%\"bG%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 76 "In fact (i i) is a consequence of (iii), as explained in the next subsection." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Explanation and examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 9 "Suppose " }{XPPEDIT 18 0 "x[k]=k/n" "6#/&%\"xG6#%\"kG*&F' \"\"\"%\"nG!\"\"" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "k=0, ` . . . `, n" "6%/%\"kG\"\"!%(~.~.~.~G%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k<>0" "6#0%\"kG\"\"!" }{TEXT -1 7 ", \+ then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[n], [k]])*x[k] = ``(n!/(k!*(n-k)!))*`.`*``(k/n);" "6#/*&-%'matrixG6#7$7#% \"nG7#%\"kG\"\"\"&%\"xG6#F,F-*(-%!G6#*&-%*factorialG6#F*F-*&-F76#F,F-- F76#,&F*F-F,!\"\"F-F?F-%\".GF--F36#*&F,F-F*F?F-" }{TEXT 271 1 " " }} {PARA 258 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = (n-1)!/((k-1)!*(n-k)!);" "6#/%!G*&-%*factorialG6#, &%\"nG\"\"\"F+!\"\"F+*&-F'6#,&%\"kGF+F+F,F+-F'6#,&F*F+F1F,F+F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = matrix([[n-1], [k-1]]);" "6#/%!G-% 'matrixG6#7$7#,&%\"nG\"\"\"F,!\"\"7#,&%\"kGF,F,F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = matrix([[m], [j]]);" "6#/%!G-%'matrixG6#7$7#%\"mG7 #%\"jG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {XPPEDIT 18 0 "m=n-1" "6#/%\"mG,&%\"nG\"\"\"F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "j=k-1" "6#/%\"jG,&%\"kG\"\"\"F'!\"\"" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\" \"!" }{TEXT -1 8 ", then " }{XPPEDIT 18 0 "matrix([[n], [k]])*x[k] =0 " "6#/*&-%'matrixG6#7$7#%\"nG7#%\"kG\"\"\"&%\"xG6#F,F-\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(matrix( [[n], [k]])*(1-t)^(n-k)*t^k*x[k],k = 0 .. n) = Sum(matrix([[m], [j]])* (1-t)^(m-j)*t^(j+1),j = 0 .. m)" "6#/-%$SumG6$**-%'matrixG6#7$7#%\"nG7 #%\"kG\"\"\"),&F0F0%\"tG!\"\",&F-F0F/F4F0)F3F/F0&%\"xG6#F/F0/F/;\"\"!F --F%6$*(-F)6#7$7#%\"mG7#%\"jGF0),&F0F0F3F4,&FDF0FFF4F0)F3,&FFF0F0F0F0/ FF;F " 0 "" {MPLTEXT 1 0 100 "alias(C=binomial):\nn := 4;\nSum(C (n,k)*(1-t)^(n-k)*t^k*k/n,k=0..n);\n``=value(%);\n``=simplify(rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$,$**-%\"CG6$\"\"%%\"kG\"\"\"),&F-F-%\"tG!\"\", &F+F-F,F1F-)F0F,F-F,F-#F-F+/F,;\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,**&),&\"\"\"F)%\"tG!\"\"\"\"$F)F*F)F)*(F,F))F(\"\"#F))F*F/F )F)*(F,F)F(F))F*F,F)F)*$)F*\"\"%F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G%\"tG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "n := 5;\nSum(C(n,k)*(1-t)^(n-k)*t^k*k/n,k=0..n);\n ``=value(%);\n``=simplify(rhs(%));\nn := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6 $,$**-%\"CG6$\"\"&%\"kG\"\"\"),&F-F-%\"tG!\"\",&F+F-F,F1F-)F0F,F-F,F-# F-F+/F,;\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*&),&\"\"\"F) %\"tG!\"\"\"\"%F)F*F)F)*(F,F))F(\"\"$F))F*\"\"#F)F)*(\"\"'F))F(F1F))F* F/F)F)*(F,F)F(F))F*F,F)F)*$)F*\"\"&F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G%\"tG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Us ing (ii), we see that the first equation of (i) becomes " }{XPPEDIT 18 0 "x = t" "6#/%\"xG%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "Hence the parametric equations (i) reduce to the single equatio n:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=Sum(matrix([ [n], [k]])*(1-x)^(n-k)*x^k*y[k],k = 0 .. n)" "6#/%\"yG-%$SumG6$**-%'ma trixG6#7$7#%\"nG7#%\"kG\"\"\"),&F1F1%\"xG!\"\",&F.F1F0F5F1)F4F0F1&F$6# F0F1/F0;\"\"!F." }{TEXT -1 15 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Suppose that the " } {TEXT 281 1 "y" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "y[0],y[1],` . . . `,y[n]" "6&&%\"yG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 52 " are obtained as values when a real valued function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " is applied to the " } {TEXT 277 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0], x[1], ` . . . `, x[n]" "6&&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x[k]=k/n" "6#/&%\"xG6#%\"kG*&F'\"\"\"% \"nG!\"\"" }{TEXT -1 10 " for each " }{TEXT 278 1 "k" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "y[k]=f(k/n)" "6#/ &%\"yG6#%\"kG-%\"fG6#*&F'\"\"\"%\"nG!\"\"" }{TEXT -1 11 ", for each " }{TEXT 280 1 "k" }{TEXT -1 19 ", and (iv) becomes " }{XPPEDIT 18 0 "y= B(n,x)" "6#/%\"yG-%\"BG6$%\"nG%\"xG" }{TEXT -1 9 ", where " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "B(n,x) = Sum(matrix([[n], \+ [k]])*(1-x)^(n-k)*x^k*f(k/n),k = 0 .. n);" "6#/-%\"BG6$%\"nG%\"xG-%$Su mG6$**-%'matrixG6#7$7#F'7#%\"kG\"\"\"),&F4F4F(!\"\",&F'F4F3F7F4)F(F3F4 -%\"fG6#*&F3F4F'F7F4/F3;\"\"!F'" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 274 24 "________________________" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "is ca lled the " }{TEXT 260 20 "Bernstein polynomial" }{TEXT -1 30 " associa ted with the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "An an imation procedure for Bezier curves and Bernstein polynomials: " } {TEXT 0 6 "bezier" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "bezier: usage" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 264 4 " " } {TEXT -1 14 "bezier( pts ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 34 " pts - a list of points " } {XPPEDIT 18 0 "[[x[0], y[0]], [x[1], y[1]], ` . . . `, [x[n], y[n]]]; " "6#7&7$&%\"xG6#\"\"!&%\"yG6#F(7$&F&6#\"\"\"&F*6#F/%(~.~.~.~G7$&F&6#% \"nG&F*6#F6" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{TEXT 260 2 "OR" }{TEXT -1 18 " a list of values " } {XPPEDIT 18 0 "[y[1],y[2],` . . . `,y[n]]" "6#7&&%\"yG6#\"\"\"&F%6#\" \"#%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 63 " In the 2nd case, the x values are taken to be " } {XPPEDIT 18 0 "x[0]=0, x[1]=1/n, x[2]=2/n,` . . . `,x[n]=1" "6'/&%\"xG 6#\"\"!F'/&F%6#\"\"\"*&F+F+%\"nG!\"\"/&F%6#\"\"#*&F2F+F-F.%(~.~.~.~G/& F%6#F-F+" }{TEXT -1 9 ". \n " }}{PARA 257 "" 0 "" {TEXT -1 12 "De scription:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "bezier" }{TEXT -1 187 " provides an animation which shows how a Bezier curve determined by a list of poin ts is related to these points, or, simply plots the points together wi th the corresponding Bezier curve.\n" }}{PARA 0 "" 0 "" {TEXT 265 8 "O ptions:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 18 "animate=true/ false" }}{PARA 0 "" 0 "" {TEXT -1 54 "With the option \"animate=true\" , which is the default, " }{TEXT 0 6 "bezier" }{TEXT -1 85 " construct s an animation to show how the Bezier curve is related to the input po ints." }}{PARA 0 "" 0 "" {TEXT -1 33 "With the option \"animate=false \", " }{TEXT 0 6 "bezier" }{TEXT -1 102 " just plots the input points \+ together with the corresponding Bezier curve determined by these point s. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "fr ames=n\nThe number of frames for the animation. The default value of \+ \"frames\" is 40." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "color,colour = " }{XPPEDIT 18 0 "[clr[1], clr[2], ` . . . `, clr[n]];" "6#7&&%$clrG6#\"\"\"&F%6#\"\"#%(~.~.~.~G&F%6#%\"nG" } {TEXT -1 2 "\n " }{XPPEDIT 18 0 "clr[1],` . . . `,clr[n];" "6%&%$clrG6 #\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 82 " are the colours for the va rious line segments used to construct the Bezier curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "pointcolor, pointcol our=clr" }}{PARA 0 "" 0 "" {TEXT -1 61 "The colour for the points. The default is \"pointcolor=black\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "curvecolor, curvecolour=clr" }}{PARA 0 " " 0 "" {TEXT -1 60 "The colour for the curve. The default is \"curveco lor=green\"." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 29 "With the \+ option \"info=true\", " }{TEXT 0 6 "bezier" }{TEXT -1 32 " gives the p arametric equations " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x=p(t),``],[y=q(t),``])" "6#-%*PIECEWISEG6$7$/%\"xG-%\"p G6#%\"tG%!G7$/%\"yG-%\"qG6#F,F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "for the Bezier curve, whe n the input is a list of points, and the cartesian equation " } {XPPEDIT 18 0 "y=p(x)" "6#/%\"yG-%\"pG6#%\"xG" }{TEXT -1 50 " of the c urve, when the input is a list of values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Ot her options for plot can be used just as with the plot command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 154 "To make the procedure active open the subsection, place \+ the cursor anywhere after the prompt [ > and press [Enter].\nYou can \+ then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 " bezier: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4499 "bezier := proc(pts::list)\n loc al m,n,i,j,k,ct,t,frms,p,pp,thick,Options,temp,clr,crv,links,\n \+ u,v,pt,ps,lns,mn,prntflg,x,y,xvals,ptclr,crvclr,anmt,pnts;\n\n pp \+ := pts;\n m := nops(pp);\n mn := m-1;\n if m<3 then\n error \"the 1st argument, %1, is invalid .. it should be a list of at least 3 points, or real values\",pts;\n end if;\n pnts := type(pp[1],li st);\n if not pnts then\n xvals := [seq(k/mn,k=0..mn)];\n \+ pp := zip((x,y)->[x,y],xvals,pp);\n end if;\n p := evalf(pp);\n \+ if pnts then\n for i to m do\n if nops(pp[i])<>2 then\n \+ error \"the %-1 point must have two components\",i;\n \+ end if; \n if not type(p[i,1],float) or not type(p[i,2],floa t) then \n error \"both components of the %-1 point must ev aluate to a float\",i;\n end if;\n end do;\n else\n \+ for i to m do\n if not type(p[i,2],float) then \n \+ error \"the %-1 value must evaluate to a float\",i;\n end if; \n end do;\n end if;\n\n # Get the options we need to proces s.\n Options := [];\n n := 40;\n thick := 2;\n anmt := true;\n prntflg := false;\n clr := [brown,coral,COLOR(RGB,.4,0,.9),magent a,red,blue,tan,khaki];\n if m>9 then\n clr := [op(clr),seq(COLO R(RGB,rand()/10^12,rand()/10^12,rand()/10^12),i=1..m-9)];\n end if; \n crvclr := green;\n ptclr := black;\n if nargs >= 2 then\n \+ Options := [args[2..nargs]];\n if not type(Options,list(equatio n)) then\n error \"each optional argument must be an equation \"\n end if;\n if hasoption(Options,'animate','anmt','Option s') then\n if anmt<>true then anmt := false end if;\n end if;\n if hasoption(Options,'frames','n','Options') then\n \+ if not type(n,posint) or n < 2 then\n error \"\\\"frames \\\" must be an integer greater than 1\"\n end if;\n end \+ if;\n if hasoption(Options,'thickness','thick','Options') then\n \+ if not type(thick,posint) then\n error \"\\\"thickn ess\\\" must be a positive integer\"\n end if;\n end if; \n if hasoption(Options,'pointcolor','temp','Options') or \n \+ hasoption(Options,'pointcolour','temp','Options') then\n p tclr := temp;\n end if;\n if hasoption(Options,'curvecolor', 'temp','Options') or \n hasoption(Options,'curvecolour','temp' ,'Options') then\n crvclr := temp;\n end if;\n if ha soption(Options,'color','temp','Options') or \n hasoption(Opti ons,'colour','temp','Options') then\n if type(temp,list) then \n for i to min(nops(temp),mn) do\n clr[i] := temp[i]\n end do;\n else\n clr := [temp ,op(clr)];\n end if;\n end if;\n if hasoption(Option s,'info','prntflg','Options') then\n if prntflg<>true then prn tflg := false end if;\n end if;\n end if;\n clr := [ptclr,op( clr)];\n \n if prntflg then\n if pnts then \n print(x =sort(simplify(add(binomial(mn,j)*\n pp[j+1,1]*t^ j*(1-t)^(mn-j),j=0..mn))));\n print(y=sort(simplify(add(binomi al(mn,j)*\n pp[j+1,2]*t^j*(1-t)^(mn-j),j=0..mn))) );\n else\n print(y=sort(simplify(add(binomial(mn,j)*\n \+ pp[j+1,2]*x^j*(1-x)^(mn-j),j=0..mn))));\n end if;\n end if;\n crv := plot([add(binomial(mn,j)*p[j+1,1]*t^j*(1-t )^(mn-j),j=0..mn),\n add(binomial(mn,j)*p[j+1,2]*t^j*(1-t)^(mn- j),j=0..mn),t=0..1],\n color=crvclr,thickness=thick);\n\n i f anmt then\n frms := NULL; \n for ct from 0 to n do\n \+ u := evalf(ct/n);\n v := 1-u;\n\n for i to mn do\n \+ for k from 1 to m-i do\n pt[i,k] := [add(bino mial(i,j)*p[j+k,1]*u^j*v^(i-j),j=0..i),\n add(binomial (i,j)*p[j+k,2]*u^j*v^(i-j),j=0..i)];\n end do;\n en d do;\n\n ps := NULL;\n for i from 1 to mn do\n \+ ps := ps,seq(pt[i,j],j=1..m-i)\n end do;\n\n lns \+ := p;\n for i from 1 to mn-1 do\n lns := lns,[seq(p t[i,j],j=1..m-i)]\n end do; \n links := plot([[ps ],lns],\n color=clr,thickness=thick,\n styl e=[POINT,LINE$mn],symbol=CIRCLE);\n frms := frms,plots[display ]([crv,links]);\n end do;\n return plots[display]([frms],ins equence=true,op(Options));\n else\n links := plot([p,p],style=[ POINT,LINE],\n symbol=CIRCLE,color=clr,thickness=thick,line style=2);\n return plots[display]([crv,links]);\n end if; \ne nd proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 0 6 "bezier" }{TEXT -1 39 ": examples wi th input a list of points " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "The curves are drawn with the opti on \"" }{TEXT 279 13 "animate=false" }{TEXT -1 23 "\", to conserve mem ory'." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "3 point example" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "pts := [[0,0],[2,3],[6,0]];\nbezier(pts,info=true,animate=false); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7%7$\"\"!F'7$\"\"#\"\"$7$\" \"'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,&*&\"\"#\"\"\")%\"tGF' F(F(*&\"\"%F(F*F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,$*(\"\"' \"\"\",&%\"tGF(F(!\"\"F(F*F(F+" }}{PARA 13 "" 1 "" {GLPLOT2D 580 344 344 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3e`(>E&[)Q\"))!#>$\" 3)Q29UHA$z7!#=7$$\"3%=Jd,oTPm\"F0$\"3Ak!G?UogM#F07$$\"3Q!>ZMFo2c#F0$\" 3wF%eY?qT\\$F07$$\"3)>#[[b*=@[$F0$\"3sLba3$[Zf%F07$$\"3([!fY'4)R%[Bv\")H@(F0$\"3DRZX0*4aK)F07$$\"3J.$=1_-'3#)F 0$\"3\\$3X1N[\\9*F07$$\"3y$R5:V$Q^#*F0$\"3o<)oalc>$**F07$$\"3>3/!R=a&= 5!#<$\"3Yv()zvC\"y0\"F[o7$$\"3'>Ni0=va7\"F[o$\"3bVHJV]GD6F[o7$$\"3)G^s r]2ZB\"F[o$\"3qhC[`xT(=\"F[o7$$\"3)>,C\"F[o7$$ \"3P;ZZb5YS9F[o$\"3Q]e#)\\u(pG\"F[o7$$\"3[w`\"=95(f:F[o$\"3jqQb4s)[L\" F[o7$$\"3-cYNE#>Bm\"F[o$\"3#=.O4m/2P\"F[o7$$\"3KT/(HJ7Ny\"F[o$\"37w'Qt )e'oS\"F[o7$$\"3Q()\\S&oMC*=F[o$\"3LP]yLo$RV\"F[o7$$\"3*pkq*[!=P,#F[o$ \"3?f!Q$zJIe9F[o7$$\"3C\\xE]z$48#F[o$\"3__n%zKf7s\\\" F[o7$$\"3kGYbbwzZFF[o$\"3I9h3([Q+\\\"F[o7$$\"3%eH_a'\\SwGF[o$\"3'>6VhM DtZ\"F[o7$$\"3P_vN6Q26IF[o$\"3sUt#fpw(e9F[o7$$\"3;VvcOgfWJF[o$\"3Uotzi `IN9F[o7$$\"3y1]V(pcaF$F[o$\"3py\\%R^#e29F[o7$$\"3a,t&Qi!oAMF[o$\"3A%4 GffC5P\"F[o7$$\"3IjV'=4'pcNF[o$\"3E4pS95*HL\"F[o7$$\"3K-X1$)>f,PF[o$\" 3e\"\\1eskpG\"F[o7$$\"3#Qy]Cy*\\MQF[o$\"3Q[w*)3jUS7F[o7$$\"3o/H)3Gb:)R F[o$\"3![G^B-\"H%=\"F[o7$$\"3d8Yw$G%f@TF[o$\"3'e:c\\Pek7\"F[o7$$\"3?.q .*\\2(pUF[o$\"35()*QTF531\"F[o7$$\"3@Qq>toA;WF[o$\"3'e#))3aN#\\\"**F07 $$\"3g%ynGLK9d%F[o$\"3=JkYw:$[8*F07$$\"3*euES4nEs%F[o$\"3\"yf(>zl!4L)F 07$$\"3ym6`we5z[F[o$\"3m/]1z_RbuF07$$\"3,+9z'HEg.&F[o$\"3*))*zveF#Q`'F 07$$\"3]bo$3/!y\"=&F[o$\"3%yI%*[2)=ScF07$$\"3g#\\`E&zm]`F[o$\"3'\\?&R& p^3c%F07$$\"3Og)e<<'Q.bF[o$\"3C4UBZYfXNF07$$\"3/')*H;?Xzm&F[o$\"3B5/g8 AB6CF07$$\"3jgQ6=57FeF[o$\"3?wT3%zZ`F\"F07$$\"\"'F)F(-%'COLOURG6&%$RGB GF($\"*++++\"!\")F(-%*THICKNESSG6#\"\"#-F$6(7%F'7$$F_[lF)$\"\"$F)Fbz-F fz6&FhzF)F)F)-%&STYLEG6#%&POINTGF\\[l-%*LINESTYLEGF^[l-%'SYMBOLG6#%'CI RCLEG-F$6(Fb[l-Ffz6&Fhz$\")#)eqkF[[l$\"))eqk\"F[[lFi\\l-Fj[l6#%%LINEGF \\[lF]\\lF_\\l-%+AXESLABELSG6%Q!6\"Fa]l-%%FONTG6#%(DEFAULTG-%%VIEWG6$F f]lFf]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "4 point example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "pts := [[0,0],[2,3],[5,2],[6 ,0]];\nbezier(pts,animate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$ptsG7&7$\"\"!F'7$\"\"#\"\"$7$\"\"&F)7$\"\"'F'" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$\"\"!F)F(7$$\"3! Ggj\")e\"Rul!#>$\"3av\"RE&Hdm'*F-7$$\"3(*=zjU?x@8!#=$\"3&>>$o[1/0>F37$ $\"3&oh03wJ_!>F3$\"3yPc!*[7q)p#F37$$\"3'\\Mlxd!e$\\#F3$\"3U^nxUlGrMF37 $$\"3fmSSe3(R$QF3$\"3\\uNeBhxK^F37$$\"3kiuL/Un0_F3$\"3dP3[$)*\\,q'F37$ $\"3i$*HI&)zV\"f'F3$\"3qH/&o$36d\")F37$$\"3_Tq)oKEI*yF3$\"3k8klP-n<%*F 37$$\"3yM4@VqDc#*F3$\"3?o$fn%\\Hj5!#<7$$\"35cM)=#y5o5FY$\"3CbZo-b^z6FY 7$$\"3!HzA+[y9@\"FY$\"3qpRKOc&fG\"FY7$$\"35XN)H*[;FY7$$\"3UW`Rt?XP>FY$\"3u+\"G\"H' Q7o\"FY7$$\"3WK!Rx\"*[*p?FY$\"3WV(\\dzq*HNFY$\"3[yjv*R!Qb=FY7$$\"3)3]p+ZVtk$FY$\"3#)p'4=T\">K =FY7$$\"35GhWrpe$y$FY$\"3SS&o_zZ7!=FY7$$\"3A,x$z*\\[ARFY$\"3k7L&fv7Iw \"FY7$$\"3AT^*\\\\xj0%FY$\"3Y54q1\\FY$\"3-EpKQ(R vE\"FY7$$\"3]]xynb,K]FY$\"3yH2*Q^FY$\"3c;ulfM/ &4\"FY7$$\"3+:\"3nz*4Z_FY$\"3/%\\=^e;0***F37$$\"3_0Z\\c.^Z`FY$\"3;R$\\ Q5'*[-*F37$$\"3soPZw#*=YaFY$\"37=)f)y1R(*zF37$$\"3&fMq*[J))RbFY$\"3xON $f`1,%pF37$$\"381me#ou@i&FY$\"3I.aZ'Q)ROfF37$$\"3DM6pey$=r&FY$\"3'QY\" =\"Qi)\\ZF37$$\"3%4XpF9rwy&FY$\"3%*)zcJ92ul$F37$$\"3-3C(Q0CQ'eFY$\"3#) Rl " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "4 point example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "pts := [[0,0 ],[5,1],[2,4],[7,3]];\nbezier(pts,animate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7&7$\"\"!F'7$\"\"&\"\"\"7$\"\"#\"\"%7$\"\"(\"\"$ " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6% 7S7$$\"\"!F)F(7$$\"31)>$))*G-s:$!#=$\"3)*y$eF5-!=o!#>7$$\"3QYAP$)RYEdF -$\"35!p!GCW^=8F-7$$\"3_8ib:\\uE%)F-$\"3uL\"3V?/(z?F-7$$\"3CC%4Mlz^4\" !#<$\"3sD\\<@ez!*GF-7$$\"3)3/FOw&=G8F>$\"3t()fNkA>RPF-7$$\"3eG8;P;sG:F >$\"3#[3CDA!QfXF-7$$\"3&3\"H7$zu9s\"F>$\"3#=(=<0,mRaF-7$$\"3$Q`!*ebcd! >F>$\"3W=\">k&)y*zjF-7$$\"3S:h+,]5v?F>$\"3dee?W;>WtF-7$$\"30-O)R'zJNAF >$\"3!4!4n%QW/O)F-7$$\"3#*o'HC_maO#F>$\"3=<,#3QsJF*F-7$$\"3A>'Rm3q0]#F >$\"3^9\"zpH8<.\"F>7$$\"3)G1FN=+]i#F>$\"3L#3cQ9Sz8\"F>7$$\"3N!yc)Qh$\"3E*o!HRuHT7F>7$$\"3G&H,PP!pFGF>$\"3fkINFvrN8F>7$$\"3f8*4wp&HHHF>$ \"3k<26H#Q$[9F>7$$\"3py`ron**3IF>$\"3$*[(\\M-[Pa\"F>7$$\"3eEHtT7H&4$F> $\"31;EA6rQa;F>7$$\"3!Qn=KF@n;$F>$\"3A;0c<2r^7$$\"3euc'>CS1C$F>$\"3 ]E]Q8>\\d=F>7$$\"3X&\\)4gTb2LF>$\"3e:eHZ$Hp&>F>7$$\"33'RM$\"3% Q%G2-i**e?F>7$$\"3-Un#[^yYV$F>$\"3Gf'Gj'*44:#F>7$$\"31VfKH#>&)\\$F>$\" 3C\"\\/n6yxC#F>7$$\"3?wi'4t@[c$F>$\"3?wko&\\QbM#F>7$$\"3jW\"p)e4KBOF>$ \"3*=+kK@rzU#F>7$$\"3#zYhDZS\")o$F>$\"3dbAk@Q!R^#F>7$$\"3#\\ro0Ayxv$F> $\"3)HSv3$4(*)f#F>7$$\"3kl!R3qH%HQF>$\"3/%GSPD3#yEF>7$$\"3B0))\\]:\"H! RF>$\"3qh&))\\h43v#F>7$$\"3'z(3S(Gq-*RF>$\"3xmCG6#pj#GF>7$$\"3YaQ$)=c \"[2%F>$\"3!)edEG>S*)GF>7$$\"3I\"G!Q)4OB<%F>$\"3yR\"[pGv7&HF>7$$\"3?7t XC;'zE%F>$\"3(pXbGJ/A+$F>7$$\"3AKH#4\"yF\"Q%F>$\"3qU?R_`'>0$F>7$$\"3+M UC[K3(\\%F>$\"3-9'*e_e$G4$F>7$$\"3e5BpT$=&GYF>$\"3K,0&*\\Z1HJF>7$$\"3k qOh%zF\"oZF>$\"3u9=T4BvdJF>7$$\"3d2RHm*op#\\F>$\"3SI7H:&4.=$F>7$$\"33$ ea#z\")4$4&F>$\"3O6%*)HuuW>$F>7$$\"3qK_EPPBx_F>$\"3!y`(>u*\\4?$F>7$$\" 3;?U`La'\\Z&F>$\"3E#p:[rN!*>$F>7$$\"3cw1>^WqqcF>$\"3#=@t,!*p'*=$F>7$$ \"3m5LGRX]7fF>$\"3ooAj.7jpJF>7$$\"3A?3'[\"))QXhF>$\"3x-*QbH$)H9$F>7$$ \"3eZb.)zY=T'F>$\"3$\\\")>Ypo^5$F>7$$\"3+\"z)4TyC&o'F>$\"3EP!RB-!efIF> 7$$\"\"(F)$\"\"$F)-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESSG6# \"\"#-F$6(7&F'7$$\"\"&F)$\"\"\"F)7$$Fa[lF)$\"\"%F)Fbz-Fhz6&FjzF)F)F)-% &STYLEG6#%&POINTGF^[l-%*LINESTYLEGF`[l-%'SYMBOLG6#%'CIRCLEG-F$6(Fd[l-F hz6&Fjz$\")#)eqkF][l$\"))eqk\"F][lF`]l-Fa\\l6#%%LINEGF^[lFd\\lFf\\l-%+ AXESLABELSG6%Q!6\"Fh]l%(DEFAULTG-%%VIEWG6$Fj]lFj]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "5 point example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "pts := [[0,0],[1,4],[3,6],[4,5],[6,0]];\nbezier(pts,info=true,an imate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7'7$\"\"!F'7$ \"\"\"\"\"%7$\"\"$\"\"'7$F*\"\"&7$F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,**$)%\"tG\"\"%\"\"\"F)*&\"\")F*)F(\"\"$F*!\"\"*&\"\"'F*)F (\"\"#F*F**&F)F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,(*$)% \"tG\"\"$\"\"\"!\"%*&\"#7F*)F(\"\"#F*!\"\"*&\"#;F*F(F*F*" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F( 7$$\"3]/,Xylt&**)!#>$\"3l*3)33l6IM!#=7$$\"3aDxFZw*[s\"F0$\"39l>wXs$*>j F07$$\"3)\\mfOLEkp#F0$\"3/&*yiBhUi%*F07$$\"3&**RsM`$p;PF0$\"3cq/>YD'3D \"!#<7$$\"3%*='pnQL-x%F0$\"3%z+,#zE8U:F@7$$\"3!\\0809`wx&F0$\"3tvP]acO ,=F@7$$\"3O:'>FD7([oF0$\"3'f%*[$*y>'e?F@7$$\"3EiWs'3?J)zF0$\"3p0#*H_R] 7BF@7$$\"3.rBYNc#y8*F0$\"3/Yd`[].`DF@7$$\"3)4%*oU_UZ.\"F@$\"3AyI57-:(y #F@7$$\"3JM@cwQ)G9\"F@$\"3Uv!=+)>*>)HF@7$$\"3CJ$=/XrhE\"F@$\"37ER::*y$ )=$F@7$$\"3Cs(f<$)p8R\"F@$\"31>1>I'p:Q$F@7$$\"3V))3Z0c88:F@$\"3'epdM&> ?aNF@7$$\"3(3T\"=1c\\C;F@$\"3cVw=QqE*p$F@7$$\"357K1#o,xv\"F@$\"3ymu@jQ .dQF@7$$\"37&=ygDy5(=F@$\"3Y%ei9&3(z(RF@7$$\"33N,T#QJM+#F@$\"34[:!3dWP 5%F@7$$\"3f\\6)ob)*37#F@$\"3lwP8D3],UF@7$$\"3Md#*G*G7+D#F@$\"3#)3:C%Qj QH%F@7$$\"3sUG-Xg5tBF@$\"3i%)=(R'y6nVF@7$$\"3$3Sn^VF;]#F@$\"3O'=R#4!R! GWF@7$$\"3Q90%Qx&o>EF@$\"3Is(e4yI)pWF@7$$\"3aP-]i%Qqu#F@$\"3g*\\Xb--&* \\%F@7$$\"3mUP/-RKzGF@$\"3r!3=Fc^J^%F@7$$\"3_O))*p<)[%*HF@$\"3\"eIAt0 \"f5XF@7$$\"3P9'eT(Q!*=JF@$\"3%e8V;A@D\\%F@7$$\"3Q=l'y_@vC$F@$\"3u\\)* Q14%pX%F@7$$\"3;(>4:?'\\tLF@$\"3%)Q)>T3w_S%F@7$$\"3?p4$y')*f&\\$F@$\"3 +%)*G4\\4#RVF@7$$\"3.XR[w*zRt$F@7$$\"3ma6#Q6%eiUF@$ \"3#>\\f`w_nc$F@7$$\"3Qjd+\")Gu%R%F@$\"3qLw3BpwtLF@7$$\"3*)*GX)QaXDXF@ $\"3-j:.zj!p;$F@7$$\"3_r:!R;\\Tm%F@$\"3g-_LCD&3$HF@7$$\"3]))f]aEy*z%F@ $\"3k+B6BpV%o#F@7$$\"3!GGh^nk3%\\F@$\"3?gBQ1d'GT#F@7$$\"3=qMGj0Y$3&F@$ \"3xFfH\\(pP7#F@7$$\"3e+:3)REr@&F@$\"3AB1U$)zcS=F@7$$\"3+:G(Re!yt`F@$ \"31nNptC3&\\\"F@7$$\"3#Gjhx@at^&F@$\"3YC " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "5 point example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "pts := [[0,0],[5,2],[1,5],[4,6],[6,0]];\nbezier(pts,animate=fals e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7'7$\"\"!F'7$\"\"&\"\"#7 $\"\"\"F)7$\"\"%\"\"'7$F/F'" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7Z7$$\"\"!F)F(7$$\"3[%RKp+U%4T!#=$\"3y[hRL? .rF8$\" 3#z(=/Yt&4D\"F87$$\"3P:(=g!fu4@F8$\"3.&f`SJE6W\"F87$$\"3v\"=Rll]WC#F8$ \"3ZO?86RRI;F87$$\"3aT(*)>G@:O#F8$\"3SlC3YD(R#=F87$$\"3'4UT=Xu([CF8$\" 3soG&Hj')G*>F87$$\"3#po;$yCeJDF8$\"3#QdL9720=#F87$$\"3pmf)>v\"p+EF8$\" 3i0#*Hf;QlBF87$$\"3#G[#=qx>cEF8$\"3O%=!GHsSRDF87$$\"3A8`\\H\\)))p#F8$ \"3_\"H*oX8E$p#F87$$\"3mQM%[d@@u#F8$\"3!3\\wNR#>qGF87$$\"3W-p(zyVRx#F8 $\"3mT*=*)QgU,$F87$$\"3HU%3VWJs!GF8$\"3Q(pemYqU<$F87$$\"3+1M\")Rf![$GF 8$\"3/ym8;4O3LF87$$\"3)H2n7YTX'GF8$\"3'[%[)HuwkW$F87$$\"33*oR^'*)o$*GF 8$\"3+lp+\"oD%oNF87$$\"3bxEE#4#GEHF8$\"3c(>DPaL[o$F87$$\"3G'>z&)pA#fHF 8$\"33Y;?#*3@\"y$F87$$\"3m7>p!G9!**HF8$\"39`/8*)>-tQF87$$\"3MU2Z>,0YIF 8$\"3!f/#y&>'3aRF87$$\"3k3]TD,Z#4$F8$\"3[@n4*3$*>,%F87$$\"3%>+]&3z**[J F8$\"3CsO)[`l/1%F87$$\"3Omt1K,*4=$F8$\"31oVF@@]zSF87$$\"3QhAWK)z\\@$F8 $\"3y#Q;`g!H%4%F87$$\"3E8W0\\cA]KF8$\"3=Qr\"**zHX5%F87$$\"3?5ltZb\\(G$ F8$\"3\"fv**y'3Z5TF87$$\"33*o84?DbK$F8$\"3T)HiV'*=?6%F87$$\"3)p+VRcEbO $F8$\"3@b\")p\"eH$4TF87$$\"3+5\">y6T:Y$F8$\"33$Qx6oqw3%F87$$\"3Ec5e)># RcNF8$\"3A02*)Hl(*[SF87$$\"31@LP;`omOF8$\"3YTUuVYc')RF87$$\"3)4LE]W?Yx $F8$\"3LqmE')>L5RF87$$\"3+]'f.>_6!RF8$\"3o7EJ]yh/QF87$$\"3m9GST#*3GSF8 $\"3g5*=ok/Ho$F87$$\"3iUPR.XcoTF8$\"3UuAS4TlJNF87$$\"3n<(*HN!GJJ%F8$\" 3!RA>Ge#3fLF87$$\"3oHGh%o'\\rWF8$\"3M8>7x)37:$F87$$\"3NBAjx95IYF8$\"34 *[e\\5xO#HF87$$\"3knV(zHr\"F87$$\"3;?pdF8)fR&F8$\"3FN70#)y\\N:F87$$\"3wM4!pA6#zaF8$\"3 _#eLZ]?;N\"F87$$\"3gr3*zWNyc&F8$\"3mJA36PHZ6F87$$\"3YE7Nzk:ccF8$\"3lYh RJJ\"HM*F-7$$\"3,r;())zF-u&F8$\"3cBeCi5n@sF-7$$\"3*GPE`5nO#eF8$\"3-i$) R$**f%=]F-7$$\"3<>H%>&4U7fF8$\"3)))z&GxK.eDF-7$$\"\"'F)F(-%'COLOURG6&% $RGBGF($\"*++++\"!\")F(-%*THICKNESSG6#\"\"#-F$6(7'F'7$$\"\"&F)$Fa]lF)7 $$\"\"\"F)Ff]l7$$\"\"%F)Fe\\lFd\\l-Fh\\l6&Fj\\lF)F)F)-%&STYLEG6#%&POIN TGF^]l-%*LINESTYLEGF`]l-%'SYMBOLG6#%'CIRCLEG-F$6(Fd]l-Fh\\l6&Fj\\l$\") #)eqkF]]l$\"))eqk\"F]]lFa_l-Fb^l6#%%LINEGF^]lFe^lFg^l-%+AXESLABELSG6%Q !6\"Fi_l%(DEFAULTG-%%VIEWG6$F[`lF[`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "6 point example" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "pts := [[1 ,0],[1,1],[-1,1],[-1,-1],[1,-1],[1,0]];\nbezier(pts,animate=false);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7(7$\"\"\"\"\"!7$F'F'7$!\"\"F' 7$F+F+7$F'F+F&" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6'-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3wE[3PT24**!#=$\"3Vw#)eRkOT5F/ 7$$\"3cFzRk,A%p*F/$\"3)y<:vz#)e'=F/7$$\"3eBw$*)p4o*eSF/7$ $\"3/=`x$4Afh(F/$\"3X3[#3gBWa%F/7$$\"3v2,Bqe4=^\\F/7$$\" 3!py>\"G$3$\\hF/$\"3/vH'*)pQ,F&F/7$$\"3S^'zdU%)QN&F/$\"3!o0r;5Cp[&F/7$ $\"3)Q#\\5W4z>XF/$\"3%)e&**zp$G2cF/7$$\"3]%p8pO:Ny$F/$\"3i**yeX>%*HcF/ 7$$\"3%p.,s]'=lHF/$\"3sleE)yAfc&F/7$$\"3#G-P1x$)o;#F/$\"3I5kz#QO5T&F/7 $$\"3:phIPLjI9F/$\"3?4o=a49\"=&F/7$$\"3Y3GlYYs#)z!#>$\"3Z\"H>T#ya3\\F/ 7$$\"3;(G#3w)HU+\"F`p$\"3TjO@-U!=^%F/7$$!3/gXwh[^zVF`p$\"34![qw%eq>TF/ 7$$!3O*pY)>Z`f**F`p$\"3&odx(HKJ1OF/7$$!3.&43Qd$>B9F/$\"3xT^>?e12JF/7$$ !3z50$f,7Z\"=F/$\"3P.f\")4;h=DF/7$$!3#>maxt/%3@F/$\"3Q;%f4f9i#>F/7$$!3 c4kVJI@HBF/$\"3?@C>x(QIG\"F/7$$!3AxWH+A(GX#F/$\"3G5qD5B)=x'F`p7$$!3)Q> Wpjv**\\#F/$\"3\"4c8TklCa\"!#?7$$!3)pl5PO'e`CF/$!3i_t.](f0s'F`p7$$!3j@ /@sg_MBF/$!3qG=h%Q#=j7F/7$$!3t>fJNa%\\7#F/$!3i)f(H/oJ')=F/7$$!3'\\I[a# eQA=F/$!3@Y=jj?C0DF/7$$!3[(p#3Z#3]W\"F/$!3m<9%yG;#yIF/7$$!3bT\")z>&er+ \"F/$!3K*fPNd*e%f$F/7$$!3#3'=K8oCGWF`p$!3)\\fJMrWd6%F/7$$\"3e\"\\NhIv_ G\"F`p$!3!3%yx/0-IXF/7$$\"3_ocaSTa%)zF`p$!3'*oql:tj3\\F/7$$\"3GU'e)yJ! >X\"F/$!3Qab@nN/*=&F/7$$\"3A0'y)z?33AF/$!3*=%*y$*3`8U&F/7$$\"3C#)H<(\\ 30&HF/$!3!)4m)RJ7Rc&F/7$$\"3aQ?c,4B[PF/$!37d:isi6HcF/7$$\"3Yyue<AWhF/$!3K S'fmNq=F&F/7$$\"34zT\\o+17pF/$!3O/a<>Zn`\\F/7$$\"36XkRvmGGwF/$!31O$=BO Cg`%F/7$$\"3d!ynx8#oK#)F/$!3;0j3IG!=1%F/7$$\"3pDR.M%oV%))F/$!3'Ga$*zO! p6MF/7$$\"3LS8')*p(f,$*F/$!3@Epf<*HXt#F/7$$\"3#[J0St(*pn*F/$!3@+>_9Tw9 >F/7$$\"30p8XA#Q'4**F/$!3syC!)3U@Q5F/F'-%'COLOURG6&%$RGBGF+$\"*++++\"! \")F+-%*THICKNESSG6#\"\"#-F$6(7(F'7$F(F(7$$!\"\"F*F(7$Fd[lFd[l7$F(Fd[l F'-Fez6&FgzF*F*F*-%&STYLEG6#%&POINTGF[[l-%*LINESTYLEGF][l-%'SYMBOLG6#% 'CIRCLEG-F$6(Fa[l-Fez6&Fgz$\")#)eqkFjz$\"))eqk\"FjzFj\\l-F[\\l6#%%LINE GF[[lF^\\lF`\\l-%+AXESLABELSG6%Q!6\"Fb]l%(DEFAULTG-%%VIEWG6$Fd]lFd]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "7 p oint example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "In this example we plot the Bezier curve associated with the 6 \+ vertices of a regular hexagon." }}{PARA 0 "" 0 "" {TEXT -1 115 "There \+ are actually 7 points because the point (1,0) appears as both the firs t and last point in the list of points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "h := Pi/3; \npts := [se q([cos(i*h),sin(i*h)],i=0..6)];\nbezier(pts,animate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$%#PiG#\"\"\"\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$ptsG7)7$\"\"\"\"\"!7$#F'\"\"#,$*$-%%sqrtG6#\"\"$F' F*7$#!\"\"F+F,7$F4F(7$F3,$F-F37$F*F7F&" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$\"\"\"\"\"!$F*F*7$$\"3_+lY 0@Rk'*!#=$\"3or35'f'y3b!#>7$$\"3S4iiQe]7$*F/$\"3oyg-`I#42\"F/7$$\"31j! Q8KnU**)F/$\"37.1]>TP)\\\"F/7$$\"3%H@\"GeN$em)F/$\"3#ffS3n)f->F/7$$\"3 y.j9Qu$[*yF/$\"35A7akwUFFF/7$$\"3K)=P()pwB3(F/$\"3MB_G7U0TMF/7$$\"3an# )Hoi$zC'F/$\"3WMyzeW7PSF/7$$\"3!3z^;EG*faF/$\"37,JI#Rn-\\%F/7$$\"3w?p] d0oPYF/$\"3#y[bG4\"Gh[F/7$$\"3#*[ArIk'*)y$F/$\"3@As=+5@V^F/7$$\"3'pHZd 23H&HF/$\"3F5Zx6/5D`F/7$$\"3lh*[ADa:6#F/$\"3iBd/RsX8aF/7$$\"3I&R)HIU[# R\"F/$\"3cP-SiHN7aF/7$$\"3Knj**HhHVhF2$\"3%>xY&)yDvK&F/7$$!3%)3=$zJY)p 7F2$\"3k2:-!>1)e^F/7$$!3myW#=#>^uzF2$\"3.ly_W\")pA\\F/7$$!3WB^#Q+\"[k8 F/$\"3y_6Il-l]YF/7$$!31gOsou$=)>F/$\"3UO.NiJ9iUF/7$$!3uTdn_CF/$\"3O O.LF/$\"3\"e\"Q Vx'Rv\"HF/7$$!3v0@&eI=yj$F/$\"3Ep-5.%H:O#F/7$$!3/f%*yCXY()QF/$\"3.)o&> T#RT!=F/7$$!3+K/PVK[uSF/$\"3*e]iA$zw+7F/7$$!3w740,M(*yTF/$\"3[@S.6Y&[L 'F27$$!3\"[KoOWH(=UF/$\"3Dr%y@rzEW\"!#?7$$!39h*4LNw&zTF/$!3I\\k_\"=GoG 'F27$$!3%Rrq!yi(*ySF/$!354qZPA;#=\"F/7$$!3wq8pFQ\\,RF/$!3kkI!['RmmF/$!3)=tORO\\)zUF/7$$!3S7RDL*=VO\"F/$!3Aj=-A(Q2l%F/7$$!3Oxo` r2V#y(F2$!3D3%Q)3CoI\\F/7$$!3hI0V[t66*)Fjs$!31`k0iamp^F/7$$\"36\")=Z+V ]0gF2$!3h#fEyf+_K&F/7$$\"3)32K&yO]e8F/$!3I(eUZZ.0T&F/7$$\"3.kaUrO;I@F/ $!3*))['4:u`7aF/7$$\"3Mb;(RqxM'HF/$!31P`aReQB`F/7$$\"3_PY14[\\$y$F/$!3 'oTGVW2Z9&F/7$$\"3eauV_!\\>j%F/$!3)><%4u@_j[F/7$$\"3k!=ck\\D]Z&F/$!3%= 1$=,:^#[%F/7$$\"3%*Q$QRTdPC'F/$!31(\\A2V'zRSF/7$$\"3+pW,g`Z3rF/$!3xVjc ApH?MF/7$$\"3uu#Hf#zZeyF/$!3[\"eq/-EDw#F/7$$\"3a\"*=mv'eTi)F/$!3kSTc=@ W^>F/7$$\"3igPmzbBv*)F/$!3s]P1yZxA:F/7$$\"3+Eo;M,z9$*F/$!3Y^?=:crn5F/7 $$\"31eG->][l'*F/$!3#)3))Rk?x\"\\&F2F'-%'COLOURG6&%$RGBGF+$\"*++++\"! \")F+-%*THICKNESSG6#\"\"#-F$6(7)F'7$$\"3++++++++]F/$\"3a+++SSDg')F/7$$ !3++++++++]F/Fi\\l7$$!\"\"F*F+7$F\\]l$!3a+++SSDg')F/7$Fg\\lFb]lF'-Fi[l 6&F[\\lF*F*F*-%&STYLEG6#%&POINTGF_\\l-%*LINESTYLEGFa\\l-%'SYMBOLG6#%'C IRCLEG-F$6(Fe\\l-Fi[l6&F[\\l$\")#)eqkF^\\l$\"))eqk\"F^\\lFg^l-Fh]l6#%% LINEGF_\\lF[^lF]^l-%+AXESLABELSG6%Q!6\"F__l%(DEFAULTG-%%VIEWG6$Fa_lFa_ l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" " Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 6 "bezier" }{TEXT -1 35 ": examples of Bernstein polynomials" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 38 "The curves are draw n with the option \"" }{TEXT 279 13 "animate=false" }{TEXT -1 22 "\", \+ to conserve memory." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "8 point ex ample " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "pts := [0,1,0,-1,0,1,0,-1,0];\nbezier(pts,info=true,a nimate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7+\"\"!\"\"\" F&!\"\"F&F'F&F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,.*$)%\"xG \"\"(\"\"\"!#k*&\"$C#F*)F(\"\"'F*F**&F-F*)F(\"\"&F*!\"\"*&\"$7\"F*)F( \"\"$F*F**&\"#cF*)F(\"\"#F*F3*&\"\")F*F(F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 395 164 164 {PLOTDATA 2 "6'-%'CURVESG6%7cp7$$\"\"!F)F(7$$\"3 qkmmT&)G\\a!#?$\"3aq?_\"[_\\>%!#>7$$\"3kLLL3x&)*3\"F0$\"3)\\`#Q.Y>o!)F 07$$\"3$*****\\ilyM;F0$\"3U7t&>XeI;\"!#=7$$\"3Gnmm;arz@F0$\"3Afz$fa&H* [\"F;7$$\"3.++D\"y%*z7$F0$\"3I?y,*>&o))>F;7$$\"35KL$e9ui2%F0$\"3jKO<_* RhS#F;7$$\"3+)**\\(oMrU^F0$\"3wgwx*3zYy#F;7$$\"3Glmm\"z_\"4iF0$\"3%eEH h'=\\uIF;7$$\"3?lmmm6m#G(F0$\"3%>!=b!e)Q%G$F;7$$\"3[mmmT&phN)F0$\"3%pa )3jK\")>MF;7$$\"3IL$3-js.*))F0$\"3Cn2$eRs$[*\\$F;7$$\"3(*\\i:N!)eA5F;$\"3QtGNiP#**\\$F;7$$\"3QLLe*=)H\\5 F;$\"3m$QX$p[)p\\$F;7$$\"3)ommTlD))4\"F;$\"3cVD2;?&H[$F;7$$\"37++v=JN[ 6F;$\"3)zMs4iH$eMF;7$$\"3[LLL$e!)y>\"F;$\"34hF(ojrPU$F;7$$\"3/mm\"z/3u C\"F;$\"3[O96b$>*zLF;7$$\"3kLLe*ot*\\8F;$\"3MnsbQ)[AE$F;7$$\"3c***\\7L RDX\"F;$\"3ElzSEez7JF;7$$\"31nm\"zR'ok;F;$\"3QdbA\"=ZYs#F;7$$\"3?***\\ i5`h(=F;$\"3/$R<:$pplAF;7$$\"3sLLL3En$4#F;$\"3cy\"oG>mqv\"F;7$$\"3)pmm T!RE&G#F;$\"3W.m_yoO08F;7$$\"3\")*****\\K]4]#F;$\"35vD,y%*[#=)F07$$\"3 $******\\PAvr#F;$\"3?o^?-]SQPF07$$\"3())****\\nHi#HF;$\"33j:nm-n^V!#@7 $$\"3jmm\"z*ev:JF;$!3K3(e]LRsr#F07$$\"3'[LL$347TLF;$!3I@r$)=vb\">&F07$ $\"3QLL$3xxlV$F;$!3cTa6FpplfF07$$\"3,LLLLY.KNF;$!3YyQRPvMwlF07$$\"3bl; HdO2VOF;$!3C.=!)z+W$3(F07$$\"3'3+]7o7Tv$F;$!3g)*y@4N:xtF07$$\"3)zm\"HK 5S_QF;$!3ySNvN&4aY(F07$$\"33NLL$Q*o]RF;$!3)4\\P-%[r*R(F07$$\"3ql;H#GF& eSF;$!3s0Lg^vcgrF07$$\"3y++D\"=lj;%F;$!33kMY<78fnF07$$\"3=,+vV&R*f&F07$$\"3cML$e9Ege%F;$!3&QSY!4VXURF07$$\"3%QL$eR\"3Gy%F;$ !3#)pm%e[&GV@F07$$\"3-mm;/T1&*\\F;$!3W7$oB;ie$\\Fht7$$\"3&em;zRQb@&F;$ \"3V1$)GP!*RF@F07$$\"3\\***\\(=>Y2aF;$\"3U'\\Vn%)ek)QF07$$\"3.lm;zXu9c F;$\"3!*[\\:!Rln]&F07$$\"3l******\\y))GeF;$\"3%y%34^C&zt'F07$$\"3I*** \\ibOO$fF;$\"3eI![a/*pOrF07$$\"3'*)***\\i_QQgF;$\"3+&*elI(eKQ(F07$$\"3 ?+](=-N(RhF;$\"3wI;w(4;dY(F07$$\"3@***\\7y%3TiF;$\"3y&>K**fb^Q(F07$$\" 3;**\\P4kh`jF;$\"3aS$QsbZ_4(F07$$\"3+)***\\P![hY'F;$\"3i^C'pf'R'e'F07$ $\"3)em;/risc'F;$\"30!*35eWrPfF07$$\"3uLLL$Qx$omF;$\"3[*))H[Tib5&F07$$ \"3#4++]P+V)oF;$\"31\"p%*Hu4lr#F07$$\"3Knm\"zpe*zqF;$!3=5]sYr+O9F-7$$ \"3%)*****\\#\\'QH(F;$!3#=KH^')F;$!3_)=+Hf%yjKF;7$$ \"3=LLe9tOc()F;$!3+jG/%3vNQ$F;7$$\"3\")*\\P%[ko/))F;$!3(=`:Uk'zDMF;7$$ \"3Ym;H#e0I&))F;$!3-YO!zfS\"fMF;7$$\"34Le9;ZK,*)F;$!39#*\\#>)4,$[$F;7$ $\"3u******\\Qk\\*)F;$!3O*fy.P+o\\$F;7$$\"37nT5SMLx*)F;$!3%\\M$=B+#** \\$F;7$$\"3RL$3-.B]+*F;$!33FhTh\"3%*\\$F;7$$\"3m*\\7.i7F.*F;$!3+\"o%R( fX^\\$F;7$$\"31nmT5ASg!*F;$!3qGT\\4?,([$F;7$$\"3f**\\i!R\"y:\"*F;$!3)) ))pdH\\keMF;7$$\"3CLL$3dg6<*F;$!34htVkCK8MF;7$$\"3)3+](oTAq#*F;$!3C!p6 hq\"z'G$F;7$$\"3ImmmmxGp$*F;$!3y&*[hce$p4$F;7$$\"3sK$eRA5\\Z*F;$!3H_Vw @I*z\"GF;7$$\"37***\\7oK0e*F;$!3Q?0'34(y_CF;7$$\"3C+++]oi\"o*F;$!334eQ 5LV:?F;7$$\"3A++v=5s#y*F;$!3mEs:sIL&[\"F;7$$\"3;+D1k2/P)*F;$!3sYS3jh!) f6F;7$$\"35+]P40O\"*)*F;$!3K!**f9guX/)F07$$\"31+voa-oX**F;$!3YyzD&3R@= %F07$$\"\"\"F)F(-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESSG6#\" \"#-F$6(7+F'7$$\"3+++++++]7F;Fbfl7$$\"3++++++++DF;F(7$$\"3+++++++]PF;$ !\"\"F)7$$\"3++++++++]F;F(7$$\"3+++++++]iF;Fbfl7$$\"3++++++++vF;F(7$$ \"3+++++++]()F;F[hlFafl-Fefl6&FgflF)F)F)-%&STYLEG6#%&POINTGF[gl-%*LINE STYLEGF]gl-%'SYMBOLG6#%'CIRCLEG-F$6(Fagl-Fefl6&Fgfl$\")#)eqkFjfl$\"))e qk\"FjflF[jl-F\\il6#%%LINEGF[glF_ilFail-%+AXESLABELSG6%Q!6\"Fcjl-%%FON TG6#%(DEFAULTG-%%VIEWG6$FhjlFhjl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "11 point example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "pts := [0 ,4,6,5,3,2,1,1,3,2,0];\nbezier(pts,info=true,animate=false,pointcolor= blue,curvecolor=red);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7-\"\" !\"\"%\"\"'\"\"&\"\"$\"\"#\"\"\"F,F*F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,4*$)%\"xG\"#5\"\"\"!#R*&\"$5%F*)F(\"\"*F*F**&\"%N5F*)F(\" \")F*!\"\"*&\"$g*F*)F(\"\"(F*F**&\"$c(F*)F(\"\"&F*F4*&\"$I'F*)F(\"\"%F *F**&\"$?\"F*)F(\"\"$F*F4*&\"#!*F*)F(\"\"#F*F4*&\"#SF*F(F*F*" }}{PARA 13 "" 1 "" {GLPLOT2D 453 340 340 {PLOTDATA 2 "6'-%'CURVESG6%7gn7$$\"\" !F)F(7$$\"3YLLL3x&)*3\"!#>$\"3y>v!Q/k5D%!#=7$$\"3inmm;arz@F-$\"3cC%=\\ ([@!G)F07$$\"3.++D\"y%*z7$F-$\"3kUu07m/g6!#<7$$\"3yKL$e9ui2%F-$\"3_ktD eA\\u9F;7$$\"3p)**\\(oMrU^F-$\"3-=xo??(o!=F;7$$\"3^imm\"z_\"4iF-$\"3N' HHqNAm6#F;7$$\"3!Qmmm;hEG(F-$\"3(yzniKibS#F;7$$\"35lmmT&phN)F-$\"35$*> 0**>orEF;7$$\"3F/+v=ddC%*F-$\"3ux?[QB69HF;7$$\"3QLLe*=)H\\5F0$\"3JN-SX y]MJF;7$$\"37++v=JN[6F0$\"3;lqsG-`>LF;7$$\"3wlm\"z/3uC\"F0$\"3U7#Rr=&Q '[$F;7$$\"3c***\\7LRDX\"F0$\"3!fk)3*=%>wPF;7$$\"3Mnm\"zR'ok;F0$\"3!R&y d&H%4,SF;7$$\"3O)**\\i5`h(=F0$\"3[H;**)[*RbTF;7$$\"3;n;HdG\"\\)>F0$\"3 h\"*4N&Gp'4UF;7$$\"3sLLL3En$4#F0$\"3Qm.$y=\"3[UF;7$$\"33++Dc#o%*=#F0$ \"3qE_#oE%F;7$$\"3)4+++NO#4 EF0$\"3#z:3h8!zUUF;7$$\"3$******\\PAvr#F0$\"3j^PGy*G$3UF;7$$\"3())**** \\nHi#HF0$\"3$fm>ik(z:TF;7$$\"3jmm\"z*ev:JF0$\"3\\J44_QO1SF;7$$\"3'[LL $347TLF0$\"331o\"z^*=^QF;7$$\"3,LLLLY.KNF0$\"3ZTcW+,[.PF;7$$\"3U,+D\"o 7Tv$F0$\"3$*=$=^$*Q'=NF;7$$\"33NLL$Q*o]RF0$\"3)R&pAAx!zM$F;7$$\"3A++D \"=lj;%F0$\"3#>bB[_Fw:$F;7$$\"3%G+]PaRtgEw(HF;7$$\"3mNL$e 9Ege%F0$\"31i5^^$p\\z#F;7$$\"3uKLeR\"3Gy%F0$\"3b:O6R')*[j#F;7$$\"3cmm; /T1&*\\F0$\"35M>PYiKtCF;7$$\"3ukm\"zRQb@&F0$\"3!>avG$HN?BF;7$$\"3\\*** \\(=>Y2aF0$\"3#>')3z\"**H,AF;7$$\"39mm;zXu9cF0$\"3]-q\\a3())3#F;7$$\"3 l******\\y))GeF0$\"3O*3CxRE8*>F;7$$\"3'*)***\\i_QQgF0$\"3G@+@#=ZY\">F; 7$$\"3@***\\7y%3TiF0$\"3'>b/'H9(z&=F;7$$\"3+)***\\P![hY'F0$\"3O!eURg_V \"=F;7$$\"3kKLL$Qx$omF0$\"3#yP;)y50\"z\"F;7$$\"3!)*****\\P+V)oF0$\"3w) p!*z3$[!y\"F;7$$\"3?mm\"zpe*zqF0$\"3Q-D*=b]5y\"F;7$$\"3%)*****\\#\\'QH (F0$\"3'[;H " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "13 point example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 41 "13 equally spaced points along the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\"\"%\"xGF'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "h := Pi/12;\npts := [seq(sin(h*i),i=0..12)];\nbezier(pts,animate=f alse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$*&\"#7!\"\"%#PiG\"\" \"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7/\"\"!-%$sinG6#,$*&\"# 7!\"\"%#PiG\"\"\"F/#F/\"\"#,$*&F1F-F1F0F/,$*&F1F-\"\"$F0F/-F(6#,$*(\" \"&F/F,F-F.F/F/F/F7F4F2F0F'F&" }}{PARA 13 "" 1 "" {GLPLOT2D 520 286 286 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3inmm;arz@!#>$\"3%4d ;W.&z5nF-7$$\"3[LL$e9ui2%F-$\"3\"[*[u%QhUC\"!#=7$$\"3!Rmm;z_\"4iF-$\"3 9kwh\"Q(4v=F57$$\"35lmmT&phN)F-$\"3G')>XP2\\$\\#F57$$\"3_LLe*=)H\\5F5$ \"33T#*)ye![!4$F57$$\"3wlm\"z/3uC\"F5$\"3n/'))=.;fi$F57$$\"3c***\\7LRD X\"F5$\"3gEQt!)z\\gTF57$$\"3inm\"zR'ok;F5$\"3mGN%z[!e!p%F57$$\"3k)**\\ i5`h(=F5$\"3uGA+!HrV>&F57$$\"3WLLL3En$4#F5$\"3?$eTJg=ao&F57$$\"3Umm;/R E&G#F5$\"3+^5KL\"HR4'F57$$\"3\")*****\\K]4]#F5$\"3)[$[GQilDlF57$$\"3$* *****\\PAvr#F5$\"3h5bVa-!z#pF57$$\"3J)****\\nHi#HF5$\"3a!*eb]sx%G(F57$ $\"3=nm\"z*ev:JF5$\"3'4g&\\2g&=e(F57$$\"3UNLL347TLF5$\"3lRlau'*p+zF57$ $\"3,LLLLY.KNF5$\"3-Mi\"pDO39)F57$$\"3U,+D\"o7Tv$F5$\"33k8v=k\"[Q)F57$ $\"33NLL$Q*o]RF5$\"37'*[g/&y%o&)F57$$\"3A++D\"=lj;%F5$\"3'R$\\fa_`M()F 57$$\"3%G+]PaR([@(3+*F57$$\"37nm;/T1&*\\F5$\"3nJZT*eF. -*F57$$\"3&em;zRQb@&F5$\"3K&3fuKn6+*F57$$\"3Q)**\\(=>Y2aF5$\"3O<$QHAq= &*)F57$$\"3#Rmm\"zXu9cF5$\"3=!=%fb?mk))F57$$\"3l******\\y))GeF5$\"3BbP uh#pxt)F57$$\"3&y***\\i_QQgF5$\"3'*\\qXUjyx&)F57$$\"35)**\\7y%3TiF5$\" 3')\\T'R:n'*Q)F57$$\"3+)***\\P![hY'F5$\"3O'y8s2))H9)F57$$\"3_JLL$Qx$om F5$\"3yFt%R#*G!))yF57$$\"3!)*****\\P+V)oF5$\"39D3#porZG7'Hp.>F57$$ \"3A++D\"oK0e*F5$\"3ea_jB%y'z7F57$$\"3A++v=5s#y*F5$\"3)HYs'\\9n*o'F-7$ $\"\"\"F)F(-%'COLOURG6&%$RGBGF($\"*++++\"!\")F(-%*THICKNESSG6#\"\"#-F$ 6(7/F'7$$\"3^*****HLLLL)F-$\"3y*****4X!>)e#F57$$\"3%******pmmmm\"F5$\" 3++++++++]F57$$\"3++++++++DF5$\"3=+++5y1rqF57$$\"3z*****HLLLL$F5$\"3a+ ++SSDg')F57$$\"3A+++nmmmTF5$\"37+++j#e#f'*F57$Fj[lFbz7$$\"3M+++LLLLeF5 Fi\\l7$$\"3m*****pmmmm'F5Fd\\l7$$\"3++++++++vF5F_\\l7$$\"3M+++LLLL$)F5 Fj[l7$$\"3m*****pmmm;*F5Fe[lFaz-Fez6&FgzF)F)F)-%&STYLEG6#%&POINTGF[[l- %*LINESTYLEGF][l-%'SYMBOLG6#%'CIRCLEG-F$6(Fa[l-Fez6&Fgz$\")#)eqkFjz$\" ))eqk\"FjzF]_l-F^^l6#%%LINEGF[[lFa^lFc^l-%+AXESLABELSG6%Q!6\"Fe_l-%%FO NTG6#%(DEFAULTG-%%VIEWG6$Fj_lFj_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "49 point example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "49 equally spaced poin ts along the curve " }{XPPEDIT 18 0 "y = sin*x;" "6#/%\"yG*&%$sinG\"\" \"%\"xGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "h := Pi/48;\npts := [seq(sin(h*i),i =0..48)]:\nbezier(pts,animate=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG,$*&\"#[!\"\"%#PiG\"\"\"F*" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3-K5`;arz@!#>$\" 3Q&p=l&o*G#oF-7$$\"3kxu!f9ui2%F-$\"3!4N**o@%=r7!#=7$$\"3[B%4Az_\"4iF-$ \"3Py>BeCjD>F57$$\"3!pBuEaphN)F-$\"3MiB?E;StDF57$$\"3Mi8r*=)H\\5F5$\"3 6cI!4?FW?$F57$$\"3ObH0[!3uC\"F5$\"3!oGUMm/]x$F57$$\"33SWQJ$RDX\"F5$\"3 5!p**4Q>*[VF57$$\"353@/)R'ok;F5$\"3z3'pG#z4A\\F57$$\"3g]pN1J:w=F5$\"3W 3Ms@VaqaF57$$\"378jS3En$4#F5$\"3xH`&*oZ^3gF57$$\"3Mtl>/RE&G#F5$\"3m'4+ $\\0dekF57$$\"3:!)Q(\\K]4]#F5$\"3vZ%)*3WRm$pF57$$\"3A%\\C\\PAvr#F5$\"3 S:\"*4j%yUQ(F57$$\"3-*z**[nHi#HF5$\"3MD8R*G'>$y(F57$$\"3+]N#y*ev:JF5$ \"3JA]f\"3Al6)F57$$\"3(f)))G347TLF5$\"3u/>]h$=bZ)F57$$\"30>\"fLjM?`$F5 $\"3U*pny'>uY()F57$$\"3QuhP\"o7Tv$F5$\"3Wa1:OY/B!*F57$$\"3kz%[NQ*o]RF5 $\"3%3z,7PD:B*F57$$\"3!)o[a\"=lj;%F5$\"3&=WO3bz.U*F57$$\"3W*y\"4W&RYh-'e%F5$\"3'\\GnjXZbm*F57$$\"3)\\DJ*R\" 3Gy%F5$\"3d[q$y$f)Rs*F57$$\"3%)p<\\/T1&*\\F5$\"3G!H=&>x>Y(*F57$$\"3sI. A)RQb@&F5$\"38=(R:!HKC(*F57$$\"3tsf/>>Y2aF5$\"3Mfj+t?1o'*F57$$\"3@UuYz Xu9cF5$\"3u@J@@tgo&*F57$$\"3\\PQJ]y))GeF5$\"3#4G`c!41C%*F57$$\"3fW\\#G E&QQgF5$\"3;Xx%H3-@C*F57$$\"3oCzd\"y%3TiF5$\"3#)oo6(\\X&G!*F57$$\"3c-N #y.[hY'F5$\"3kf0@>c<\\()F57$$\"3IFRl$Qx$omF5$\"3mD!*4MuAh%)F57$$\"3#\\ NG`P+V)oF5$\"3X;b&HJFk6)F57$$\"3g6TE)pe*zqF5$\"3e*4gUmT=x(F57$$\"31dHP D\\'QH(F5$\"3y;4'))3\"ehtF57$$\"3%Hlo\\,M^\\(F5$\"3#R>aD!o-XpF57$$\"3J m\">m?=bq(F5$\"3U^%R()QM'zkF57$$\"3:&R^'3s?6zF5$\"3=\")G)Ra`n*fF57$$\" 3>(H'[JXaE\")F5$\"3A'yLN;,PY&F57$$\"3rft\"o'*RRL)F5$\"3*pN&)4qOd#\\F57 $$\"31VWCa<.Y&)F5$\"3=>Beh=&GN%F57$$\"3I+Zh9tOc()F5$\"31!Q9&=WFkPF57$$ \"3e\"=5+&Qk\\*)F5$\"39?WbwJ^2KF57$$\"3kf]$3dg6<*F5$\"39g3f$o\\Jb#F57$ $\"3/XommxGp$*F5$\"3G\\w*y#>Xb>F57$$\"3c.+D\"oK0e*F5$\"3(3f())[Lv28F57 $$\"3A++v=5s#y*F5$\"3G?F(3](H,oF-7$$\"\"\"F)F(-%'COLOURG6&%$RGBGF($\"* ++++\"!\")F(-%*THICKNESSG6#\"\"#-F$6(7SF'7$$\"3')*****HLLL3#F-$\"3c*** **H#HJSlF-7$$\"3!)*****pmmm;%F-$\"3-+++A>E08F57$$\"3+++++++]iF-$\"3#** ****4A.4&>F57$$\"3^*****HLLLL)F-$\"3y*****4X!>)e#F57$$\"3$******pmm;/ \"F5$\"38+++aYR9KF57$$\"3+++++++]7F5$\"3w*****\\KMo#QF57$$\"31+++LLLe9 F5$\"3,+++-p)GU%F57$$\"3%******pmmmm\"F5$\"3++++++++]F57$$\"3+++++++v= F5$\"3y*****4L-db&F57$$\"31+++LLL$3#F5$\"3k*******G9w3'F57$$\"3%****** pmm;H#F5$\"3=+++_\"eMf'F57$$\"3++++++++DF5$\"3=+++5y1rqF57$$\"3z*****H LL$3FF5$\"37+++v!)R=vF57$$\"3@+++nmm;HF5$\"3^+++/M`LzF57$$\"3+++++++DJ F5$\"3Z*****R7'p9$)F57$$\"3z*****HLLLL$F5$\"3a+++SSDg')F57$$\"3@+++nmm TNF5$\"3e*****\\TF(o*)F57$$\"3+++++++]PF5$\"3*)*****\\K&zQ#*F57$$\"3y* ****HLL$eRF5$\"3'******\\H,$p%*F57$$\"3A+++nmmmTF5$\"37+++j#e#f'*F57$$ \"3+++++++vVF5$\"3:+++/G&y!)*F57$$\"3y*****HLLLe%F5$\"3))*****Rh[W\"** F57$$\"3A+++nmm\"z%F5$\"3E+++K#*ey**F57$Fh]lFbz7$$\"3M+++LLL3_F5Fcbl7$ $\"3m*****pmmmT&F5F^bl7$$\"3+++++++DcF5Fial7$$\"3M+++LLLLeF5Fdal7$$\"3 l*****pmm;/'F5F_al7$$F^\\lF5Fj`l7$$\"3M+++LLLekF5Fe`l7$$\"3m*****pmmmm 'F5F``l7$$\"3+++++++voF5F[`l7$$\"3M+++LLL$3(F5Ff_l7$$\"3m*****pmm;H(F5 Fa_l7$$\"3++++++++vF5F\\_l7$$\"3M+++LLL3xF5Fg^l7$$\"3m*****pmmm\"zF5Fb ^l7$$\"3+++++++D\")F5F]^l7$$\"3M+++LLLL$)F5Fh]l7$$\"3l*****pmm;a)F5Fc] l7$$\"3+++++++]()F5F^]l7$$\"3M+++LLLe*)F5Fi\\l7$$\"3m*****pmmm;*F5Fd\\ l7$$\"3+++++++v$*F5F_\\l7$$\"3M+++LLL$e*F5Fj[l7$$\"3m*****pmm;z*F5Fe[l Faz-Fez6&FgzF)F)F)-%&STYLEG6#%&POINTGF[[l-%*LINESTYLEGF][l-%'SYMBOLG6# %'CIRCLEG-F$6(Fa[l-Fez6&Fgz$\")#)eqkFjz$\"))eqk\"FjzF\\hl-F]gl6#%%LINE GF[[lF`glFbgl-%+AXESLABELSG6%Q!6\"Fdhl-%%FONTG6#%(DEFAULTG-%%VIEWG6$Fi hlFihl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 14 "Given 5 points" }{XPPEDIT 18 0 "``(x[0],y[0]), ``(x[1],y[ 1]), ``(x[2],y[2]), ``(x[3],y[3]), ``(x[4],y[4])" "6'-%!G6$&%\"xG6#\" \"!&%\"yG6#F)-F$6$&F'6#\"\"\"&F+6#F1-F$6$&F'6#\"\"#&F+6#F8-F$6$&F'6#\" \"$&F+6#F?-F$6$&F'6#\"\"%&F+6#FF" }{TEXT -1 78 ", use Maple to check t hat the associated Bezier curve has parametric equations" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x = (1-t)^4*x[0]+4*( 1-t)^3*t*x[1]+6*(1-t)^2*t^2*x[2]+4*(1-t)*t^3*x[3]+t^4*x[4], ``],[y = ( 1-t)^4*y[0]+4*(1-t)^3*t*y[1]+6*(1-t)^2*t^2*y[2]+4*(1-t)*t^3*y[3]+t^4*y [4], ``]);" "6#-%*PIECEWISEG6$7$/%\"xG,,*&,&\"\"\"F,%\"tG!\"\"\"\"%&F( 6#\"\"!F,F,**F/F,*$,&F,F,F-F.\"\"$F,F-F,&F(6#F,F,F,**\"\"'F,*$,&F,F,F- F.\"\"#F,F-F=&F(6#F=F,F,**F/F,,&F,F,F-F.F,F-F6&F(6#F6F,F,*&F-F/&F(6#F/ F,F,%!G7$/%\"yG,,*&,&F,F,F-F.F/&FJ6#F2F,F,**F/F,*$,&F,F,F-F.F6F,F-F,&F J6#F,F,F,**F:F,*$,&F,F,F-F.F=F,F-F=&FJ6#F=F,F,**F/F,,&F,F,F-F.F,F-F6&F J6#F6F,F,*&F-F/&FJ6#F/F,F,FG" }{TEXT -1 3 ". " }{TEXT 273 0 "" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 87 "x01 := (1-t)*x0+t*x1;\nx12 := (1-t)*x1+t*x2;\nx23 : = (1-t)*x2+t*x3;\nx34 := (1-t)*x3+t*x4;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 17 "For the functio n " }{XPPEDIT 18 0 "f(x)=x-x^2" "6#/-%\"fG6#%\"xG,&F'\"\"\"*$F'\"\"#! \"\"" }{TEXT -1 20 ", use the procedure " }{TEXT 0 6 "bezier" }{TEXT -1 115 " to construct the associated Bernstein polynomials based on 6, 11, 26 and 51 equally spaced points between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "h := 1/5; \nf := x -> x-x^2;\npts := [seq(f(h*i),i=0..5)];\nbezier(pts,info=true ,animate=false);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 20 "Code for \+ animations " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 26 "Code for first animation " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 986 "n := 25:\nalias(C=C);\np1 := plot([[0,0],[2,3],[6,0]],color=brown,thicknes s=2):\nt1 := plots[textplot]([[-0.2,0,`A`],[2,3.15,`B`],[6.25,0,`C`]]) :\nfrms := NULL:\nfor i from 0 to n do\n tt := evalf(i/n); \n pt1 \+ := [2*tt,3*tt];\n pt2 := [2+4*tt,3-3*tt];\n pt3 := [2*tt*(2+tt),6* tt*(1-tt)];\n t2 := plots[textplot]([[op(1,pt1)-0.18,op(2,pt1)+(1-2* tt)*0.07,`P`],\n [op(1,pt2)+0.18,op(2,pt2)+tt*0.07,`Q`],\n \+ [op(1,pt3)+(1-2*tt)*0.1,op(2,pt3)-0.1,`R`]]):\n p2 := plo t([[pt1,pt2,pt3],[pt1,pt2]],color=[black,red],\n thickne ss=2,style=[point,line],symbol=circle);\n if i<>0 then\n p3 := p lot([2*t*(2+t),6*t*(1-t),t=0..tt],color=green,thickness=2);\n end if ;\n if i=0 then\n frms := frms,plots[display]([t1,p1,p2]);\n \+ elif i=n then\n frms := frms,plots[display]([t1,p1,p2,p3]);\n e lse\n frms := frms,plots[display]([t1,t2,p1,p2,p3]);\n end if; \nend do:\nplots[display](frms,insequence=true,axes=none,title=`quadra tic Bezier curve animation`);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 24 "Code for 2nd animation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1372 "n := 25:\np1 := \+ plot([[0,0],[2,3],[5,2],[6,0]],color=brown,thickness=2):\nt1 := plots[ textplot]([[-0.2,0,`A`],[2,3.15,`B`],[5.2,2.05,`C`],[6.25,0,`D`]]):\nf rms := NULL:\nfor i from 0 to n do \n tt := evalf(i/n);\n pt0 := [ 2*tt,3*tt];\n pt1 := [2+3*tt,3-tt];\n pt2 := [5+tt,2-2*tt];\n pt 3 := [tt*(4+tt),2*tt*(3-2*tt)];\n pt4 := [2+2*tt*(3-tt),(tt+3)*(1-tt )];\n pt5 := [3*tt*(tt+1)*(2-tt),3*tt*(tt-1)*(tt-3)];\n t2 := plot s[textplot]([[op(1,pt0)-0.18,op(2,pt0)+(1-2*tt)*0.07,`P`],\n \+ [op(1,pt1)+(2-2*tt)*0.1,op(2,pt1)+(1+tt)*0.1,`Q`],\n [op( 1,pt2)+0.2,op(2,pt2)+(2*tt-1)*0.1,`R`],\n [op(1,pt3)+(1-2*t t)*0.2,op(2,pt3)-0.02+tt*(tt-1)*0.4,`S`],\n [op(1,pt4)+0.05 -tt*0.18,op(2,pt4)-0.1,`T`],\n [op(1,pt5)+(1-2*tt)*0.2,op(2 ,pt5)+tt*(tt-1)*0.4,`U`]]):\n p2 := plot([[pt0,pt1,pt2,pt3,pt4,pt5], [pt0,pt1,pt2],[pt3,pt4]],\n color=[black,red,coral],thickness=2, style=[point,line$2],\n symbol=circle);\n if i<>0 then\n \+ p3 := plot([3*t*(t+1)*(2-t),3*t*(t-1)*(t-3),t=0..tt],\n c olor=green,thickness=2);\n end if;\n if i=0 then\n frms := fr ms,plots[display]([t1,p1,p2]);\n elif i=n then\n frms := frms,p lots[display]([t1,p1,p2,p3]);\n else\n frms := frms,plots[displ ay]([t1,t2,p1,p2,p3]);\n end if;\nend do:\nplots[display](frms,inseq uence=true,axes=none,title=`cubic Bezier curve animation`);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 "\004" }}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }