{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "moi" -1 256 "Times" 1 12 0 0 1 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" -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 Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Lucida Sans" 1 12 0 0 255 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 "R3 Font 2" -1 257 1 {CSTYLE " " -1 -1 "Courier" 1 10 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 {EXCHG {PARA 18 "" 0 "" {TEXT 257 29 "D\351velopp\351e d'une c ourbe plane" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "with(plots): \nps := (u,v) -> u[1]*v[1]+u[2]*v[2]:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Centre du cercle circonscrit \340 3 points" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "circ := proc(A,B,C)\n local x,y,M ,sol;\n M := [x,y];\n sol := solve(\{ps(2*M-A-B,A-B)=0, ps(2*M-A-C,A -C)=0\},\{x,y\});\n subs(sol,M)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "circ([0,0],[1,0],[2,1]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#\"\"\"\"\"##\"\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Centre approch\351 de courbure et limite" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "centre_app := (M,t,h) -> circ(M(t-h ),M(t),M(t+h)):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "M := t ->[t,t^2]: Ih := (t,h) -> centre_app(M,t,h): Ih(t,h): I0 := t -> subs( h=0,Ih(t,h)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "plot([[M( t)[1],M(t)[2],t=-5..5],\n [Ih(t,0.5)[1],Ih(t,0.5)[2],t=-5..5],\n \+ [Ih(t,0.1)[1],Ih(t,0.1)[2],t=-5..5],\n [I0(t)[1],I0(t)[2],t= -5..5]],\n color=[red,blue,green,black],view=[-5..5,0..5]);" }} {PARA 13 "" 1 "" {GLPLOT2D 490 282 282 {PLOTDATA 2 "6'-%'CURVESG6$7S7$ $!\"&\"\"!$\"#DF*7$$!1LLLe%G?y%!#:$\"1))4j<'znG#!#97$$!1mmT&esBf%F0$\" 1c'Gjf))*3@F37$$!1LL$3s%3zVF0$\"1&fB#*HQw\">F37$$!1ML$e/$QkTF0$\"16CC: '3Ut\"F37$$!1nmT5=q]RF0$\"1`H[zW!3c\"F37$$!1ML3_>f_PF0$\"1hS)ej%>39F37 $$!1++vo1YZNF0$\"1?Gj>xWe7F37$$!1ML3-OJNLF0$\"15UU#oJC6\"F37$$!1++v$*o 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os(u),sin(u)]):\nplot([[M(t)[1],M(t)[2],t=-0.5..0.5],\n [Mc(0,u)[ 1],Mc(0,u)[2],u=-Pi..Pi]],\n color=[red,blue],view=[-0.5..0.5,0.. 0.5]);\nplot([[M(t)[1],M(t)[2],t=-1.5..1.5],\n [Mc(0.5,u)[1],Mc(0 .5,u)[2],u=-Pi..Pi]],\n color=[red,blue],view=[-2..1.5,-0.5..3]); " }}{PARA 13 "" 1 "" {GLPLOT2D 462 240 240 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$!3++++++++]!#=$\"3++++++++DF*7$$!3MLLLe%G?y%F*$\"3k()4j<'znG#F*7$ $!3OmmT&esBf%F*$\"3gb'Gjf))*3@F*7$$!3KLL$3s%3zVF*$\"3>&fB#*HQw\">F*7$$ !33LL$e/$QkTF*$\"3x5CC:'3Ut\"F*7$$!3!pm;/\"=q]RF*$\"3)G&H[zW!3c\"F*7$$ !3SLL3_>f_PF*$\"3CgS)ej%>39F*7$$!3))***\\(o1YZNF*$\"3w>Gj>xWe7F*7$$!3] LL3-OJNLF*$\"3<5UU#oJC6\"F*7$$!3C++v$*o%Q7$F*$\"3AD:f:%>%e(*!#>7$$!3am mm\"RFj!HF*$\"3;p>b2*QnW)FZ7$$!3JLL$e4OZr#F*$\"3%*3/-q?zptFZ7$$!3=+++v '\\!*\\#F*$\"3m1w6y#\\_C'FZ7$$!33+++DwZ#G#F*$\"3&3kD'3Tq4_FZ7$$!3-+++D .xt?F*$\"3\\01&3OB0I%FZ7$$!3OLL3-TC%)=F*$\"3=FeBOeP]NFZ7$$!3!omm;4z)e; F*$\"3_D)o2%)z=v#FZ7$$!3+nmmm`'zY\"F*$\"3'>!GtNu8<#FZ$\"3$QGDJUIsr%!#@7$$!3vDMLLe* 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lexion).\nV\351rification :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "centre := proc(M,t)\n local dx,dy,d2x,d2y;\n dx := diff(M(t)[1 ],t); dy := diff(M(t)[2],t);\n d2x := diff(dx,t); d2y := diff(d y,t);\n evalm(M(t) + (dx^2+dy^2)/(dx*d2y-d2x*dy)*[-dy,dx])\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "M := t -> [t^2,t^3]; Ic := \+ t -> centre(M,t);\nIc(t):\nplot([[M(t)[1],M(t)[2],t=-5..5],\n [Ic (t)[1],Ic(t)[2],t=-5..5]],\n color=[red,blue],view=[-5..5,-5..5]) :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$%)operatorG%&a rrowGF(7$*$)9$\"\"#\"\"\"*$)F/\"\"$F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IcGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%'centreG6$%\" MG9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "M := t -> [t ,t^3]; Ic := t -> centre(M,t);\nIc(t);\nplot([[M(t)[1],M(t)[2],t=-2..2 ],\n [Ic(t)[1],Ic(t)[2],t=-2..2]],\n color=[red,blue],view=[ -5..5,-5..5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$% )operatorG%&arrowGF(7$9$*$F-\"\"$F(F(6\"" 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S'*)eT'Fbo7$$!13$\\2Y)y%*=Fhq$\"15]ZP;]%)yFbo7$$!1X]@(RY'>EFbo$\"1'3\" eRpnF**Fbo7$$!1(p$f2LfxrFbo$\"1Nc$[D&e\"H\"F07$$!1`#oK&y&yP\"F0$\"1O6W TY1S;F07$$!1K&)G7#[xY#F0$\"1')G\"fFFG6#F07$$!1\"o&4x%\\t&RF0$\"1Hl-P@U _EF07$$!1/)HAe;(QhF0$\"1PZ&4.NEK$F07$$!1eBTiHM`!*F0$\"1y-]n8y!4%F07$$! 1\">)zEs:98F[\\l$\"1+u2NNKB]F07$$!1^A&*fOMJ=F[\\l$\"1=^$z9RT0'F07$$!1% *\\tE6o7DF[\\l$\"1P1-[Vu^sF07$$!1K&\\p\"*p6P$F[\\l$\"1>[Kh^Z\"f)F07$$! 13\\Z>6(*\\VF[\\l$\"1C6'eq2O'**F07$$!18mL$))))[t&F[\\l$\"1*>$G%fX6<\"F [\\l7$$!1J2%4h6/D(F[\\l$\"1O]VX(QTM\"F[\\l7$$!1xvA-*H+?*F[\\l$\"1F.;hM yY:F[\\l7$$!1h5 " 0 "" {MPLTEXT 1 0 226 "M := t - > [x(t),y(t)];\nnormale := D(x)(t)*(X-x(t))+D(y)(t)*(Y-y(t)) = 0;\nnor mder := (D@@2)(x)(t)*(X-x(t))+(D@@2)(y)(t)*(Y-y(t)) = D(x)(t)^2+D(y)(t )^2;\nsol := solve(\{normale,normder\},\{X,Y\});\nsimplify(subs(sol,[X ,Y])-(M(t)+MI));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6 $%)operatorG%&arrowGF(7$-%\"xG6#9$-%\"yGF/F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 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i^m-F$6$7$7$$\"1ddA%pB^'=F.$!1l'[_A0\\'>F@7$$!1.c*e7ak%>Fe`m$!1u4)3D9C U\"F@Fi^m-F$6$7$7$$\"1jAja=iOGF.$!1xiK\\&[y\">F@7$$!1>V\"[E)RZoFe`m$!1 ]\\0\\xkA8F@Fi^m-F$6$7$7$$\"1-)HrUx(zPF.$!1lalk$H;&=F@7$$!1h[$zN=+i\"F .$!1zEaK#>.>\"F@Fi^m-F$6$7$7$$\"1rP+8n;&o%F.$!11.W6$4pw\"F@7$$!1p)z`\" HH&3$F.$!1kuH2THM5F@Fi^m-F$6$7$7$$\"13;zhLuVbF.$!1.yW%)[`k;F@7$$!1LK)p )4H6^F.$!1$ol[/*GZ')F.Fi^m-F$6$7$7$$\"1YjUf(GpM'F.$!1u>6v*Gba\"F@7$$!1 AD*4.'HqwF.$!1.1,O!=?#pF.Fi^m-F$6$7$7$$\"1+E\"Hu(p'3(F.$!1%yS6l!369F@7 $$!1+$G!*>4x1\"F@$!1(p#yGO8o_F.Fi^m-F$6$7$7$$\"1'\\-^yecv(F.$!1VYuvK`i 7F@7$$!1Q&*3'G9&*R\"F@$!1zH8+LPtPF.Fi^m-F$6$7$7$$\"1)f\"R[y7Z$)F.$!1v_ >&3r85\"F@7$$!1sdaipuWb))F.$ !1Z^F)eV?H*F.7$$!1CV18m7$3#F@$!1(*)R;%=I/:F.Fi^m-F$6$7$7$$\"1f.W2Vyu#* F.$!1?Zg'HLvZ(F.7$$!1UKsBg\\$R#F@$!1h@BEHGRyFe`mFi^m-F$6$7$7$$\"1hO]mG q,'*F.$!1<&yR'*4$)e&F.7$$!1%QYhn?cl#F@$!1*HTMz@AF$Fe`mFi^m-%%VIEWG6$;$ !\"%F*$\"\"%F*F]`p" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" " Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57 " "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" }} }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Cycloides :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 430 "M := t -> (1+1/6)*[cos(t),sin(t)] + (1/6)*[ cos(7*t),sin(7*t)];\nC := t -> centre(M,t);\n### WARNING: calls to `C` for generating C code should be replaced by codegen[C]\nmap(simplify, C(t));\n### WARNING: calls to `C` for generating C code should be repl aced by codegen[C]\n### WARNING: calls to `C` for generating C code sh ould be replaced by codegen[C]\nplot([[M(t)[1],M(t)[2],t=0..2*Pi],[C(t )[1],C(t)[2],t=0..2*Pi]],color=[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$%)operatorG%&arrowGF(7$,&-%$cosG6#,$ 9$\"\"(#\"\"\"\"\"'-F/6#F2#F3F6,&-%$sinGF0F4-F%\"CGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%'c entreG6$%\"MG9$F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6# 7$,$*&,**$-%$cosG6#%\"tG\"\"'\"#K*$F+\"\"%!#c*$F+\"\"#\"#G!\"(\"\"\"F8 F+F8#!\"\"F2,$*&,*F*\"\")F1!#5F4\"\"$F:F8F8-%$sinGF-F8F:" }}{PARA 13 " " 1 "" {GLPLOT2D 320 306 306 {PLOTDATA 2 "6$-%'CURVESG6$7]q7$$\"1LLLLL LL8!#:\"\"!7$$\"1L3?t_\"=J\"F*$\"1\"32f)e%pc\"!#;7$$\"1+M^ky]^7F*$\"1W *e\"G%*)o&HF17$$\"1tFKgj]w6F*$\"1Sx\"*>eF8RF17$$\"1G\\%fZ1>4\"F*$\"1M5 Z'=`7e%F17$$\"1!Q&*3Rf++\"F*$\"1da'[5-)))\\F17$$\"1PdMKZ#>E*F1$\"1-)ek nEB5&F17$$\"1jh+Zk\"H!))F1$\"1*=aO;'p^]F17$$\"1><_-jJg')F1$\"1sg)oEO++ 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5\"*F1$\"1`=nkEIv=F17$$!1[@[f>([w)F1$\"1lu@Y\"4#R5F17$$!1H$yzwI8N)F1$ \"1SoV#f\\[%\\F^r7$$!125GJ`!p%zF1$\"1m>&4QAzp\"F^r7$$!1$Q$\\qr*[i(F1$ \"174E_S**\\BFir7$$!1O2$\\Y_-](F1$\"1G\"yU_tz3#F`bl7$$!1#pa;GPDh(F1$!1 k\"3s5?`+#Fir7$$!1QeuR()F1$!1q[59bNi**F^r7$$!1/XmC)**e3*F1$!15umC&)o(y\"F1 7$$!1uCi!3guB*F1$!1s!H0@!QmFF17$$!1N?E(fSZ6*F1$!1())\\()pM^(QF17$$!1!G 9f3dWo)F1$!1(Gdo(H`d\\F17$$!19t`F1$!1**>%oj#G qqF17$$!1i6t# ))F17$$!1A7yV\"4W6#F1$!1/fJNf@Z%*F17$$!1nF`6J-R6F1$!1..V76h[)*F17$$!1J H*3&4A]`Fir$!1#*)oT%Gn****F17$$\"16sS5A0&3\"F1$!1K\")Q!R!*G')*F17$$\"1 AN\\q>&*3@F1$!1&*)p0'4K]%*F17$$\"1TTqWq11HF1$!1nE+2*4C&))F17$$\"1FBf(3 )[]MF1$!1xbqY&G^;)F17$$\"1e9VRJE\\PF1$!1Wbsj3.tuF17$$\"144bo<#o#QF1$!1 K*f!R-iApF17$$\"1pF1;]n(y$F1$!1Hs&*\\j4)f'F17$$\"173>5>+]PF1$!1-#y\"HQ >&\\'F17$$\"1A;'\\]Hd#QF1$!1._Oq$*p%e'F17$$\"1#GK\\prM5%F1$!10>ll-(yy' F17$$\"1(\\RaItvh%F1$!1$oDE+U$GqF1$!1M#R'G*=\"4mF17$$\"1 5N&y:*y`zF1$!11)f))*\\_;fF17$$\"1F%G$)4=Zo)F1$!1I:XG*oq&\\F17$$\"1]A^0 [D$4*F1$!1OX7A\\7eRF17$$\"1)p)pnvPQ#*F1$!1^')[B-GGHF17$$\"1:=mLwr>\"*F 1$!1sy[Mx`5>F17$$\"1zj&fPCiy)F1$!1d()46_?w5F17$$\"1bjLusRj$)F1$!1W@UTG Eo]F^r7$$\"1LvRr7BZzF1$!1$)4'zr-**p\"F^r7$$\"1iIC_Ui>wF1$!1A'3L*Qf+AFi r7$Fdjl$!1Q^wiI!f8%!#S-Fjil6&F\\jlF+F+F]jl" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "On v\351rifie que les courbes sont semblables \+ : il faut faire tourner la rouge de -Pi/6 et faire une homoth\351tie d e rapport |C(0)|." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "### W ARNING: calls to `C` for generating C code should be replaced by codeg en[C]\nmap(simplify,subs(t=0,C(t)));\nsim := (M,a,k) -> k*[cos(a)*M[1] -sin(a)*M[2],sin(a)*M[1]+cos(a)*M[2]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$#\"\"$\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$simGR6%%\"MG%\"aG%\"kG6\"6$%)operatorG%&arrowGF**&9&\"\"\"7$,&*&- %$cosG6#9%F0&9$6#F0F0F0*&-%$sinGF6F0&F96#\"\"#F0!\"\",&*&FF0F0F0F*F*6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "Mr : = t -> sim(M(t),-Pi/6,3/4);\n### WARNING: calls to `C` for generating \+ C code should be replaced by codegen[C]\n### WARNING: calls to `C` for generating C code should be replaced by codegen[C]\nplot([[Mr(t)[1],M r(t)[2],t=0..2*Pi],[C(t)[1],C(t)[2],t=0..2*Pi]],color=[red,blue]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MrGR6#%\"tG6\"6$%)operatorG%&arrowG F(-%$simG6%-%\"MG6#9$,$%#PiG#!\"\"\"\"'#\"\"$\"\"%F(F(6\"" }}{PARA 13 "" 1 "" {GLPLOT2D 326 326 326 {PLOTDATA 2 "6$-%'CURVESG6$7]q7$$\"1)QWy .a-m)!#;$!1+++++++]F*7$$\"1WxpW[43\"*F*$!1S)Ro1Y:!RF*7$$\"1RUMjbhP#*F* $!1'y*>r%)fsFF*7$$\"1B.i+964\"*F*$!1$Hirk^,(=F*7$$\"12XOX065))F*$!1NUI a!Q!>6F*7$$\"1\")*))G2xjO)F*$!1Ag2zp+*4&!#<7$$\"1Z^G$3p\"HzF*$!1:RkWQj \"f\"FF7$$\"16ZC9![?h(F*$!1#=i$=g2#*>!#=7$$\"1Blf)QIBB%zF*$\"11QC[k=q;FF7$$ \"17!f(om*)[$)F*$\"11DB5M;?\\FF7$$\"1#y$y!em_w)F*$\"1=^l'G#))R5F*7$$\" 1Jq\"fMJ(*4*F*$\"1:fWp%=f$=F*7$$\"1_?'*)3o$Q#*F*$\"1*ey3D.9\"GF*7$$\"1 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F*$!1**>%oj#GqqF*7$$!1i6t#))F*7$$!1A7yV\"4W6#F*$!1/fJNf@Z%*F*7$$!1nF`6J-R6F*$!1..V7 6h[)*F*7$$!1JH*3&4A]`FQ$!1#*)oT%Gn****F*7$$\"16sS5A0&3\"F*$!1K\")Q!R!* G')*F*7$$\"1AN\\q>&*3@F*$!1&*)p0'4K]%*F*7$$\"1TTqWq11HF*$!1nE+2*4C&))F *7$$\"1FBf(3)[]MF*$!1xbqY&G^;)F*7$$\"1e9VRJE\\PF*$!1Wbsj3.tuF*7$$\"144 bo<#o#QF*$!1K*f!R-iApF*7$$\"1pF1;]n(y$F*$!1Hs&*\\j4)f'F*7$$\"173>5>+]P F*$!1-#y\"HQ>&\\'F*7$$\"1A;'\\]Hd#QF*$!1._Oq$*p%e'F*7$$\"1#GK\\prM5%F* $!10>ll-(yy'F*7$$\"1(\\RaItvh%F*$!1$oDE+U$GqF*$!1M#R'G*= \"4mF*7$$\"15N&y:*y`zF*$!11)f))*\\_;fF*7$$\"1F%G$)4=Zo)F*$!1I:XG*oq&\\ F*7$$\"1]A^0[D$4*F*$!1OX7A\\7eRF*7$$\"1)p)pnvPQ#*F*$!1^')[B-GGHF*7$$\" 1:=mLwr>\"*F*$!1sy[Mx`5>F*7$$\"1zj&fPCiy)F*$!1d()46_?w5F*7$$\"1bjLusRj $)F*$!1W@UTGEo]FF7$$\"1LvRr7BZzF*$!1$)4'zr-**p\"FF7$$\"1iIC_Ui>wF*$!1A '3L*Qf+AFQ7$Fijl$!1Q^wiI!f8%!#S-F^jl6&F`jlFdjlFdjlFajl" 1 2 0 1 0 2 9 0 4 2 1 0.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "V\351rification formelle" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "M := t -> evalm((1+a)*[cos( t),sin(t)] + a*[cos(t+t/a),sin(t+t/a)]);\nC := t -> centre(M,t);\nmap( simplify,C(t));\nmap(simplify,subs(t=0,\"));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%&evalmG6#,&*& ,&\"\"\"F2%\"aGF2F27$-%$cosG6#9$-%$sinGF7F2F2*&F3F27$-F66#,&F8F2*&F8F2 F3!\"\"F2-F:F>F2F2F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGR6# %\"tG6\"6$%)operatorG%&arrowGF(-%'centreG6$%\"MG9$F(F(6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$,$*&,(*&%\"aG\"\"\"-%$cosG6#*(%\" tGF,,&F,F,F+F,F,F+!\"\"F,F,*&-F.6#F1F,F+F,F3F5F3F,,&F+\"\"#F,F,F3F3,$* &,(*&F+F,-%$sinGF/F,F,*&-F>F6F,F+F,F3F@F3F,F7F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7$*$,&%\"aG\"\"#\"\"\"F+!\"\"\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "Mr := t -> sim(M(t+a*Pi),-a *Pi,1/(1+2*a));\n### WARNING: calls to `C` for generating C code shoul d be replaced by codegen[C]\nzero := evalm(Mr(t)-C(t));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#MrGR6#%\"tG6\"6$%)operatorG%&arrowGF(-%$simG6 %-%\"MG6#,&9$\"\"\"*&%\"aGF4%#PiGF4F4,$F5!\"\"*$,&F6\"\"#F4F4F9F(F(6\" " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%zeroG-%'VECTORG6#7$,**&,&%\"aG \"\"#\"\"\"F.!\"\",&*&-%$cosG6#*&F,F.%#PiGF.F.,&*&,&F.F.F,F.F.-F36#,&% \"tGF.F5F.F.F.*&F,F.-F36#,(F=F.F5F.*&F " 0 "" {MPLTEXT 1 0 52 "z1 := map(expand,zero): z2 := map(s implify,z1,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z2G-%'VECTORG6 #7$\"\"!F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Inesp\351r\351 !!! \+ R\351cr\351ation : d\351velopp\351es successives d'une hypocycloide " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "C0 := t -> (9/8)*[cos(t) ,sin(t)] + (1/8)*[cos(9*t),-sin(9*t)];\nfor i from 1 to 5 do\n C||i : = map(simplify,centre(C||(i-1),t));\n C||i := unapply([C||i[1],C||i[2 ]],t)\nod:\nplot([seq([C||i(t)[1],C||i(t)[2],t=0..2*Pi],i=0..5)],color =[red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C0GR6#%\"tG6\"6$%) operatorG%&arrowGF(,&7$-%$cosG6#9$-%$sinGF0#\"\"*\"\")*&#\"\"\"F6F97$- F/6#,$F1F5,$-F3F " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "34 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }