{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 13 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "From last time: An affine transformation A is represented as vector with 5 elements.\nFirst com pute image of a point in R^2 under A:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "transform_point := proc(A, x)\n evalf( [\nx[1]*A[1] *cos(A[5]) - x[2]*A[2]*sin(A[5]) + A[3],\n x[1]*A[1]*sin(A[5]) + x[ 2]*A[2]*cos(A[5]) + A[4]] )\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "A polygon is a list of the vertices, next apply map A to the po lygon:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "transform_polygo n:= proc(A, vertices)\n [seq( transform_point(A, vertices[i]), i=1... nops(vertices) ) ]\n end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "To draw the nth level approximation to your fractal, the following proc. is basically a loop which applies each of the affine trans. for your \+ fractal to each of the polygons at the (n-1) level:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "IFS:= proc( n, list_of_transforms, base_vertices )\n local vertices, j, i, k,s;\n vertices := base_ver tices;\n for j from 1 to n do\n s:= NULL;\n for i from 1 to nop s(list_of_transforms) do\n s := s,\n seq( transform_polygon( list_of_transforms[i], \n [vertices][k] ), k=1...nops([vert ices]))\n od;\n vertices := s;\n od;\n plots[polygonplot]([ver tices], axes=none, scaling=constrained, color=blue);\n end:" }}} {EXCHG {PARA 256 "" 1 "" {TEXT -1 182 "To draw the fractal from the co urse web page, with reflection, in the affine trans. for the lower rig ht quadrant, need to reflect about the y-axis, so replace A[1]=.5 with A[1]= -.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "reflectdrag on:= [ [.5, .5, -.25, .25, -Pi/2],\n [.5, .5, -.25, - .25, 0],\n [-.5, .5, .25, -.25, Pi] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "base_square:= [ [-.5, -.5], [.5, -. 5], [.5, .5], [-.5, .5] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IFS(7, reflectdragon, base_square);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG6gcs7&7$$!%v$*!\"&$\"'voH!\"'7$ F($\"(vo/$!\"(7$$!(Dc,\"F1F/7$F3F+7&7$F($\"%DJ!\"%7$F3F8F2F.7&F27$$!'v $4\"F-F/7$F>F8F;7&F@7$$!(v=<\"F1F87$FCF/F=7&7$$!$D\"!\"$F87$FHF/FEFB7& FE7$FCF+7$FHF+FK7&FNFM7$FC$\"(D1*GF17$FHFQ7&7$F>F+7$F>FQFPFM7&FP7$FC$ \"&D\"GF*7$F>FYFV7&FenFX7$FC$\"(vVt#F17$F>Fhn7&7$FHFY7$FHFhnFgnFX7&Fgn 7$FC$\"'DcEF-7$FHF`oF]o7&Fbo7$FH$\"(D\"yDF17$FCFeoF_o7&7$FH$\"#D!\"#7$ FCFjoFgoFdo7&Fgo7$F>Feo7$F>FjoF]p7&F`pF_p7$F3Feo7$F3Fjo7&7$F>F`o7$F3F` oFbpF_p7&Fbp7$F(Feo7$F(F`oFfp7&7$$!&D\"yF-Fjo7$$!'v$f)F1Fjo7$F_qFeo7$F 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eflection:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Koch := [ [1 /3, 1/3, 0,0,0], [1/3, 1/3, 1/3, 0, Pi/3],\n [1/3, -1/3, 2/ 3, 0, 2*Pi/3], [1/3, 1/3, 2/3, 0,0] ]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Or (TJ's suggestion), don't use reflection in 3rd piece, \+ just rotate -60 shift:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 " altKoch := [ [1/3, 1/3, 0,0,0], [1/3, 1/3, 1/3, 0, Pi/3],\n \+ [1/3, 1/3, 1/2, sqrt(3)/6, -Pi/3], [1/3, 1/3, 2/3, 0,0] ]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "base_line:= [ [0,0], [1,0] ] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IFS(5, Koch, base_line );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGON SG6\\[o7$7$$\"\"!F)F(7$$\"+PjA:T!#7F(7$F*7$$\"+2&RG<'F-$\"+`0*Qc$F-7$7 $$\"+tEXI#)F-F(F/7$F57$$\"+,zcM7!#6F(7$F97$$\"+=#H.W\"F<$\"+]0*Qc$F-7$ F>7$F:$\"+.6yFrF-7$7$$\"+M04Y;Fk3$Fy\"Fyq'fFk)F\"FezF]r7$Fa]l7$F`\\l$\"+(QBZ\\#F<7$7$Fj[l$\"+KVLQ@F9Fez$\"+]?RrnF<7$7$$\"+bp<*R\"FezFeblFjbl7$F`cl7$$\"+\"pC! 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MonethirdCantor, base_square);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG6*7&7$$!+_=&=& =!#6F(7$$\"+_=&=&=F*F(7$F,F,7$F(F,7&7$$\"+dbbbbF*F(7$$\"+g#f#f#*F*F(7$ F5F,7$F2F,7&7$$\"+Pq.P?!#5F(7$$\"+3uS2CF=F(7$F?F,7$F;F,7&7$$\"+yxxxFF= F(7$$\"+[\"[\"[JF=F(7$FHF,7$FEF,7&7$$\"+#[\"[\"['F=F(7$$\"+_=&=&oF=F(7 $FQF,7$FNF,7&7$$\"+BAAAsF=F(7$$\"+$f#f#f(F=F(7$FZF,7$FWF,7&7$$\"+/Pq.( )F=F(7$$\"+vS2u!*F=F(7$F]oF,7$FjnF,7&7$$\"+XWWW%*F=F(7$$\"+:[\"[\")*F= F(7$FfoF,7$FcoF,-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%'CO LOURG6&%$RGBG$\"\"!FgpFfp$\"*++++\"!\")" 1 2 0 1 10 0 2 6 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "IFS(4, MonethirdCantor, base_square);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG627&7$$!+2&RG< '!#7F(7$$\"+2&RG<'F*F(7$F,F,7$F(F,7&7$$\"+_=&=&=!#6F(7$$\"+`(>k3$F4F(7 $F6F,7$F2F,7&7$$\"+dM7!z'F4F(7$$\"+g8pC!)F4F(7$F?F,7$Fk3`#FJF(7$FenF,7$FXF,7&7$$\"+oXB,H 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%!o*F*F(7$FeoF,7$FboF,-%*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINED G-%'COLOURG6&%$RGBG$\"\"!FfpFep$\"*++++\"!\")" 1 2 0 1 10 0 2 6 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "IFS(6, Cantor(.4), base_square);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%)POLYGONSG6\\o7&7$$!&![?!\"(F (7$$\"&![?F*F(7$F,F,7$F(F,7&7$$\"&g4%F*F(7$$\"&?>)F*F(7$F5F,7$F2F,7&7$ $\"'?J8F*F(7$$\"'!3u\"F*F(7$F>F,7$F;F,7&7$$\"'gX>F*F(7$$\"'?bBF*F(7$FG F,7$FDF,7&7$$\"'?NOF*F(7$$\"'![/%F*F(7$FPF,7$FMF,7&7$$\"'g\\UF*F(7$$\" '?fYF*F(7$FYF,7$FVF,7&7$$\"'?r^F*F(7$$\"'!3e&F*F(7$F\\oF,7$FinF,7&7$$ \"'g&y&F*F(7$$\"'?&>'F*F(7$FeoF,7$FboF,7&7$$\"'?&R*F*F(7$$\"'![!)*F*F( 7$F^pF,7$F[pF,7&7$$\"(g4+\"F*F(7$$\"(?>/\"F*F(7$FgpF,7$FdpF,7&7$$\"(?J 4\"F*F(7$$\"(!3M6F*F(7$F`qF,7$F]qF,7&7$$\"(gX:\"F*F(7$$\"(?b>\"F*F(7$F iqF,7$FfqF,7&7$$\"(?NK\"F*F(7$$\"(![k8F*F(7$FbrF,7$F_rF,7&7$$\"(g\\Q\" F*F(7$$\"(?fU\"F*F(7$F[sF,7$FhrF,7&7$$\"(?rZ\"F*F(7$$\"(!3=:F*F(7$FdsF 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