{VERSION 5 0 "IBM INTEL NT" "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 }{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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 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 "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 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 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warni ng" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Error" 7 8 1 {CSTYLE "" -1 -1 "" 0 1 255 0 255 1 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 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 "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 "Normal" -1 256 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 18 "" 0 "" {TEXT -1 39 "Bielles Trois barres-20020104-R F err\351ol" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 " Version finale \340 peu pr\350s satisfai sante" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "tiges BA=b, AD=d, DC=c, B C=a\narticulations consid\351r\351es comme fixes : B et C donc tige BC fixe" }}{PARA 0 "" 0 "" {TEXT -1 455 "Le cercle de centre B de rayon \+ b (support de la trajectoire de A) coupe la droite BC en B1 \340 gauch e et B2 \340 droite.\nLe cercle de centre C de rayon c (support de la \+ trajectoire de D) coupe la droite BC en C1 \340 gauche et C2 \340 droi te.\nLes conditions de passage en ces points sont respectivement :\npB 1:= abs(a+b-c)<=d and d<=a+b+c) : pB2:= abs(c-abs(a-b))<=d and c+abs (a-b)>=d:\npC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c- b)<=d and a+c+b>=d :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "On fixe \+ sur la tige AD un triangle APD tel que le vecteur AP soit d\351duit d u vecteur AD par une similitude d'angle \"angle\" et de rapport \"r\". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 370 "En mettant partout un peti t epsilon = 0.0001, en plus \340 t1, en moins \340 t2 tout marche !!!! \nDes cas superflus peuvent \352tre supprim\351s (j'ai laiss\351 pour pouvoir afficher dans quels cas on est)\nSeule l'animation dans les c as o\371 la courbe rouge est trac\351e en deux arcs n'est pas dans le \+ bon ordre : je n'ai pas modifi\351 car \347a compliquerait beaucoup la r\351daction des display" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 99 "restart:\nwith(plots):\nsim:=(V,t,r) -> [r*V[1]*cos (t)-r*V[2]*sin(t),r*V[1]*sin(t)+r*V[2]*cos(t)]: \n" }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4600 "bielle:=proc(a,b,c,d,angle,r) # donner des r\351els par exemple b=8./3 et non 8/3\n local U,A,B,C, D1,D2,P1,P2,lieu_P1,lieu_P2,\n AD1,AD2,n,cercle_A,cercle_D, t t,debut,\n artiFixe,arti1,arti2,barres1,barres2,triangle1,tri angle2,mobile1,mobile2,k,\n pB1,pB2,pC1,pC2,t1,t2,t12,delta,d elta2,cas,eps;\n tt:=time():\n n:=7: # nre de frames\n pB1:= abs( a+b-c)<=d and d=d: \n pC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d a nd a+c+b>d : \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n \+ if b>c then\n printf(\"%s%a%s,%a\\n\",\n \"Par raison \+ de sym\351trie, on se limite \340 b <= c or b=\",b,\" c=\",c):\n el if d>=a+b+c or a>=b+c+d or c>=a+b+d then\n print(\"M\351canisme \+ impossible ou sans mouvement possible\")\n else\n\n debut:=0;t 12:=t1: eps:=0.0001:\n \n if pB1 and pB2 then\n cas:= \"pB1 and pB2\":\n t1:=-evalf(Pi): \n t2:=t 1+2*evalf(Pi): t12:=-t1:\n delta:=(t2-t1)/2: delta2:=delta: \n if not(pC1) then debut:=-n fi;\n elif pB1 and not(pB 2) then\n cas:=\"pB1 and not(pB2)\":\n t1:=eva lf(arccos((a^2+b^2-(c-d)^2)/2/a/b ))+eps:\n t2:=evalf(2*Pi -t1):\n delta:=(t2-t1)/2: t12:=t2: delta2:=-delta: debut:= 0 :\n if pC2 then \n cas:=cat(cas,\" and pC2\"): \n else cas:=cat(cas,\" and not(pC2)\") fi;\n \+ \n elif pB2 and not(pB1) then\n cas :=\"pB2 and not(pB1)\":\n t1:=-evalf(arccos((a^2+b^2-(c+d)^2 )/2/a/b ))+ eps:\n t2:=-t1: debut:=0: delta:=(t2-t1)/2: \n \+ t12:=t2:delta2:=-delta:\n if not(pC2) then cas:=c at(cas,\" and not pC2)\"):\n else cas:=cat(cas,\" and not \+ pC2)\"): # ne se pr\351sente pas ??\n fi;\n else # not (pB1) and not(pB2)\n cas:=\"not(pB1) and not(pB2)\":\n \+ t1:=evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )) +eps: \n t2:=eva lf(arccos((a^2+b^2-(c+d)^2)/2/a/b ))-eps:\n debut:=0: delta:=( t2-t1)/2: t12:=t2:delta2:=-delta:\n fi;\n\n U:=simplify([solve (( b*cos(t)-a-c*cos(w) )^2 + (b*sin(t)-c*sin(w) )^2-d^2,w)]):\n B:= [0,0]: C:=[a,0]: A:=[b*cos(t),b*sin(t)]:\n cercle_A:=plot([op(A),t= 0..2*Pi],color =blue,thickness=1):\n cercle_D:=plot([a+c*cos(w),c*s in(w),w=0..2*Pi],color =blue,thickness=1): \n D1:=[a+c*cos(U[1]) ,c*sin(U[1])]: AD1:=(D1-A):\n D2:=[a+c*cos(U[2]),c*sin(U[2])]: AD2: =(D2-A):\n P1:=simplify(expand(A+sim(AD1,evalf(angle),r))):\n P2 :=simplify(expand(A+sim(AD2,evalf(angle),r))):\n lieu_P1:=plot([op( P1),t=t1..t2],color =COLOR(RGB,1,0,0),thickness=1):\n lieu_P2:=plot ([op(P2),t=t1..t2],color =COLOR(RGB,0,1,0),thickness=1):\n#print(\"t1, t2\",t1,t2,cas);\n# --------------------------- \n# si Maple 7, rempl acer partout \"CIRCLE\" par \"CIRCLE,16\"\n# ------------------------- --\n artiFixe:= PLOT(POINTS(C,SYMBOL(CIRCLE,16),COLOR(RGB,0,0,0))), \n PLOT(POINTS(B,SYMBOL(CIRCLE,16),COLOR(RGB,0,0,0))):\n barres1:='PLOT(CURVES(evalf( subs(t=t1+k*delta/n,[B,A,D1,C]) ),THI CKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=t12+k*delta2/n ,[B,A,D2,C]) ),THICKNESS(2) ))':\n arti1:= 'PLOT(POINTS(op(evalf(su bs(t=t1+k*delta/n,[A,D1]))),SYMBOL(CIRCLE,16))),\n PLOT(PO INTS(op(evalf(subs(t=t1+k*delta/n,[P1]))), SYMBOL(BOX,14),COLOR(RGB,0. 6,0,0)))':\n arti2:='PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[A, D2]))),SYMBOL(CIRCLE,16))),\n PLOT(POINTS(op(evalf(subs(t= t12+k*delta2/n,[P2]))),SYMBOL(BOX,14),COLOR(RGB,0,0.6,0)))':\n tria ngle1:=display([seq(PLOT(POLYGONS(evalf( subs(t=t1+k*delta/n,[D1,P1,A] ) ),THICKNESS(2) ),\n COLOR(RGB,1,0.5,0.5)),k=debut. .debut+2*n)],insequence=true):\n triangle2:=display([seq(PLOT(POLYG ONS(evalf( subs(t=t12+k*delta2/n,[D2,P2,A]) ),THICKNESS(2) ),\n \+ COLOR(RGB,0.5,1,0.5)),k=debut..debut+2*n)],insequence=true ):\n mobile1:=display([display([seq(display([eval(barres1),eval(art i1)]),k=debut..debut+2*n)],\n insequence =true,color=black), triangle1]):\n mobile2:=display([display([seq(d isplay([eval(barres2),eval(arti2)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle2]): \n# print(display([lieu_P1,lieu_P2]));\n#plotsetup(gif,plotoutput=\"c:/aaC /bielleRoberts.gif\",plotoptions=\"width=300,height=300\");\nprint(dis play([cercle_A,cercle_D,lieu_P1,lieu_P2,artiFixe, di splay([mobile1,eval(mobile2)],insequence=true)],axes=none,scaling=cons trained));\n#plotsetup(inline);\nfi;\n printf(\" Temps mis %a second es\",time()-tt)\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the nam e changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4630 "restart:sim:=(V,t,r) -> [r*V[1]*cos(t)-r*V[2]*sin(t ),r*V[1]*sin(t)+r*V[2]*cos(t)]: \nbielle:=proc(a,b,c,d,angle,r) # d onner des r\351els par exemple b=8./3 et non 8/3\n local U,A,B,C,D1 ,D2,P1,P2,lieu_P1,lieu_P2,\n AD1,AD2,n,cercle_A,cercle_D, tt, debut,\n artiFixe,arti1,arti2,barres1,barres2,triangle1,trian gle2,mobile1,mobile2,k,\n pB1,pB2,pC1,pC2,t1,t2,t12,delta,del ta2,cas,eps;\n tt:=time():\n n:=7: # nre de frames\n pB1:= abs(a+ b-c)<=d and d=d:\n \+ pC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d and \+ a+c+b>d : \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n if b>c then\n printf(\"%s%a%s,%a\\n\",\n \"Par raison de \+ sym\351trie, on se limite \340 b <= c or b=\",b,\" c=\",c):\n elif \+ d>=a+b+c or a>=b+c+d or c>=a+b+d then\n print(\"M\351canisme imp ossible ou sans mouvement possible\")\n else\n\n debut:=0;t12: =t1: eps:=0.0001:\n \n if pB1 and pB2 then\n cas:=\"pB 1 and pB2\":\n t1:=-evalf(Pi): \n t2:=t1+2* evalf(Pi): t12:=-t1:\n delta:=(t2-t1)/2: delta2:=delta:\n \+ if not(pC1) then debut:=-n fi;\n elif pB1 and not(pB2) th en\n cas:=\"pB1 and not(pB2)\":\n t1:=evalf(ar ccos((a^2+b^2-(c-d)^2)/2/a/b ))+eps:\n t2:=evalf(2*Pi-t1): \n delta:=(t2-t1)/2: t12:=t2: delta2:=-delta: debut:=0 :\n if pC2 then \n cas:=cat(cas,\" and pC2\"):\n \+ else cas:=cat(cas,\" and not(pC2)\") fi;\n \+ \n elif pB2 and not(pB1) then\n cas:=\"pB 2 and not(pB1)\":\n t1:=-evalf(arccos((a^2+b^2-(c+d)^2)/2/a/ b ))+ eps:\n t2:=-t1: debut:=0: delta:=(t2-t1)/2: \n \+ t12:=t2:delta2:=-delta:\n if not(pC2) then cas:=cat(cas ,\" and not pC2)\"):\n else cas:=cat(cas,\" and not pC2)\" ): # ne se pr\351sente pas ??\n fi;\n else # not(pB1) \+ and not(pB2)\n cas:=\"not(pB1) and not(pB2)\":\n t1:= evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )) +eps: \n t2:=evalf(arc cos((a^2+b^2-(c+d)^2)/2/a/b ))-eps:\n debut:=0: delta:=(t2-t1) /2: t12:=t2:delta2:=-delta:\n fi;\n\n U:=simplify([solve(( b*c os(t)-a-c*cos(w) )^2 + (b*sin(t)-c*sin(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]: A:=[b*cos(t),b*sin(t)]:\n cercle_A:=plot([op(A),t=0..2*P i],color =blue,thickness=1):\n cercle_D:=plot([a+c*cos(w),c*sin(w), w=0..2*Pi],color =blue,thickness=1): \n D1:=[a+c*cos(U[1]),c*sin (U[1])]: AD1:=(D1-A):\n D2:=[a+c*cos(U[2]),c*sin(U[2])]: AD2:=(D2-A ):\n P1:=simplify(expand(A+sim(AD1,evalf(angle),r))):\n P2:=simp lify(expand(A+sim(AD2,evalf(angle),r))):\n lieu_P1:=plot([op(P1),t= t1..t2],color =COLOR(RGB,1,0,0),thickness=1):\n lieu_P2:=plot([op(P 2),t=t1..t2],color =COLOR(RGB,1,0,0),thickness=1):\n#print(\"t1,t2\",t 1,t2,cas);\n# --------------------------- \n# si Maple 7, remplacer p artout \"CIRCLE\" par \"CIRCLE,16\"\n# ---------------------------\n \+ artiFixe:= PLOT(POINTS(C,SYMBOL(CIRCLE,16),COLOR(RGB,0,0,0))),\n \+ PLOT(POINTS(B,SYMBOL(CIRCLE,16),COLOR(RGB,0,0,0))):\n ba rres1:='PLOT(CURVES(evalf( subs(t=t1+k*delta/n,[B,A,D1,C]) ),THICKNESS (2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=t12+k*delta2/n,[B,A, D2,C]) ),THICKNESS(2) ))':\n arti1:= 'PLOT(POINTS(op(evalf(subs(t=t 1+k*delta/n,[A,D1]))),SYMBOL(CIRCLE,16))),\n PLOT(POINTS(o p(evalf(subs(t=t1+k*delta/n,[P1]))),COLOR(RGB,1,0,0)))':\n arti2:=' PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[A,D2]))),SYMBOL(CIRCLE,16) )),\n PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[P2]))),C OLOR(RGB,1,0,0)))':\n triangle1:=display([seq(PLOT(POLYGONS(evalf( \+ subs(t=t1+k*delta/n,[D1,P1,A]) ),THICKNESS(2) ),\n C OLOR(RGB,1,0.1,0.1)),k=debut..debut+2*n)],insequence=true):\n trian gle2:=display([seq(PLOT(POLYGONS(evalf( subs(t=t12+k*delta2/n,[D2,P2,A ]) ),THICKNESS(2) ),\n COLOR(RGB,1,0.1,0.1)),k=debut ..debut+2*n)],insequence=true):\n mobile1:=display([display([seq(di splay([eval(barres1),eval(arti1)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle1]):\n mob ile2:=display([display([seq(display([eval(barres2),eval(arti2)]),k=deb ut..debut+2*n)],\n insequence=true,colo r=black), triangle2]): \n#print(display([lieu_P1,lieu_P2]));\n#plot setup(gif,plotoutput=\"c:/aaC/bielleRoberts.gif\",plotoptions=\"width= 300,height=300\");\nprint(display([cercle_A,cercle_D,lieu_P1,lieu_P2,a rtiFixe, display([mobile1,eval(mobile2)],insequence= true)],axes=none));\n#plotsetup(inline);\nfi;\n printf(\" Temps mis \+ %a secondes\",time()-tt)\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning , the name changecoords has been redefined\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 " Voici les cas qui posent probl\350me (mail du 29/01/0 3)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a:=4:k:=2.9:b:=a*k/3;c :=b:d:=b:\n bielle(a,b,c,d,Pi/3,1);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+nmmmQ!\"*" }}{PARA 6 "" 1 "" {TEXT -1 52 " a=4 ; b=3.8666 66667 ; c=3.866666667 ; d=3.866666667" }}{PARA 8 "" 1 "" {TEXT -1 45 " Plotting error, non-numeric vertex definition" }}{PARA 6 "" 1 "" {TEXT -1 26 " Temps mis 43.919 secondes" }}}{EXCHG {PARA 0 "" 0 "" {GLPLOT2D 249 131 131 {PLOTDATA 2 "6&-%(ANIMATEG6@7,-%'CURVESG6$7&7$$ 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"Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}}{PARA 6 "" 1 "" {TEXT -1 25 " Temps mis 4.140 secondes" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 939 "Cas du paral l\351logramme et du contre-parall\351logramme (a = d et b = c) : seul \+ cas o\371 P et Q sont \340 r\351volution compl\350te ; la courbe se d \351compose en un cercle et une quartique bicirculaire rationnelle, qu i est une podaire d'ellipse quand a > b ou d'hyperbole quand a < b (l a bielle est l'axe focal d'une conique \340 centre roulant sur une con ique \351gale).\n#D'autre part, si cela vous dit, je vous propose de r egarder le cas du contre-parall\351logramme a = d et b = c.\ndans ce c as les extr\351mit\351s de la bielle sont les foyers d'une ellipse ou \+ d'une hyperbole roulant sur une conique \351gale : le m\351canisme a d \351ja \351t\351 vu dans \nhttp://www.mathcurve.com/courbes2d/delaunay /delaunay.shtml mais sous une autre forme.\n\nL'int\351ret est que l'o n obtient ainsi toutes les quartiques bicirculaires unicursales.\nhttp ://www.mathcurve.com/courbes2d/quarticbicirculairerationnelle/quarticb icirculairerationnelle.shtml\nCe serait bien de visualiser les deux co niques..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:=4: d:=a: b:=5: c:=b:\nbielle(a,b,c,d,Pi/3,1);\n\n " }}{PARA 6 "" 1 "" {TEXT -1 22 " a=4 ; b=5 ; c=5 ; d=4" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%(ANIMATEG6@7,-%'CURVESG6 $7&7$$\"\"!F-F,7$$!\"&F-$\"+3Q.^?!#=7$$!\"\"F-F17$$\"\"%F-F,-%*THICKNE SSG6#\"\"#-%'POINTSG6%F.F4-%'SYMBOLG6$%'CIRCLEG\"#;-F?6%7$$!+********H !\"*$\"+<;5kMFK-FB6$%$BOXG\"#9-%&COLORG6&%$RGBG$\"\"'F6F-F--%)POLYGONS G6%7%F4FHF.F:-FS6&FU\"\"\"$\"\"&F6Fin-F(6%7S7$$FjnF-F,7$$\"3aX0)p=\"=` \\!#<$\"3b@$)y74REoF37$$\"3uc:T\\5!p$[Fco$\"3aiuzzWkm7Fco7$$\"3&y^I%RK HCYFco$\"3!o;GC;b:!>Fco7$$\"3@I%)Rfa`EVFco$\"3Kdnsp/@1DFco7$$\"3eFvV;g 7_RFco$\"3Ow9,NGziIFco7$$\"3([n>V3(GTNFco$\"3i\"=Y/Nr(HNFco7$$\"3gk'fJ 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\"+QV%[]%FK$!+)p=%p@FK7$$\"+QV%[])FKFcjoF7F:-F?6%F`joFejoFA-F?6%7$$\"+ RV%[]'FK$\"+D>r#FKFNFifp-FY6%7%F^[qFg[ qFijpF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$$\"+wYg76FKFcgo7$$!*57 d4)FK$!+Fqvm8FKF7F:-F?6%Fc\\qFf\\qFA-F?6%7$$!+_')R')GFK$!+S9N&y%FKFNFi fp-FY6%7%Ff\\qF_]qFc\\qF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$$\"+ 7!\\u6$FK$!+5u:4RFK7$$!*%pnX\\FK$!+K.j!>#FKF7F:-F?6%F[^qF`^qFA-F?6%7$$ !+tWZo#FKF7F :-F?6%F`dqFedqFA-F?6%7$$\"+u(G(*z#FK$!+%eM/#yFhcqFNFifp-FY6%7%FedqF^eq F`dqF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$$\"+kYg76FKF[`p7$$!*77d 4)FK$\"+Cqvm8FKF7F:-F?6%FjeqF]fqFA-F?6%7$$\"+?@V*=$FK$\"+c^/c9FKFNFifp -FY6%7%F]fqFffqFjeqF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$$!+uYg76 FK$\"+g&RY([FK7$$!*$f(4A*FK$\"+Qvy\"z)F^hpF7F:-F?6%FbgqFggqFA-F?6%7$$ \"+C%>GW#FK$\"+GJ*=/$FKFNFifp-FY6%7%FggqF`hqFbgqF:F^gpF[oFj^lF^hlFfgmF \\boFabo7,-F(6$7&F+7$$!+9!\\u6$FK$\"+3u:4RFK7$Feip$\"+qEbN`F^hpF7F:-F? 6%F\\iqFaiqFA-F?6%7$$\"+s^2*y)F^hp$\"+!oX)zSFKFNFifp-FY6%7%FaiqFhiqF\\ iqF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$$!+TV%[]%FK$\"+#p=%p@FK7$ Fjgp$\"+(f3W`#F^hpF7F:-F?6%FdjqFijqFA-F?6%7$$!+:I#**3\"FK$\"+m-G_UFKFN Fifp-FY6%7%FijqF`[rFdjqF:F^gpF[oFj^lF^hlFfgmF\\boFabo7,-F(6$7&F+7$F/$! +U,J:6Fco7$F\\fp$!+\"\\L#R7F3F7F:-F?6%F\\\\rF_\\rFA-F?6%7$$!+3+++IFK$ \"+5;5kMFKFNFifp-FY6%7%F_\\rFf\\rF\\\\rF:F^gpF[oFj^lF^hlFfgmF\\boFabo- %*AXESSTYLEG6#%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!6\"F i]r-%%FONTG6#%(DEFAULTG-%%VIEWG6$F^^rF^^r" 1 2 0 1 10 0 2 9 1 1 1 1.000000 33.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" }}} {PARA 6 "" 1 "" {TEXT -1 24 " Temps mis .532 secondes" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 " Cas marchant bien" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 40 "# cercles non s\351cants de \+ rayons in\351gaux " }{TEXT -1 3 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "a:=4:b:=1.5:c:=2:d:=a+c-b:#cas limite droite \nbielle (a,b,c,d,Pi/3,1);bielle(a,b,c,d,Pi,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a:=4:b:=1.5:c:=2:d:=a+b-c:#cas limite gauche \nbielle (a,b,c,d,Pi/4,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:=4:b:=1.5:c:=2:d:=a+c-b-1.3: \nbielle(a,b,c,d,Pi/3, 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a:=4:b:=1.5:c:=2:d:= a+c-b+2: \nbielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 37 "# cercles non s\351cants de rayons \351gaux" }{TEXT -1 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a:=4:b:=4/3.:c:=b:d:=6:bie lle(a,b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a:=4 :b:=1.5:c:=1.5:d:=a+c-b:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 53 "a:=4:b:=1.5:c:=1.5:d:=a+c-b+2:bielle(a,b,c,d,P i/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:=4:b:=1.5:c:=1 .5:d:=a+c-b-0.2:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:=4:b:=1.5:c:=1.5:d:=a+c-b-1.5:bielle(a,b,c,d,Pi/3,1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a:=4:b:=1.5:c:=1.5:d: =a+c-b-2:bielle(a,b,c,d,Pi/3,1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 47 "# cercles tangents ext\351ri eurs de rayons in\351gaux" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "a:=4:b:=1.5:c:=2.5:d:=a+b-c:bielle(a,b,c,d,Pi/3,1); #d=2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a:=4:b:=1.5:c:=2.5: d:=a+c-b:bielle(a,b,c,d,Pi/3,1);#d=5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:=4:b:=1.5:c:=2.5:d:=1.5:bielle(a,b,c,d,Pi/3,1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:=4:b:=1.5:c:=2.5:d:=4:bie lle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:= 4:b:=1.5:c:=2.5:d:=7:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 262 45 "# cercles tangents ext\351rieurs de rayons \351gaux" } {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a:=4:b:=2: c:=2:d:=a+b-c:bielle(a,b,c,d,Pi/3,1);#d=4" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "a:=4:b:=2:c:=2:d:=3.6:bielle(a,b,c,d,Pi/3,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=2:d:=2:bielle(a ,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:= 2:c:=2:d:=5:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=2:d:=6:bielle(a,b,c,d,Pi/3,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "# Watt\na:=10:b:=3: c:=b:d: =evalf(sqrt(a^2-4*b^2)):bielle(a,b,c,d,0,1/2);\na:=4:b:=2:c:=2:d:=3:bi elle(a,b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 35 "# cercle s s\351cants de rayons in\351gaux" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 58 "a:=4:b:=1.5:c:=3.5:d:=a+b-c-0.5: \n bielle(b ,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=4:b:= 2:c:=3:d:=a+b-c:bielle(a,b,c,d,Pi/3,1);#d:=3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=4:b:=2:c:=3:d:=a-b+c:bielle(a,b,c,d,Pi/3,1);# d:=5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "a:=4:b:=2:c:=3:d:=1 :bielle(a,b,c,d,*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a:=4:b:=2:c:=3:d:=0.95:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=3:d:=4:bielle(a,b,c,d,Pi/3,1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=3:d:=6:bie lle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:= 4:b:=2:c:=3:d:=8:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#lemniscate 2\na:=sqrt(2.):b:=1:c:=a:d:=b:\nbielle(a, b,c,d,Pi,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 33 "# cercles s\351c ants de rayons \351gaux" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=4:b:=3:c:=3:d:=a+b-c:bielle(a,b,c,d,Pi/3,1);#d:=4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a:=4:b:=3:c:=3:d:=2:bie lle(b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a:=4. 3:b:=3:c:=3:d:=2.7:bielle(a,b,c,d,Pi/3,1);# allure de rideau" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "a:=4:b:=3:c:=3:d:=1.5:bielle (a,b,c,d,Pi/3,1);# allure de roberts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=3:c:=3:d:=5:bielle(a,b,c,d,Pi/3,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=3:c:=3:d:=7:bielle(a ,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "#Watt\na :=4:b:=3:c:=3:d:=7:bielle(a,b,c,d,0,1/2);\na:=4:b:=3:c:=3:d:=2:bielle( a,b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "#lemnisc ate 1 \na:=sqrt(2.):b:=1:c:=b:d:=a:\nbielle(a,b,c,d,0,1/2.);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#limacon\na:=1:b:=sqrt(2.):c :=b:d:=a:\nbielle(a,b,c,d,0,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 265 33 "# cercles s\351cants \+ de rayons \351gaux" }{TEXT -1 5 " \340 a" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "a:=4:b:=4:c:=4:d:=a+b-c:bielle(a,b,c,d,Pi/3,1);#d:= 4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=4:c:=4:d:=3:bi elle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a: =4:b:=4:c:=4:d:=7:bielle(a,b,c,d,Pi/3,1);# pb d'animation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=4:c:=4:d:=9:bielle(a,b,c,d, Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "#Watt\na:=4:b:= 4:c:=4:d:=4:bielle(a,b,c,d,0,1/2);\na:=4:b:=4:c:=4:d:=3:bielle(a,b,c,d ,0,1/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 33 "# cercles s\351cants de rayons \351gaux" }{TEXT -1 4 " > a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=4:b:=6:c:=6:d:=a+b-c:bielle(a,b,c,d,Pi/3,1);#d:=4 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a:=4:b:=6:c:=6:d:=6:bie lle(a,b,c,d,0,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "a:=4:b :=6:c:=6:d:=5:bielle(a,b,c,d,Pi/3,1);# pb d'animation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:=4:b:=6:c:=6:d:=10:bielle(a,b,c,d ,Pi/3,1);#d:=4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 267 30 "# cercles tangents int\351rieurs \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a:=4:b:=2:c:=6:d:=a-b+c :bielle(a,b,c,d,Pi/3,1);#d=8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=6:d:=6:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=6:d:=2:bielle(a,b,c,d,Pi/3,1); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "a:=4:b:=2:c:=6:d:=10:bi elle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 33 "# un ce rcle contenu dans l'autre " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=4:b:=2:c:=8:d:=a-b+c:bielle(a,b,c,d,Pi/3,1);#d=10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=8:d:=8:bielle(a,b,c,d, Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=7: d:=4:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a:=4:b:=2:c:=8:d:=6:bielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "a:=4:b:=2:c:=8:d:=12:bielle(a,b,c,d,Pi/3,1); " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 " Quelques cas o\371 l'animat ion est incorrecte" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=4:b :=4:c:=6:d:=a-b+c-3: bielle(a,b,c,d,Pi/3,1);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "# animation incorrecte pour d>4\na:=4:b:=5:c:=b:d:=6:k:=1.5:\nbielle(a,b,c,d,arccos(-1/2/k),k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "a:=4:b:=4:c:=2*b+1:d:=a+c -b-4:k:=1.5:\nbielle(a,b,c,d,arccos(-1/2/k),k);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 " " 0 "" {TEXT -1 58 "Programme origine de R. Ferr\351ol - mail du 04/01 /03 - 14h08" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 13 "Texte du mail" }} {PARA 0 "" 0 "" {TEXT -1 451 "J'ai travaill\351 aussi sur les courbes \+ du trois barres\nhttp://www.mathcurve.com/courbes2d/troisbarres/troisb arre.shtml\nci-dessous le fichier maple correspondant.\nla m\351thode \+ est assez artisanale :\nl'\351quation reliant t et theta (=th en maple ) a plusieurs solutions, mais\nfsolve n'en donne qu'une.\nj'ai donc ut ilis\351 la solution pr\351c\351dente pour r\351duire l'amplitude de\n recherche de la solution suivante.\n\nMais lorsqu'il y a un croisement , ca ne marche pas..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1312 " restart:with(plots):\na:=2:b:=1.8;c:=1.1;mini:=max(2*a-b-c,-2*a+b-c,-2 *a-b+c);maxi:=2*a+b+c;d:=(mini+maxi)/2;\n pi:=evalf(Pi):n:=40:e:=1:th: =0:\n\nt:=fsolve((2*a+b*cos(th+tt)-c*cos(th-tt))^2+(b*sin(th+tt)-c*sin (th-tt))^2=d^2,tt=0..pi);\nquad[0]:=plot([[a,0],[a+b*cos(th+t),b*sin(t h+t)],[-a+c*cos(th-t),c*sin(th-t)],[-a,0],[a,0]],color=black,axes=none ):th0:=th:\n for k to n+1 do\n #print(k,th*180/pi,t*180/pi):\n M[k]:=[ a+b*cos(th+t),b*sin(th+t)]:N[k]:=[-a+c*cos(th-t),c*sin(th-t)]:\n R[k]: =(M[k]+N[k])/2:U[k]:=(M[k]-N[k])/2:V[k]:=[-U[k][2],U[k][1]]:\n\nquad[k ]:=display(plot([[[a,0],M[k],N[k],[-a,0],[a,0]],[M[k]+V[k],M[k]-V[k]], \n [N[k]+V[k],N[k]-V[k]],[R[k]+V[k], R[k]-V[k]]],color=black,axes=none,thickness=3),\n \+ pointplot([[0,0],(M[k]+N[k])/2]),symbol=circle) :\n\nth:=th0+k*pi /n:t:=fsolve((2*a+b*cos(th+tt)-c*cos(th-tt))^2+(b*sin(th+tt)-c*sin(th- tt))^2=d^2,tt=t-e..t+e)\n od:\nm:=n+1:\ndisplay(plot([[a+b*cos(tt),b*s in(tt),tt=0..2*Pi],[-a+c*cos(tt),c*sin(tt),tt=0..2*Pi]],color=blue),\n plot([[seq(R[k],k=1..m)],[seq(R[k]+V[k],k=1..m)],[seq(R[k] -V[k],k=1..m)],\n [seq(M[k]+V[k],k=1..m)],[seq(N[k] +V[k],k=1..m)],[seq(M[k]-V[k],k=1..m)],\n [seq(N[k] -V[k],k=1..m)]],color=red),display(seq(quad[k],k=1..m),insequence=true ));" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " Esculier - mail 050103 - 20h30" }}{EXCHG {PARA 0 "" 0 "" {TEXT 257 13 "Texte du mail" }}{PARA 0 "" 0 " " {TEXT -1 542 "Entre deux balades, je me suis pench\351 sur la bielle \340 trois barres ; le pg ci-joint ne traite pas le cas o\371 la plus petite barre ne fait pas un tour complet autour de son extr\351mit \351 fixe et s'arr\352te si les valeurs des deux premiers param\350tre s ne sont satisfaisantes.\nDe nombreuses imperfections subsistent sans doute !\n\nLes trajectoires des extr\351mit\351s des tiges sont assez surprenantes ! J'ai mis n=20 comme nombre de frames pour que ce soit \+ assez fluide et pas trop long \340 calculer ; il y a int\351r\352t \+ \340 faire executer en boucles.\n \nCordialement" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 91 "tiges articul\351es BA o AD o DC, B et C points f ixes, BA=b, D=d, DC=c, CB=a (au lieu de 2*a)" }}{PARA 0 "" 0 "" {TEXT -1 70 "On suppose b<=c et l'origine en B ( ainsi BA est simple \340 pa ram\351trer )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "restart: \nwith(plots):\n# tour complet du cercle_A si a+b-c <= d <= a+c-b\n# t our complet des deux si a=d et b=c\n# cas non trait\351 : mouvement al ternatif sur cercle_A si d > a+c-b \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1243 "bielle:=proc( c,d,partie) \n # ------- partie prend les val eurs 1 ou 2 -------------------\n # ------- selon la solution de solve choisie -----------------\n local U,u,A,B,C,DD,P,Q,lieu_P,lieu_Q,A D,n,cercle_A,cercle_D,j,a,b;\n a:=4; b:=1.5;\n printf(\" a=%a ; b= %a ; c=%a ; d=%a\\n\",a,b,c,d);\n if d<= a+c-b and d>= a+b-c and ( pa rtie=1 or partie =2) then\n U:=simplify([solve(( b*cos(t)-a-c*cos(w ) )^2 + (b*sin(t)-c*sin(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n \+ A:=[b*cos(t),b*sin(t)]:\n cercle_A:=plot([op(A),t=0..2*Pi],color = red): \n u:=U[partie]:\n DD:=[a+c*cos(u),c*sin(u)]:\n AD:= combine(DD-A):\n P:=[A[1]-AD[2],A[2]+AD[1]]:\n Q:=[DD[1]-AD[2]/2 ,DD[2]+AD[1]/2]:\n lieu_P:=plot([op(P),t=0..2*Pi],color =black,line style=2):\n lieu_Q:=plot([op(Q),t=0..2*Pi],color =black,linestyle=2 ):\n cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =red):\n \+ n:=20:\n print(display([cercle_A,cercle_D,lieu_P,lieu_Q,\n \+ display([seq(PLOT(CURVES(evalf( subs(t=2*k*Pi/n,[B,A,P,A,DD,Q,DD,C ]) ) ),\n COLOR(RGB,0,0,1),THICKNESS(2)),k=0..n)],in sequence=true)],\n scaling=constrained,axes=none)); \n else print(\" valeur incorrecte des param\350tres : il faut a+b-c <= d <= a+c-b\") fi;\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, t he name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 98 "a:=4:b:=1.5:c:=2:d:=a+b-c: # cas limite\nprintf(\" \+ %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n bielle(c,d,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "a:=4:b:=1.5:c:=2:d:=a+b-c+0.2:\nprintf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n bielle(c,d,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "a:=4:b:=1.5:c:=2:d:=a+b-c :\nprintf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n bielle(c,d,2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "a:=4:b:=1.5:c:=2:d:=a+b-c+1: #cas limite \nprintf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\nbielle(c, d,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "a:=4:b:=1.5:c:=2:d :=a+b-c+1: #cas limite\nprintf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b); \nbielle(c,d,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "#cas b =c a=d les deux cercles sont enti\350rement parcourus\na:=4:b:=1.5:c:= 1.5;d:=a+b-c;\nprintf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\nbielle(c ,d,1);\nbielle(c,d,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Ferr\351ol - mail du 060103 - 15h11" }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 13 "Texte du mail" }}{PARA 0 "" 0 "" {TEXT -1 364 "merci beaucoup pour votre envoi.\n\nIl fallait oser utiliser s olve et non fsolve !\n\nIl est maintenant clair que les deux courbes p our chaque solution sont en continuit\351.\n\nen utilisant votre fichi er, j'ai fait ce qui suit :\n\n...( mis en ex\351cutable au-dessous) . ..\n\nmais ce serait \351videmment plus clair si les deux portions de \+ courbes \351taient\nparcourues \340 la suite...." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1034 "with(plots):\na:=4:b:=1.5:c:=2:d:=a+b-c:\n \+ u:=simplify([solve(( b*cos(t)-a-c*cos(w) )^2 +(b*sin(t)-c*sin(w) ) ^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos(t),b*sin(t)]:\n cercle_A:=plot([op(A),t=0..2*Pi],color =blue):\n u1:=u[1]:u2: =u[2]:\n D1:=[a+c*cos(u1),c*sin(u1)]:D2:=[a+c*cos(u2),c*sin(u2 )]:\n U1:=(D1-A):U2:=(D2-A):\n\n V1:=[-U1[2],U1[1]]: V2:=[-U2[2],U2[1] ]:\n M1:=(A+D1)/2+V1: M2:=(A+D2)/2+V2:\n lieu_M1:=plot([op(M1) ,t=0..2*Pi],color =red):\n lieu_M2:=plot([op(M2),t=0..2*Pi],color \+ =green):\n cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =b lue):\n\n n:=30:display([cercle_A,cercle_D,lieu_M1,lieu_M2,\n \+ display([seq(display(\nPLOT(CURVES(evalf(subs(t=2*k*Pi/n ,[B,A,D1,C,D2,A]) ) ),COLOR(RGB,0,0,0),THICKNESS(2)),\nPLOT(CURVES(eva lf(subs(t=2*k*Pi/n,[A,M1,D1]) ) ),COLOR(RGB,1,0,0),THICKNESS(2)),\nPLO T(CURVES(evalf(subs(t=2*k*Pi/n,[A,M2,D2]) ) ),COLOR(RGB,0,1,0),THICKNE SS(2)))\n ,k=0..n)],insequence=true)], \+ scaling=constrained,axes=none);\n" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 33 " Esculier mail du 060103 - 19h06" }}{EXCHG {PARA 256 " " 0 "" {TEXT -1 14 "Texte du mail " }}{PARA 0 "" 0 "" {TEXT -1 261 "Vo us \351crivez : \"mais ce serait \351videmment plus clair si les deux \+ portions de courbes \351taient\nparcourues \340 la suite....\"\n \nJe \+ ne sais pas si j'ai bien compris ce que vous voulez dire par cela ; re gardez ci-dessous comment j'ai interpr\351t\351 la chose : est-ce cela ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1385 "# ----------------- --\nwith(plots):\n tt:=time():\n a:=4:b:=1.5:c:=2:d:=a+b-c:\n u:= simplify([solve(( b*cos(t)-a-c*cos(w) )^2 +\n (b*sin(t)-c*sin(w) )^2-d ^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos(t),b*sin(t)]:\n \+ cercle_A:=plot([op(A),t=0..2*Pi],color =blue):\n u1:=u[1]:u2: =u[2]:\n D1:=[a+c*cos(u1),c*sin(u1)]:D2:=[a+c*cos(u2),c*sin(u 2)]:\n U1:=(D1-A):U2:=(D2-A):\n \n V1:=[-U1[2],U1[1]]: V2:=[-U2[2],U 2[1]]:\n M1:=(A+D1)/2+V1: M2:=(A+D2)/2+V2:\n lieu_M1:=plot([ op(M1),t=0..2*Pi],color =red):\n lieu_M2:=plot([op(M2),t=0..2*Pi] ,color =green):\n cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi], color =blue):\n \n n:=30:\n display([cercle_A,cercle_D,lieu_M1,lieu_M2 ,\n display([\n display([seq(display(\n \+ PLOT(CURVES(evalf(subs(t=k*Pi/n,[B,A,D1,C]) ) ),COLOR(RGB ,0,0,0)),\n PLOT(CURVES(evalf(subs(t=k*Pi/n,[A ,M1,D1]) ) ),COLOR(RGB,1,0,0) ) ),k=-n..n)],\n \+ insequence=true,thickness=2),\n display([seq(display(\n \+ PLOT(CURVES(evalf(subs(t=k*Pi/n,[B,A,D2,C]) ) ),C OLOR(RGB,0,0,0)),\n PLOT(CURVES(evalf(subs(t=k *Pi/n,[A,M2,D2]) ) ),COLOR(RGB,0,1,0) )),k=n..3*n)],\n \+ insequence=true,thickness=2)],\n inse quence=true)], scaling=constrained,axes=none); \n \n \n time()-tt ; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 " Ferr \351ol mail du 070103 - 9h12" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 14 " Texte du mail " }}{PARA 0 "" 0 "" {TEXT -1 417 "Vous \351crivez : \"ma is ce serait \351videmment plus clair si les deux portions\nde courbes \351taient\nparcourues \340 la suite....\"\n\nJe ne sais pas si j'ai \+ bien compris ce que vous voulez dire par cela ;\nregardez ci-dessous c omment j'ai >interpr\351t\351 la chose : est-ce cela ?\n\nexactement ! \n\nmais regardez le petit bug si vous prenez\n\na:=4:b:=1.5:c:=2:d:=a +c-b:\n\n(d\351sol\351 de ne pas en faire plus : je pars au travail)\n \n\nCordialement\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " Esculier mail du 070103 -10h20" }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 14 "Texte du mail " }}{PARA 0 "" 0 "" {TEXT -1 814 "Bonjou r,\n \n\"mais regardez le petit bug si vous prenez\na:=4:b:=1.5:c:=2:d :=a+c-b:\"\n \nLe fichier que je vous ai retourn\351 ne s'appliquait q u'au cas particulier que vous m'aviez envoy\351 car ne sachant pas si \+ c'\351tait vraiment ce que vous d\351siriez, je n'avais pas trait\351 \+ le cas g\351n\351ral tout de suite.\n \nCependant comme je trouvais l' animation int\351ressante, je l'ai fait ensuite hier soir ! Je vous j oins donc le cas g\351n\351ral.\n \nAu lieu d'un triangle ADP \340 peu pr\350s r\351gulier, si l'on a un angle obtus en D la trajectoire de \+ P est moins m\351lang\351e avec les bielles et cercles (jai mis un qua tri\350me param\350tre qui est l'angle (AD,DP).\nJ'ai mis des articula tions pour agr\351menter la chose ! J'ai structur\351 un peu l'\351nor me display final car c'e\373t \351t\351 difficile pour compl\351ter ou modifier si vous d\351sirez le faire.\n \nCordialement.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "tiges articul\351es BA o AD o DC, B et C points fixes, BA=b, D=d, DC=c, CB=a (au lieu de 2*a)" }}{PARA 0 "" 0 "" {TEXT -1 70 "On suppose b<=c et l'origine en B ( ainsi BA est simpl e \340 param\351trer )" }}{PARA 0 "" 0 "" {TEXT -1 99 "\nOn fixe sur l a tige AD un triangle APD tel que l'angle (AD,DP) soit donn\351 en pa ram\350tre et AP=AD." }}{PARA 0 "" 0 "" {TEXT -1 84 "Cecii permet de \+ \"sortir\" les trajectoires de P du reste des trac\351s si (AD,DP) " 0 "" {MPLTEXT 1 0 245 "restart:\nwith(plots):\n# tour complet du cercle_A si a+b-c <= d \+ <= a+c-b\n# tour complet des deux si a=d et b=c\n# cas non trait\351 : mouvement alternatif sur cercle_A si d > a+c-b\n\nrota:=(V,t) -> [V[1 ]*cos(t)-V[2]*sin(t),V[1]*sin(t)+V[2]*cos(t)]: \n" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 2664 "bielle:=proc( b,c,d,angle)\n local U,u,A,B,C,D 1,D2,P1,P2,Q1,Q2,lieu_P1,lieu_P2,\n lieu_Q2,lieu_Q1,AD1,AD2,n ,cercle_A,cercle_D,j,a, tt,debut,\n artiFixe,arti1,arti2,barr es1,barres2,triangle1,triangle2,mobile1,mobile2,k;\n a:=4; tt:=time (): \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n printf( \" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n if b<=c and d<= a+c-b and d \+ >= a+b-c then\n U:=simplify([solve(( b*cos(t)-a-c*cos(w) )^2 + (b* sin(t)-c*sin(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos( t),b*sin(t)]:\n cercle_A:=plot([op(A),t=0..2*Pi],color =red,thickne ss=1):\n cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =red, thickness=1): \n u:=U[1]:\n D1:=[a+c*cos(U[1]),c*sin(U[1])]: \n D2:=[a+c*cos(U[2]),c*sin(U[2])]:\n AD1:=(D1-A):#combine\n \+ AD2:=(D2-A):\n P1:=simplify(D1+rota(AD1,evalf(angle))):\n P2:=si mplify(D2+rota(AD2,evalf(angle))):\n lieu_P1:=plot([op(P1),t=0..2*P i],color =COLOR(RGB,0,0.7,0),thickness=1):\n lieu_P2:=plot([op(P2), t=0..2*Pi],color =COLOR(RGB,0,0.5,1),thickness=1):\n n:=30:\n de but:=0;\n if d=a-c+b then debut:=-n fi;\n# ------------------------ --- \n# si Maple 7, remplacer partout \"CIRCLE\" par \"CIRCLE,16\"\n# ---------------------------\n artiFixe:= PLOT(POINTS(C,SYMBOL(CIRC LE),COLOR(RGB,0,0,0))),\n PLOT(POINTS(B,SYMBOL(CIRCLE),C OLOR(RGB,0,0,0))):\n barres1:='PLOT(CURVES(evalf( subs(t=k*Pi/n,[B, A,D1,C]) ),THICKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t= k*Pi/n,[B,A,D2,C]) ),THICKNESS(2) ))':\n arti1:= 'PLOT(POINTS(op(ev alf(subs(t=k*Pi/n,[A,D1]))),SYMBOL(CIRCLE)))':\n arti2:='PLOT(POINT S(op(evalf(subs(t=k*Pi/n,[A,D2]))),SYMBOL(CIRCLE)))':\n triangle1:= display([seq(PLOT(POLYGONS(evalf( subs(t=k*Pi/n,[D1,P1,A]) ),THICKNESS (2) ),\n COLOR(RGB,0,0.7,0)),k=debut..debut+2*n)],in sequence=true):\n triangle2:=display([seq(PLOT(POLYGONS(evalf( subs (t=k*Pi/n,[D2,P2,A]) ),THICKNESS(2) ),\n COLOR(RGB,0 ,0.5,1)),k=debut+2*n..debut+4*n)],insequence=true):\n mobile1:=disp lay([display([seq(display([eval(barres1),eval(arti1)]),k=debut..debut+ 2*n)],\n insequence=true,color=black), t riangle1]):\n mobile2:=display([display([seq(display([eval(barres2) ,eval(arti2)]),k=debut+2*n..debut+4*n)],\n \+ insequence=true,color=black), triangle2]): \n\n print(disp lay([cercle_A,cercle_D,lieu_P1,lieu_P2,artiFixe,\n \+ display([mobile1,eval(mobile2)],insequence=true)], scaling=constraine d,axes=none));\n\n else print(\" valeur incorrecte des param\350tres : il faut a+b-c <= d <= a+c-b\") fi;\n printf(\" Temps mis %a secon des\",time()-tt)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a :=4:b:=1.5:c:=2:d:=a+c-b:#cas limite droite \nbielle(b,c,d,Pi/4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "#cas b=c a=d les deux cercl es sont enti\350rement parcourus\na:=4:b:=1.5:c:=1.5;d:=a+b-c;\nbielle (b,c,d,Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "# cas limi te \340 gauche avec cercles s\351cants\na:=4:b:=1.5:c:=3.5:d:=a+b-c: \+ \n bielle(b,c,d,Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "# cas limite \340 droite avec cercles s\351cants\na:=4:b:=1.5:c:=3.5:d: =a-b+c: \n bielle(b,c,d,Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a:=4:b:=1.5:c:=2:d:=a+b-c+0.5:# cas quelconque\n bielle(b,c,d, Pi/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 " Ferr\351ol mail du 080103 - 12h26" }} {EXCHG {PARA 256 "" 0 "" {TEXT -1 14 "Texte du mail " }}{PARA 0 "" 0 " " {TEXT -1 718 "votre animation est superbe et je ne me lasse pas de l a regarder.\n\ndans le pg joint, j'ai fait des modifs mineures de coul eur (standard de mes figures) et j'ai mis une similitude au lieu d'une rotation pour avoir un triangle quelconque.\na la fin j'ai mis des ca s particuliers, qui ne marchent pas car b = c...\n\nle r\351sultat g \351n\351ral est que le quadrilat\350re poss\350de un pivot \340 rotat ion compl\350te ssi\nla somme du plus petit c\364t\351 et du plus gran d est <= la somme des deux autres, et les pivots \340 rotation compl \350te sont alors les extr\351mit\351s du petit c\364t\351.\nsauf quan d il y a deux couples de c^t\351s \351gaux auxquels cas il y a trois o u 4 pivots \340 rotation compl\350te).\n\nj'en suis l\340 pour l'insta nt.\n\nEn vous remerciant de votre aide" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 68 " Version Esculier (modifi\351 apr\350s r\351ponse Ferr \351ol pour trac\351 continu)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "t iges articul\351es BA o AD o DC, B et C points fixes, BA=b, AD=d, DC= c, CB=a (au lieu de 2*a)" }}{PARA 0 "" 0 "" {TEXT -1 70 "On suppose b< =c et l'origine en B ( ainsi BA est simple \340 param\351trer )" }} {PARA 0 "" 0 "" {TEXT -1 99 "\nOn fixe sur la tige AD un triangle APD tel que l'angle (AD,DP) soit donn\351 en param\350tre et AP=AD." }} {PARA 0 "" 0 "" {TEXT -1 84 "Cecii permet de \"sortir\" les trajectoir es de P du reste des trac\351s si (AD,DP) " 0 "" {MPLTEXT 1 0 254 "restart:\nwith(plot s):\n# tour complet du cercle_A si a+b-c <= d <= a+c-b\n# tour complet des deux si a=d et b=c\n# cas non trait\351 : mouvement alternatif su r cercle_A si d > a+c-b\n\nsim:=(V,t,r) -> [r*V[1]*cos(t)-r*V[2]*sin(t ),r*V[1]*sin(t)+r*V[2]*cos(t)]: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2641 "bielle:=proc( b,c,d,angle,r)\n local U,u,A,B,C,D1,D2,P1,P2,Q1 ,Q2,lieu_P1,lieu_P2,\n lieu_Q2,lieu_Q1,AD1,AD2,n,cercle_A,cer cle_D,j,a, tt,debut,\n artiFixe,arti1,arti2,barres1,barres2,t riangle1,triangle2,mobile1,mobile2,k;\n a:=4; tt:=time(): \n print f(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n printf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n if b<=c and d<= a+c-b and d >= a+b-c then \n U:=simplify([solve(( b*cos(t)-a-c*cos(w) )^2 + (b*sin(t)-c*sin(w ) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos(t),b*sin(t)]: \n cercle_A:=plot([op(A),t=0..2*Pi],color =blue,thickness=1):\n \+ cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =blue,thickness=1 ): \n u:=U[1]:\n D1:=[a+c*cos(U[1]),c*sin(U[1])]:\n D2:=[a +c*cos(U[2]),c*sin(U[2])]:\n AD1:=(D1-A):#combine\n AD2:=(D2-A): \n P1:=simplify(D1+sim(AD1,evalf(angle),r)):\n P2:=simplify(D2+s im(AD2,evalf(angle),r)):\n lieu_P1:=plot([op(P1),t=0..2*Pi],color = COLOR(RGB,1,0,0),thickness=1):\n lieu_P2:=plot([op(P2),t=0..2*Pi],c olor =COLOR(RGB,0,1,0),thickness=1):\n n:=15:\n debut:=0;\n i f d=a-c+b then debut:=-n fi;\n# --------------------------- \n# si Ma ple 7, remplacer partout \"CIRCLE\" par \"CIRCLE,16\"\n# ------------- --------------\n artiFixe:= PLOT(POINTS(C,SYMBOL(CIRCLE),COLOR(RGB, 0,0,0))),\n PLOT(POINTS(B,SYMBOL(CIRCLE),COLOR(RGB,0,0,0 ))):\n barres1:='PLOT(CURVES(evalf( subs(t=k*Pi/n,[B,A,D1,C]) ),THI CKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=k*Pi/n,[B,A,D2 ,C]) ),THICKNESS(2) ))':\n arti1:= 'PLOT(POINTS(op(evalf(subs(t=k*P i/n,[A,D1]))),SYMBOL(CIRCLE)))':\n arti2:='PLOT(POINTS(op(evalf(sub s(t=k*Pi/n,[A,D2]))),SYMBOL(CIRCLE)))':\n triangle1:=display([seq(P LOT(POLYGONS(evalf( subs(t=k*Pi/n,[D1,P1,A]) ),THICKNESS(2) ),\n \+ COLOR(RGB,1,0,0)),k=debut..debut+2*n)],insequence=true): \n triangle2:=display([seq(PLOT(POLYGONS(evalf( subs(t=k*Pi/n,[D2,P 2,A]) ),THICKNESS(2) ),\n COLOR(RGB,0,1,0)),k=debut+ 2*n..debut+4*n)],insequence=true):\n mobile1:=display([display([seq (display([eval(barres1),eval(arti1)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle1]):\n \+ mobile2:=display([display([seq(display([eval(barres2),eval(arti2)]),k= debut+2*n..debut+4*n)],\n insequence=tr ue,color=black), triangle2]): \n\n print(display([cercle_A,cer cle_D,lieu_P1,lieu_P2,artiFixe,\n display([mobile1 ,eval(mobile2)],insequence=true)],axes=none));\n\n else print(\" val eur incorrecte des param\350tres : il faut a+b-c <= d <= a+c-b\") fi; \n printf(\" Temps mis %a secondes\",time()-tt)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a:=4:b:=1.5:c:=2:d:=a+c-b:#cas limi te droite \nbielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:=4:b:=1.5:c:=2:d:=a: \nbielle(b,c,d,2*Pi/3,1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "#cas b=c a=d les deux cercl es sont enti\350rement parcourus\na:=4:b:=1.5:c:=1.5;d:=a+b-c;\nbielle (b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# cas \+ limite \340 gauche avec cercles s\351cants\na:=4:b:=1.5:c:=3.5:d:=a+b- c: \n bielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# cas limite \340 droite avec cercles s\351cants\na:=4:b:=1.5:c: =3.5:d:=a-b+c: \n bielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "a:=4 :b:=2:c:=7:d:=a:#cas cercle droite englobe cercle gauche \nbielle(b,c, d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "a:=4:b:=4. 5:c:=b:d:=a/2:angle:=arccos(2*d/b):r:=sqrt((b/d)^2-0.25):#roberts \nbi elle(b,c,d,-2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a :=10:b:=3:c:=b:d:=sqrt(a^2-4*b^2):#watt \nbielle(b,c,d,0,1/2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a:=4:b:=5:c:=b:d:=2:#tchebyc hev \nbielle(b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a:=4:k:=2:b:=a*k/3;c:=b:d:=b:r:=evalf(sqrt(3)/2):#rideau\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 " Esculier mail du 100103 - 19h20 " }}{EXCHG {PARA 256 "" 0 "" {TEXT -1 14 "Texte du mail " }}{PARA 0 " " 0 "" {TEXT -1 462 "Bonsoir,\n \nJ'ai assay\351 de faire quelques cho se qui marche dans tout cas avec possibilit\351 de mouvement mais il n 'est pas certain que j'ai bien distingu\351 tous les cas ! \nD'autre p art, par suite des approximations de calculs, le trac\351 peut ne pas \+ se r\351aliser : c'est le cas pour certains exemples que vous aviez \+ soumis ; th\351oriquement pas de probl\350mes et ... pratiquement j'ai bien cherch\351 avant de trouver \340 quel endroit le blocage se prod uisait !\n \nCordialement" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 93 " Ve rsion avec discussion ( il n'est pas certain que tous les cas particul iers vont marcher !)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "tiges BA=b , AD=d, DC=c, BC=a\narticulations consid\351r\351es comme fixes : B et C donc tige BC fixe" }}{PARA 0 "" 0 "" {TEXT -1 455 "Le cercle de cen tre B de rayon b (support de la trajectoire de A) coupe la droite BC e n B1 \340 gauche et B2 \340 droite.\nLe cercle de centre C de rayon c \+ (support de la trajectoire de D) coupe la droite BC en C1 \340 gauche \+ et C2 \340 droite.\nLes conditions de passage en ces points sont respe ctivement :\npB1:= abs(a+b-c)<=d and d<=a+b+c) : pB2:= abs(c-abs(a-b ))<=d and c+abs(a-b)>=d:\npC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : \+ pC2:=abs(a+c-b)<=d and a+c+b>=d :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "On fixe sur la tige AD un triangle APD tel que le vecteur PD \+ soit d\351duit du vecteur AD par une similitude d'angle \"angle\" et d e rapport \"r\"." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 93 " Version avec discussion ( il n'est pas c ertain que tous les cas particuliers vont marcher !)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "tiges BA=b, AD=d, DC=c, BC=a\narticulations con sid\351r\351es comme fixes : B et C donc tige BC fixe" }}{PARA 0 "" 0 "" {TEXT -1 455 "Le cercle de centre B de rayon b (support de la traje ctoire de A) coupe la droite BC en B1 \340 gauche et B2 \340 droite.\n Le cercle de centre C de rayon c (support de la trajectoire de D) coup e la droite BC en C1 \340 gauche et C2 \340 droite.\nLes conditions de passage en ces points sont respectivement :\npB1:= abs(a+b-c)<=d and \+ d<=a+b+c) : pB2:= abs(c-abs(a-b))<=d and c+abs(a-b)>=d:\npC1:=abs(b- abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d and a+c+b>=d :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "On fixe sur la tige AD un tri angle APD tel que le vecteur PD soit d\351duit du vecteur AD par une s imilitude d'angle \"angle\" et de rapport \"r\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "restart:\nwith(plots):\nsim:=(V,t,r) -> [r*V[ 1]*cos(t)-r*V[2]*sin(t),r*V[1]*sin(t)+r*V[2]*cos(t)]: \n" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 4877 "bielle:=proc( b,c,d,angle,r) # donner de s r\351els par exemple b=8./3 et non 8/3\n local U,u,A,B,C,D1,D2,P1 ,P2,Q1,Q2,lieu_P1,lieu_P2,\n lieu_Q2,lieu_Q1,AD1,AD2,n,cercle _A,cercle_D,j,a, tt,debut,\n artiFixe,arti1,arti2,barres1,bar res2,triangle1,triangle2,mobile1,mobile2,k,\n pB1,pB2,pC1,pC2 ,t1,t2,t12,delta,delta2,cas,eps;\n a:=4.; tt:=time():\n n:=15: # n re de frames\n pB1:= abs(a+b-c)<=d and d=d:\n pC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d and a+c+b>d : \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n printf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b) ;\n if b>c then\n printf(\"%s%a%s,%a\\n\",\n \"Par rai son de sym\351trie, on se limite \340 b <= c or b=\",b,\" c=\",c):\n \+ elif d>=a+b+c or a>=b+c+d or c>=a+b+d then\n print(\"M\351cani sme impossible ou sans mouvement possible\")\n else\n\n debut: =0;t12:=t1: \n if pB1 and pB2 then #---MARCHE----\n c as:=\"pB1 and pB2\": \n t1:=-evalf(Pi): t2:=-t1: t12: =-t1:\n delta:=(t2-t1)/2: delta2:=delta:\n if not( pC1) then debut:=-n fi;\n elif pB1 and not(pB2) then\n#---------- ----MARCHE MMMAAALLL-------------------------------\n eps:=0. 0001:\n cas:=\"pB1 and not(pB2)\":\n t1:=evalf(arcco s((a^2+b^2-(c-d)^2)/2/a/b )):\n if pC2 then \n ca s:=cat(cas,\" and pC2\"):\n debut:=0 :\n t2:=e valf(2*Pi-t1-eps): \n # t2:=evalf(2*Pi-arccos((a^2+b^2-c^ 2)/2/a/b ))-0.1: \n delta:=(t2-t1)/2: \n t12:= t2: delta2:=-delta:\n else\n cas:=cat(cas,\" and not(pC2)\"):\n t2:=evalf(Pi+arccos((-a^2-b^2+(c-d)^2)/2/a /b )):\n t1:=t1+eps:t2:=t2-eps:# --------ET ALORS CA MARCH E !!!!\n debut:=0: \n delta:=(t2-t1)/2: \n \+ t12:=t2: delta2:=-delta:\n fi;\n#----------------- -------------------------------------\n \+ \n elif pB2 and not(pB1) then #---MARCHE----\n ca s:=\"pB2 and not(pB1)\":\n if not(pC2) then \n \+ t1:=-evalf(arccos((a^2+b^2-(c+d)^2)/2/a/b )): t2:=-t1:\n \+ cas:=cat(cas,\" and not pC2)\"):\n else cas:=cat(cas,\" and \+ not pC2)\"):\n fi;\n debut:=0: delta:=(t2-t1)/2: \+ \n t12:=t2:delta2:=-delta:\n else # not(pB1) and n ot(pB2) #---MARCHE----\n cas:=\"not(pB1) and not(pB2)\":\n \+ t1:=evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )): \n t2:=eval f(arccos((a^2+b^2-(c+d)^2)/2/a/b )):\n debut:=0:\n del ta:=(t2-t1)/2:\n t12:=t2:delta2:=-delta:\n fi;\n\n U:= simplify([solve(( b*cos(t)-a-c*cos(w) )^2 + (b*sin(t)-c*sin(w) )^2-d^2 ,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos(t),b*sin(t)]:\n cerc le_A:=plot([op(A),t=0..2*Pi],color =blue,thickness=1):\n cercle_D:= plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =blue,thickness=1): \n \+ u:=U[1]:\n D1:=[a+c*cos(U[1]),c*sin(U[1])]:\n D2:=[a+c*cos(U[2 ]),c*sin(U[2])]:\n AD1:=(D1-A): AD2:=(D2-A):\n P1:=simplify(e xpand(D1+sim(AD1,evalf(angle),r))):\n P2:=simplify(expand(D2+sim(AD 2,evalf(angle),r))):\n lieu_P1:=plot([op(P1),t=t1..t2],color =COLOR (RGB,1,0,0),thickness=1):\n lieu_P2:=plot([op(P2),t=t1..t2],color = COLOR(RGB,0,1,0),thickness=1):\n\n# --------------------------- \n# s i Maple 7, remplacer partout \"CIRCLE\" par \"CIRCLE,16\"\n# --------- ------------------\n\n artiFixe:= PLOT(POINTS(C,SYMBOL(CIRCLE),COLO R(RGB,0,0,0))),\n PLOT(POINTS(B,SYMBOL(CIRCLE),COLOR(RGB ,0,0,0))):\n barres1:='PLOT(CURVES(evalf( subs(t=t1+k*delta/n,[B,A, D1,C]) ),THICKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=t1 2+k*delta2/n,[B,A,D2,C]) ),THICKNESS(2) ))':\n arti1:= 'PLOT(POINTS (op(evalf(subs(t=t1+k*delta/n,[A,D1]))),SYMBOL(CIRCLE)))':\n arti2: ='PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[A,D2]))),SYMBOL(CIRCLE)) )':\n triangle1:=display([seq(PLOT(POLYGONS(evalf( subs(t=t1+k*delt a/n,[D1,P1,A]) ),THICKNESS(2) ),\n COLOR(RGB,1,0,0)) ,k=debut..debut+2*n)],insequence=true):\n triangle2:=display([seq(P LOT(POLYGONS(evalf( subs(t=t12+k*delta2/n,[D2,P2,A]) ),THICKNESS(2) ), \n COLOR(RGB,0,1,0)),k=debut..debut+2*n)],insequence =true):\n mobile1:=display([display([seq(display([eval(barres1),eva l(arti1)]),k=debut..debut+2*n)],\n inseq uence=true,color=black), triangle1]):\n mobile2:=display([display([ seq(display([eval(barres2),eval(arti2)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle2]): \+ \n\n#print(\"t1,t2\",t1,t2,cas);\n#print(display([lieu_P1,lieu_P2])) ;\n\n print(display([cercle_A,cercle_D,lieu_P1,lieu_P2,artiFixe,\n \+ display([mobile1,eval(mobile2)],insequence=true)],ax es=none,scaling=constrained));\n\nfi;\n printf(\" Temps mis %a secon des\",time()-tt)\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the na me changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 " pB1 and not(pB2) : ne marche pas tout-\340-fait bien pour les trac\351s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Il y a un probl\350me de valeurs approch\351es, il faut \+ introduire un petit epsilon pour d\351coincer le trac\351 : j'ai bien \+ pein\351 pour trouver ce qui bloquait mais je n'y ai pas trouv\351 d'e xplications !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "#\"pB1 an d not(pB2) and not(pC2)\"\n#cas cercle droite englobe cercle gauche\na :=4:b:=2.:c:=7:d:=a: \nbielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 98 "#watt \"pB1 and not(pB2) and pC2\"\na:=10:b:= 3:c:=b:d:=evalf(sqrt(a^2-4*b^2)):\nbielle(b,c,d,0,1./2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 " Cas marchant sans probl\350me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a:=4:b:=1.5:c:=2:d:=a+c-b:#cas limite droite \nbielle (b,c,d,Pi/4,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a:=4:b:= 1.5:c:=2:d:=a+b-c:#cas limite gauche \nbielle(b,c,d,Pi/4,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a:=4:b:=1.5:c:=2: d:=a: \nbielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "#cas b=c a=d les deux cercles sont enti\350rement parcourus\n a:=4:b:=1.5:c:=1.5;d:=a+b-c;\nbielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# cas limite \340 gauche avec cercl es s\351cants\na:=4:b:=1.5:c:=3.5:d:=a+b-c: \n bielle(b,c,d,2*Pi/3,1); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "# cas li mite \340 droite avec cercles s\351cants\na:=4:b:=1.5:c:=3.5:d:=a-b+c: \n bielle(b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "#roberts\"not(pB1) and not(pB2)\"\na:=4: b:=4.6:c:=b:d:=a/2:angle :=evalf(arccos(2*d/b)):\nr:=evalf(sqrt((b/d)^2-0.25)):\nbielle(b,c,d,2 *Pi/3,1);" }}{PARA 6 "" 1 "" {TEXT -1 27 " a=4. ; b=4.6 ; c=4.6 ; d=2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "#tchebychev \"not(pB1) \+ and not(pB2)\" \na:=4:b:=5:c:=b:d:=2.01:\nbielle(b,c,d,Pi/4,2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "#rideau \"pB2 and not(pB1) \+ and not pC2)\"\na:=4:k:=2.:b:=a*k/3;c:=b:d:=b:r:=evalf(sqrt(3)/2):\nbi elle(b,c,d,-2*Pi/3,r);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " Ferr \351ol 120103 - 23h23" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 778 "Texte d u mail -\nencore merci pour ce pgme bien avanc\351.\n\n\nj'ai modifi \351 la d\351finition du triangle, rajout\351 a dans les param\350tres de la proc\351dure, et corrig\351 les valeurs particuli\350res donn \351es.\n\npour la lemniscate, cela va bien ainsi que pour l'une des f igures d\351nomm\351es rideau\" tr\350s r\351ussie.\n\nje suis en trai n de modifier ma page pour prendre vos notations (je voulais que cela \+ concorde avec les courbes de watt, mais prendre a,b,c,d est en effet p lus clair).\n\nje vais aussi faire tracer la construction compl\350te \+ montrant que la m\352me courbe est obtenue par deux autres trois barre s (ce qui fait que si l'on change la bielle et une manivelle, l'une de s courbes emmen\351e par la nouvelle bielle est semblable \340 l'ancie nne : voir les 2 constructions de la lemniscate)\n\nTr\350s cordialeme nt" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 93 " Version avec discussion ( il n'est pas certain que tous les cas particuliers vont marcher !)" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "tiges BA=b, AD=d, DC=c, BC=a\nart iculations consid\351r\351es comme fixes : B et C donc tige BC fixe" } }{PARA 0 "" 0 "" {TEXT -1 455 "Le cercle de centre B de rayon b (suppo rt de la trajectoire de A) coupe la droite BC en B1 \340 gauche et B2 \+ \340 droite.\nLe cercle de centre C de rayon c (support de la trajecto ire de D) coupe la droite BC en C1 \340 gauche et C2 \340 droite.\nLes conditions de passage en ces points sont respectivement :\npB1:= abs( a+b-c)<=d and d<=a+b+c) : pB2:= abs(c-abs(a-b))<=d and c+abs(a-b)>=d :\npC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d an d a+c+b>=d :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "On fixe sur la t ige AD un triangle APD tel que le vecteur AP soit d\351duit du vecteu r AD par une similitude d'angle \"angle\" et de rapport \"r\"." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "restart:\nwith(plots):\nsim: =(V,t,r) -> [r*V[1]*cos(t)-r*V[2]*sin(t),r*V[1]*sin(t)+r*V[2]*cos(t)]: \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4869 "bielle:=proc(a, b,c,d,ang le,r) # donner des r\351els par exemple b=8./3 et non 8/3\n local U,u,A,B,C,D1,D2,P1,P2,Q1,Q2,lieu_P1,lieu_P2,\n lieu_Q2,lieu_ Q1,AD1,AD2,n,cercle_A,cercle_D,j, tt,debut,\n artiFixe,arti1, arti2,barres1,barres2,triangle1,triangle2,mobile1,mobile2,k,\n \+ pB1,pB2,pC1,pC2,t1,t2,t12,delta,delta2,cas,eps;\n tt:=time():\n \+ n:=10: # nre de frames\n pB1:= abs(a+b-c)<=d and d=d:\n pC1:=abs(b-abs(a-c))<=d and b+ abs(a-c)>=d : pC2:=abs(a+c-b)<=d and a+c+b>d : \n printf(\" a=%a ; \+ b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n printf(\" %a <= %a <= %a\\n\",a+b -c,d,a+c-b);\n if b>c then\n printf(\"%s%a%s,%a\\n\",\n \+ \"Par raison de sym\351trie, on se limite \340 b <= c or b=\",b,\" c =\",c):\n elif d>=a+b+c or a>=b+c+d or c>=a+b+d then\n print( \"M\351canisme impossible ou sans mouvement possible\")\n else\n\n \+ debut:=0;t12:=t1: \n if pB1 and pB2 then #---MARCHE----\n \+ cas:=\"pB1 and pB2\": \n t1:=-evalf(Pi): t2 :=-t1: t12:=-t1:\n delta:=(t2-t1)/2: delta2:=delta:#:\n \+ if not(pC1) then debut:=-n fi;\n elif pB1 and not(pB2) then \n#--------------MARCHE MMMAAALLL-------------------------------\n \+ eps:=0.0001:\n cas:=\"pB1 and not(pB2)\":\n t1 :=evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )):\n if pC2 then \n \+ cas:=cat(cas,\" and pC2\"):\n debut:=0 :\n \+ t2:=evalf(2*Pi-t1-eps): \n # t2:=evalf(2*Pi-arcco s((a^2+b^2-c^2)/2/a/b ))-0.1: \n delta:=(t2-t1)/2: \n \+ t12:=t2: delta2:=-delta:\n else\n cas:=c at(cas,\" and not(pC2)\"):\n t2:=evalf(Pi+arccos((-a^2-b^2 +(c-d)^2)/2/a/b )):\n t1:=t1+eps:t2:=t2-eps:# --------ET A LORS CA MARCHE !!!!\n debut:=0: \n delta:=(t2 -t1)/2: \n t12:=t2: delta2:=-delta:\n fi;\n#---- --------------------------------------------------\n \+ \n elif pB2 and not(pB1) then #---MARCHE----\n cas:=\"pB2 and not(pB1)\":\n if not(pC2) then \n \+ t1:=-evalf(arccos((a^2+b^2-(c+d)^2)/2/a/b )): t2:=-t1:\n \+ cas:=cat(cas,\" and not pC2)\"):\n else cas:=ca t(cas,\" and not pC2)\"):\n fi;\n debut:=0: delta: =(t2-t1)/2: \n t12:=t2:delta2:=-delta:\n else # no t(pB1) and not(pB2) #---MARCHE----\n cas:=\"not(pB1) and not( pB2)\":\n t1:=evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )): \n \+ t2:=evalf(arccos((a^2+b^2-(c+d)^2)/2/a/b )):\n debut:=0:\n delta:=(t2-t1)/2:\n t12:=t2:delta2:=-delta:\n fi ;\n\n U:=simplify([solve(( b*cos(t)-a-c*cos(w) )^2 + (b*sin(t)-c*si n(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b*cos(t),b*sin(t) ]:\n cercle_A:=plot([op(A),t=0..2*Pi],color =blue,thickness=1):\n \+ cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],color =blue,thickness =1): \n u:=U[1]:\n D1:=[a+c*cos(U[1]),c*sin(U[1])]:\n D2:= [a+c*cos(U[2]),c*sin(U[2])]:\n AD1:=(D1-A): AD2:=(D2-A):\n P1 :=simplify(expand(A+sim(AD1,evalf(angle),r))):\n P2:=simplify(expan d(A+sim(AD2,evalf(angle),r))):\n lieu_P1:=plot([op(P1),t=t1..t2],co lor =COLOR(RGB,1,0,0),thickness=1):\n lieu_P2:=plot([op(P2),t=t1..t 2],color =COLOR(RGB,0,1,0),thickness=1):\n\n# ------------------------ --- \n# si Maple 7, remplacer partout \"CIRCLE\" par \"CIRCLE,16\"\n# ---------------------------\n\n artiFixe:= PLOT(POINTS(C,SYMBOL(CI RCLE),COLOR(RGB,0,0,0))),\n PLOT(POINTS(B,SYMBOL(CIRCLE) ,COLOR(RGB,0,0,0))):\n barres1:='PLOT(CURVES(evalf( subs(t=t1+k*del ta/n,[B,A,D1,C]) ),THICKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=t12+k*delta2/n,[B,A,D2,C]) ),THICKNESS(2) ))':\n arti1:= 'P LOT(POINTS(op(evalf(subs(t=t1+k*delta/n,[A,D1,P1]))),SYMBOL(CIRCLE)))' :\n arti2:='PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[A,D2,P2]))) ,SYMBOL(CIRCLE)))':\n triangle1:=display([seq(PLOT(POLYGONS(evalf( \+ subs(t=t1+k*delta/n,[D1,P1,A]) ),THICKNESS(2) ),\n C OLOR(RGB,1,0,0)),k=debut..debut+2*n)],insequence=true):\n triangle2 :=display([seq(PLOT(POLYGONS(evalf( subs(t=t12+k*delta2/n,[D2,P2,A]) ) ,THICKNESS(2) ),\n COLOR(RGB,0,1,0)),k=debut..debut+ 2*n)],insequence=true):\n mobile1:=display([display([seq(display([e val(barres1),eval(arti1)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle1]):\n mobile2:=di splay([display([seq(display([eval(barres2),eval(arti2)]),k=debut..debu t+2*n)],\n insequence=true,color=black) , triangle2]): \n\n#print(\"t1,t2\",t1,t2,cas);\ndisplay([lieu_P1,l ieu_P2],scaling=constrained);\n\n print(display([cercle_A,cercle_D,li eu_P1,lieu_P2,artiFixe,\n display([mobile1,eval(mo bile2)],insequence=true)],axes=none));\n\nfi;\n printf(\" Temps mis \+ %a secondes\",time()-tt)\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning , the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 " pB1 and not(pB2) : ne marche \+ pas tout-\340-fait bien pour les trac\351s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Il y a un probl\350me de valeurs approch\351es, il faut \+ introduire un petit epsilon pour d\351coincer le trac\351 : j'ai bien \+ pein\351 pour trouver ce qui bloquait mais je n'y ai pas trouv\351 d'e xplications !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "#\"pB1 an d not(pB2) and not(pC2)\"\n#cas cercle droite englobe cercle gauche\na :=4:b:=2.:c:=7:d:=a: \nbielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 99 "#watt \"pB1 and not(pB2) and pC2\"\na:=10:b:= 3:c:=b:d:=evalf(sqrt(a^2-4*b^2)):\nbielle(a,b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#lemniscate 1\na:=sqrt(2.):b:=1:c:= b:d:=a:\nbielle(a,b,c,d,0,1/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#lemniscate 2\na:=sqrt(2.):b:=1:c:=a:d:=b:\nbielle(a,b,c,d,Pi, 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#limacon????\na:=1:b :=sqrt(2.):c:=b:d:=a:\nbielle(a,b,c,d,0,2);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 " Cas marchant sans probl\350me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a:=4:b:=1.5:c:=2:d:=a+c-b:#cas limite droite \nb ielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a :=4:b:=1.5:c:=2:d:=a+b-c:#cas limite gauche \nbielle(a,b,c,d,Pi/3,1); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a:=4:b:= 1.5:c:=2:d:=a: \nbielle(a,b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 111 "#cas b=c a=d les deux cercles sont enti\350rement \+ parcourus\na:=4:b:=1.5:c:=1.5;d:=a+b-c;\nbielle(a,b,c,d,2*Pi/3,1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "# cas limite \340 gauche av ec cercles s\351cants\na:=4:b:=1.5:c:=3.5:d:=a+b-c: \n bielle(a,b,c,d, 2*Pi/3,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "# cas limite \340 droite avec cercles s\351cants\na:=4:b:=1.5:c:=3 .5:d:=a-b+c: \n bielle(a,b,c,d,2*Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 111 "#roberts\"not(pB1) and not(pB2)\"\na:=2: b:=3:c:=b :d:=a/2:angle:=-evalf(arccos(2*b/d)):\nbielle(a,b,c,d,-Pi/3,b/d);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "#tchebychev \"not(pB1) and n ot(pB2)\" \na:=4:b:=5:c:=b:d:=2.01:\nbielle(a,b,c,d,Pi,1/2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "a:=4:k:=1.1:b:=a*k/3;c:=b:d: =b:r:=evalf(sqrt(3)/2):\nbielle(b,c,d,-2*Pi/3,r);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 95 "#rideau \"pB2 and not(pB1) and not pC2)\"\na :=4:k:=1.5:b:=a*k/3;c:=b:d:=b:\nbielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "#rideau \"pB2 and not(pB1) and not \+ pC2)\"\na:=4:k:=2.2:b:=a*k/3;c:=b:d:=b:\nbielle(a,b,c,d,Pi/3,1);" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a:=4:k:=3.:b:=a*k/3;c:=b:d:= b:\nbielle(a,b,c,d,Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "a:=4:k:=4:b:=a*k/3:c:=b:d:=b:r:=evalf(sqrt(3)/2):\nbielle(a,b,c,d, Pi/3,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 " Triple g\351n\351ration d'une courbe" }}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 40 " Esculier - Mails du 130103 et du 190103" }}{PARA 0 "" 0 "" {TEXT 272 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 270 23 "Texte du ma il du 130103" }{TEXT -1 823 "\nJ'ai trouv\351 votre mail apr\350s dine r, je suis all\351 sur votre site pour comprendre et m'y retrouver dan s la for\352t de tiges : voici ci-joint ce que je vous propose.\n Si l 'on fait tracer des triangles pleins, ils se cachent tout donc j'ai fa it tracer les contours.\nLe point E avec les approximations de calculs n'\351tait pas tout-\340-fait fixe alors le artiFixe:= PLOT(POINTS(B, C,E ... plantait, c'est pourquoi j'ai un peu trich\351 en faisant t=0 \+ dans E ( ce n'est pas math\351matique, mais efficace ! ... ).\n J'ai m is des carr\351s aux articulations fixes pour qu'on les identifie faci lement visuellement.\nJ'ai attribu\351 une couleur diff\351rente \340 \+ chaque syst\350me pour que l'on puisse s'y retrouver finalement en ral entissant suffisamment l'animation, on voit assez bien les trois g\351 n\351rations.\nCela vous donnera peut-\352tre d'autres id\351es ...\n \n" }{TEXT 273 5 "Texte" }{TEXT -1 1 " " }{TEXT 274 18 " du Mail du 19 0103" }}{PARA 0 "" 0 "" {TEXT -1 1068 "J'ai test\351 :\n... en mettant des teintes pastel pour l'int\351rieur des triangles, on peut ainsi c olorier les triangles comme sur la figure explicative de votre page we b, sans g\352ner la compr\351hension du m\351canisme car les barres re stent visibles, les triangles donnant l'impression d'\352tre transpare nts ; globalement le rendu visuel de l'animation est assez r\351ussi . .. ( question d'appr\351ciation tout-\340-fait subjective ! ) \n\nCe t ype de coloriage n'est d'ailleurs pas inint\351ressant pour le program me initial de g\351n\351ration simple quand le triangle est grand cela all\350ge consid\351rablement le dessin et permet de mieux voir qu'av ec des couleurs fonc\351es.\n \nIl vous reste donc maintenant \340 cho isir le mode( triangles pleins ou non ), les couleurs, les param\350tr es ( a, b, c, d, angle, r ), les param\350tres ( dimensions, vitesse, \+ nbre d'images ) du gif : cela fait un certain nombre de solutions pos sibles ! ... \n\nJe vous envoie le pr\351sent mail ce soir car vous m' avez dit que vous alliez vous int\351resser \340 la question lundi don c vous aurez un certain nombre d'\351l\351ments \340 disposition.\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "restart:\nwith(plots):\ns im:=(V,t,r) -> [r*V[1]*cos(t)-r*V[2]*sin(t),r*V[1]*sin(t)+r*V[2]*cos(t )]:\ncomp:=X->X[1]+I*X[2]:reel:=z->[Re(z),Im(z)]: " }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 5711 "tribielle:=proc(a, b,c,d,angle,r) # donner des \+ r\351els par exemple b=8./3 et non 8/3\n local U,u,A,B,C,D1,D2,P1,P 2,Q1,Q2,lieu_P1,lieu_P2,\n lieu_Q2,lieu_Q1,AD1,AD2,n,cercle_A ,cercle_D,j, tt,debut,\n artiFixe,arti1,arti2,barres1,barres2 ,triangle1,triangle2,mobile1,mobile2,k,\n pB1,pB2,pC1,pC2,t1, t2,t12,delta,delta2,eps,\nR1,R2,r1,r2,S1,S2,E,T1,T2,U1,U2,u1,u2,p1,p2, bb,aa,d1,d2,cc;\n tt:=time():\n n:=5: # nbre de frames\n pB1:= a bs(a+b-c)<=d and d= d:\n pC1:=abs(b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d and a+c+b>d : \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d); \n if b>c then\n printf(\"%s%a%s,%a\\n\",\n \"Par rais on de sym\351trie, on se limite \340 b <= c or b=\",b,\" c=\",c):\n \+ elif d>=a+b+c or a>=b+c+d or c>=a+b+d then\n print(\"M\351canis me impossible ou sans mouvement possible\")\n else\n debut:=0; t12:=t1: eps:=0.0001:\n \n if pB1 and pB2 then\n t1:=- evalf(Pi): \n t2:=t1+2*evalf(Pi): t12:=-t1:\n \+ delta:=(t2-t1)/2: delta2:=delta:\n if not(pC1) then debut :=-n fi;\n elif pB1 and not(pB2) then\n t1:=evalf(arc cos((a^2+b^2-(c-d)^2)/2/a/b ))+eps:\n t2:=evalf(2*Pi-t1): \n delta:=(t2-t1)/2: t12:=t2: delta2:=-delta: debut:=0 : \+ \n elif pB2 and not(pB1) then\n \+ t1:=-evalf(arccos((a^2+b^2-(c+d)^2)/2/a/b ))+ eps:\n t2:=- t1: debut:=0: delta:=(t2-t1)/2: \n t12:=t2:delta2:=-delta: \n else # not(pB1) and not(pB2)\n t1:=evalf(arccos((a^2+b ^2-(c-d)^2)/2/a/b )) +eps: \n t2:=evalf(arccos((a^2+b^2-(c+d)^ 2)/2/a/b ))-eps:\n debut:=0: delta:=(t2-t1)/2: t12:=t2:delta2: =-delta:\n fi;\n U:=simplify([solve(( b*cos(t)-a-c*cos(w) )^2 \+ + (b*sin(t)-c*sin(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n A:=[b *cos(t),b*sin(t)]:\n #cercle_A:=plot([op(A),t=0..2*Pi],color =blue, thickness=1):\n #cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi],col or =blue,thickness=1): \n u:=U[1]:\n D1:=[a+c*cos(U[1]),c*sin (U[1])]:\n D2:=[a+c*cos(U[2]),c*sin(U[2])]:\n AD1:=(D1-A): AD 2:=(D2-A):\n P1:=simplify(expand(A+sim(AD1,evalf(angle),r))):\n \+ P2:=simplify(expand(A+sim(AD2,evalf(angle),r))):\n lieu_P1:=plot([o p(P1),t=t1..t2],color =COLOR(RGB,0,0.7,0),thickness=1):\n lieu_P2:= plot([op(P2),t=t1..t2],color =COLOR(RGB,0,0.7,0),thickness=1):\n\n \+ bb:=comp(B):aa:=comp(A):cc:=comp(C):\n\n p1:=comp(P1):p2:=comp(P2): d1:=comp(D1):d2:=comp(D2):\n r1:=cc+p1-d1:R1:=reel(r1):r2:=cc+p2-d2 :R2:=reel(r2):\n S1:=reel(r1+(p1-d1)*(r1-p1)/(d1-aa)):S2:=reel(r2+( p2-d2)*(r2-p2)/(d2-aa)):\n u1:=bb+p1-aa:U1:=reel(u1):u2:=bb+p2-aa:U 2:=reel(u2):\n T1:=reel(p1+(p1-u1)*(p1-d1)/(d1-aa)):T2:=reel(p2+(p2 -u2)*(p2-d2)/(d2-aa)):\n E:=evalf(subs(t=0,T1+S1-P1)): # E n'est pa s math\351matiquement fixe\n# --------------------------- \n# si Mapl e 7, remplacer partout \"CIRCLE\" par \"CIRCLE,16\" ou vice versa\n# - --------------------------\n\n artiFixe:= PLOT(POINTS(B,E,C,SYMBOL( BOX,16),COLOR(RGB,0,0,0))):\n barres1:='PLOT(CURVES(evalf( subs(t=t 1+k*delta/n,[B,A,D1,P1,A,D1,C]) ),\n THICKNESS(2 ),COLOR(RGB,0,0,0) )),\n PLOT(CURVES(evalf( subs(t=t1+k*del ta/n,[C,R1,S1,P1,R1,S1,E]) ),\n THICKNESS(2),COL OR(RGB,1,0,0) )),\n PLOT(CURVES(evalf( subs(t=t1+k*delta/n, [E,T1,U1,P1,T1,U1,B]) ),\n THICKNESS(2),COLOR(RG B,0,0,1) )) ':\n barres2:='PLOT(CURVES(evalf( subs(t=t12+k*delta2/n ,[B,A,D2,P2,A,D2,C]) ),\n THICKNESS(2),COLOR(RGB ,0,0,0) )),\n PLOT(CURVES(evalf( subs(t=t12+k*delta2/n,[C,R 2,S2,P2,R2,S2,E]) ),\n THICKNESS(2),COLOR(RGB,1, 0,0) )),\n PLOT(CURVES(evalf( subs(t=t12+k*delta2/n,[E,T2,U 2,P2,T2,U2,B]) ),\n THICKNESS(2),COLOR(RGB,0,0,1 ) )) ':\n arti1:= 'PLOT(POINTS(op(evalf(subs(t=t1+k*delta/n,[A,D1,P 1,R1,S1,T1,U1]))),\n SYMBOL(CIRCLE,16),COLOR(RG B,1,0,0)))':\n arti2:='PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[ A,D2,P2,R2,S2,T2,U2]))),\n SYMBOL(CIRCLE,16),CO LOR(RGB,1,0,0)))':\n triangle1:=display([seq(display(\n \+ PLOT(POLYGONS(evalf( subs(t=t1+k*delta/n,[D1,P1,A]) ),\n \+ COLOR(RGB,0.9,0.9,0.9) )),\n PLOT(POLYGONS(eval f( subs(t=t1+k*delta/n,[R1,P1,S1]) ),\n COLOR(RGB,1 ,0.9,0.9) )),\n PLOT(POLYGONS(evalf( subs(t=t1+k*delta/n ,[T1,P1,U1]) ),\n COLOR(RGB,0.8,1,1)) ))\n \+ ,k=debut..debut+2*n)],insequence=true):\n triangle2:=display([s eq(display(\n PLOT(POLYGONS(evalf( subs(t=t12+k*delta2/n ,[D2,P2,A]) ),\n COLOR(RGB,0.9,0.9,0.9) )),\n \+ PLOT(POLYGONS(evalf( subs(t=t12+k*delta2/n,[R2,P2,S2]) ),\n \+ COLOR(RGB,1,0.9,0.9) )),\n PLOT(POLYGO NS(evalf( subs(t=t12+k*delta2/n,[T2,P2,U2]) ),\n CO LOR(RGB,0.8,1,1)) ))\n ,k=debut..debut+2*n)],insequence=t rue):\n mobile1:=display([display([seq(display([eval(barres1),eval( arti1)]),k=debut..debut+2*n)],\n inseque nce=true,color=black), triangle1]):\n mobile2:=display([display([se q(display([eval(barres2),eval(arti2)]),k=debut..debut+2*n)],\n \+ insequence=true,color=black), triangle2]): \+ \n\n #plotsetup(gif,plotoutput=\"c:/aaC/bielleG\351n\351250.gif\",plo toptions=\"width=250,height=250\");\n print(display([lieu_P1,lieu_P2, artiFixe, #cercle_A,cercle_D,\n display([mobil e1,mobile2],insequence=true)],\n axes=none,scaling= constrained));\n\nfi;\n printf(\" Temps mis %a secondes\",time()-tt) \nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords \+ has been redefined\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "a:=4:b:=1:c:=2:d:=2: \ntribielle(a, b,c,d,Pi/3,1.5);" }}}{EXCHG {PARA 6 "" 1 "" {TEXT -1 25 " Temps mis 6. 328 secondes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 " Ferr\351ol - Mail du 130103 - 15 h14" }}{EXCHG {PARA 4 "" 0 "" {TEXT -1 34 " Esculier - Mail du 130103 \+ - 22h56" }}{PARA 0 "" 0 "" {TEXT 271 13 "Texte du mail" }}{PARA 0 "" 0 "" {TEXT -1 227 "la page http://www.mathcurve.com/courbes2d/troisbar res/troisbarre.shtml commence \340 prendre tournure\nJ'ai essay\351 d' illustrer la triple g\351n\351ration, mais on voit un imbroglio de bar res et ce n'est pas tr\350s lisible (fichier joint)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "restart:\nwith(plots):\nsim:=(V,t,r) -> \+ [r*V[1]*cos(t)-r*V[2]*sin(t),r*V[1]*sin(t)+r*V[2]*cos(t)]:\ncomp:=X->X [1]+I*X[2]:reel:=z->[Re(z),Im(z)]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5343 "bielle:=proc(a, b,c,d,angle,r) # donner des r\351els par exemp le b=8./3 et non 8/3\n local U,u,A,B,C,D1,D2,P1,P2,Q1,Q2,lieu_P1,li eu_P2,\n lieu_Q2,lieu_Q1,AD1,AD2,n,cercle_A,cercle_D,j, tt,de but,\n artiFixe,arti1,arti2,barres1,barres2,triangle1,triangl e2,mobile1,mobile2,k,\n pB1,pB2,pC1,pC2,t1,t2,t12,delta,delta 2,cas,eps,\nR1,R2,r1,r2,S1,S2,E,T1,T2,U1,U2,u1,u2,p1,p2,bb,aa,d1,d2,cc ;\n tt:=time():\n n:=5: # nre de frames\n pB1:= abs(a+b-c)<=d an d d=d:\n pC1:=abs( b-abs(a-c))<=d and b+abs(a-c)>=d : pC2:=abs(a+c-b)<=d and a+c+b>d : \+ \n printf(\" a=%a ; b=%a ; c=%a ; d=%a\\n\",a,b,c,d);\n printf(\" %a <= %a <= %a\\n\",a+b-c,d,a+c-b);\n if b>c then\n printf(\"%s% a%s,%a\\n\",\n \"Par raison de sym\351trie, on se limite \340 \+ b <= c or b=\",b,\" c=\",c):\n elif d>=a+b+c or a>=b+c+d or c>=a+b+ d then\n print(\"M\351canisme impossible ou sans mouvement possi ble\")\n else\n\n debut:=0;t12:=t1: \n if pB1 and pB2 the n #---MARCHE----\n cas:=\"pB1 and pB2\": \n \+ t1:=-evalf(Pi): t2:=-t1: t12:=-t1:\n delta:=(t2-t1)/2: de lta2:=delta:#:\n if not(pC1) then debut:=-n fi;\n elif \+ pB1 and not(pB2) then\n#--------------MARCHE MMMAAALLL---------------- ---------------\n eps:=0.0001:\n cas:=\"pB1 and not( pB2)\":\n t1:=evalf(arccos((a^2+b^2-(c-d)^2)/2/a/b )):\n \+ if pC2 then \n cas:=cat(cas,\" and pC2\"):\n \+ debut:=0 :\n t2:=evalf(2*Pi-t1-eps): \n # t2:=evalf(2*Pi-arccos((a^2+b^2-c^2)/2/a/b ))-0.1: \n delt a:=(t2-t1)/2: \n t12:=t2: delta2:=-delta:\n else \n cas:=cat(cas,\" and not(pC2)\"):\n t2:=eva lf(Pi+arccos((-a^2-b^2+(c-d)^2)/2/a/b )):\n t1:=t1+eps:t2: =t2-eps:# --------ET ALORS CA MARCHE !!!!\n debut:=0: \n \+ delta:=(t2-t1)/2: \n t12:=t2: delta2:=-delta: \n fi;\n#---------------------------------------------------- --\n \n elif pB2 and not(pB1) \+ then #---MARCHE----\n cas:=\"pB2 and not(pB1)\":\n \+ if not(pC2) then \n t1:=-evalf(arccos((a^2+b^2-(c+d)^2) /2/a/b )): t2:=-t1:\n cas:=cat(cas,\" and not pC2)\"):\n \+ else cas:=cat(cas,\" and not pC2)\"):\n fi;\n \+ debut:=0: delta:=(t2-t1)/2: \n t12:=t2:delta2:=-de lta:\n else # not(pB1) and not(pB2) #---MARCHE----\n ca s:=\"not(pB1) and not(pB2)\":\n t1:=evalf(arccos((a^2+b^2-(c-d )^2)/2/a/b )): \n t2:=evalf(arccos((a^2+b^2-(c+d)^2)/2/a/b )): \n debut:=0:\n delta:=(t2-t1)/2:\n t12:=t2:del ta2:=-delta:\n fi;\n\n U:=simplify([solve(( b*cos(t)-a-c*cos(w ) )^2 + (b*sin(t)-c*sin(w) )^2-d^2,w)]):\n B:=[0,0]: C:=[a,0]:\n \+ A:=[b*cos(t),b*sin(t)]:\n cercle_A:=plot([op(A),t=0..2*Pi],color = blue,thickness=1):\n cercle_D:=plot([a+c*cos(w),c*sin(w),w=0..2*Pi] ,color =blue,thickness=1): \n u:=U[1]:\n D1:=[a+c*cos(U[1]),c *sin(U[1])]:\n D2:=[a+c*cos(U[2]),c*sin(U[2])]:\n AD1:=(D1-A): \+ AD2:=(D2-A):\n P1:=simplify(expand(A+sim(AD1,evalf(angle),r))):\n P2:=simplify(expand(A+sim(AD2,evalf(angle),r))):\n lieu_P1:=plo t([op(P1),t=t1..t2],color =COLOR(RGB,1,0,0),thickness=1):\n lieu_P2 :=plot([op(P2),t=t1..t2],color =COLOR(RGB,0,1,0),thickness=1):\n\np1:= comp(P1):p2:=comp(P2):bb:=comp(B):aa:=comp(A):d1:=comp(D1):d2:=comp(D2 ):\ncc:=comp(C):\nr1:=cc+p1-d1:R1:=reel(r1):r2:=cc+p2-d2:R2:=reel(r2): \nS1:=reel(r1+(p1-d1)*(r1-p1)/(d1-aa)):S2:=reel(r2+(p2-d2)*(r2-p2)/(d2 -aa)):\nu1:=bb+p1-aa:U1:=reel(u1):u2:=bb+p2-aa:U2:=reel(u2):\nT1:=reel (p1+(p1-u1)*(p1-d1)/(d1-aa)):T2:=reel(p2+(p2-u2)*(p2-d2)/(d2-aa)):\nE: =T1+S1-P1:\n# --------------------------- \n# si Maple 7, remplacer p artout \"CIRCLE\" par \"CIRCLE,16\"\n# ---------------------------\n\n artiFixe:= PLOT(POINTS(C,SYMBOL(CIRCLE),COLOR(RGB,0,0,0))),\n \+ PLOT(POINTS(B,SYMBOL(CIRCLE),COLOR(RGB,0,0,0))):\n barres 1:='PLOT(CURVES(evalf( subs(t=t1+k*delta/n,[B,A,D1,C,R1,S1,E,T1,U1,B]) ),THICKNESS(2) ))':\n barres2:='PLOT(CURVES(evalf( subs(t=t12+k*de lta2/n,[B,A,D2,C,R2,S2,E,T2,U2,B]) ),THICKNESS(2) ))':\n arti1:= 'P LOT(POINTS(op(evalf(subs(t=t1+k*delta/n,[A,D1,P1]))),SYMBOL(CIRCLE)))' :\n arti2:='PLOT(POINTS(op(evalf(subs(t=t12+k*delta2/n,[A,D2,P2]))) ,SYMBOL(CIRCLE)))':\n triangle1:=display([seq(PLOT(POLYGONS(evalf( \+ subs(t=t1+k*delta/n,[D1,P1,A,P1,U1,P1,S1]) ),THICKNESS(2) ),\n \+ COLOR(RGB,1,0,0)),k=debut..debut+2*n)],insequence=true):\n \+ triangle2:=display([seq(PLOT(POLYGONS(evalf( subs(t=t12+k*delta2/n, [D2,P2,A,P2,U2,P2,S2]) ),THICKNESS(2) ),\n COLOR(RGB ,0,1,0)),k=debut..debut+2*n)],insequence=true):\n mobile1:=display( [display([seq(display([eval(barres1),eval(arti1)]),k=debut..debut+2*n) ],\n insequence=true,color=black), trian gle1]):\n mobile2:=display([display([seq(display([eval(barres2),eva l(arti2)]),k=debut..debut+2*n)],\n inse quence=true,color=black), triangle2]): \n\n#print(\"t1,t2\",t1,t2,c as);\ndisplay([lieu_P1,lieu_P2],scaling=constrained);\n\n print(displ ay([cercle_A,cercle_D,lieu_P1,lieu_P2,artiFixe,\n \+ display([mobile1,eval(mobile2)],insequence=true)],axes=none));\n\nfi; \n printf(\" Temps mis %a secondes\",time()-tt)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a:=4:b:=1.5:c:=2:d:=a+c-b:#cas limi te droite \nbielle(a,b,c,d,Pi/3,1);" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23 3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }