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::::::::::::::::::::::::::::::::::::::::::::::::::Ro_WlmfImU`uCWqfJaEa ;tVmvI]uZ]CR=gNwuyyqy;:::::::::::::::::::::::::::::::B:Zy=:::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::JUYqJisZ`qFumr`cj?ko^fPXaholtngsYxoyyyuy::::: ::::::::::::::::::::::::::Z::xI::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::m IwxS_HrF=Nu`SYDq " 0 "" {MPLTEXT 1 0 28 "alias(app=animepolaireplot);" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 14 "Initialisation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1779 "animepolaireplot:=proc(Lieu::equation,Inte rvalle::equation,Sequences::posint,Mode::name,Axes::name)\nlocal a,Ani mation,b,Courbe,d,k,r,Opts,P,Pas,Rayon,Var;\noption `Copyright f\351vr ier 2001 Pierre Lantagne`;\nRayon:=lhs(Lieu);\nOpts:=[args[6..nargs]]; \nVar:=lhs(Intervalle);\na:=evalf(lhs(rhs(Intervalle)));\nb:=evalf(rhs (rhs(Intervalle)));\nr:=unapply(rhs(Lieu), Var);\nPas:=seq(a+((b-a)/Se quences)*k,k=0..Sequences);\nd:=max(seq(abs(evalf(r(i))),i=Pas));\nif \+ Axes=polaire then\n P:=plots[coordplot](polar,[0..ceil(d),0..2*Pi],\n \+ grid=[13,25], \n color=[khaki,khaki],\n lin estyle=[5,5], \n labelling=front,\n font=[TIMES,ROMA N,10],\n scaling=constrained,\n view=[-ceil(d)..ceil (d),-ceil(d)..ceil(d)])\n elif Axes=cartesien then\n P:=plots[coordplo t](cartesian,[-ceil(d)..ceil(d),-ceil(d)..ceil(d)],\n grid=[2 1,21],\n color=[khaki,khaki],\n linestyle=[5,5], \n \+ labelling= middle,\n font=[TIMES,ROMAN,10],\n \+ scaling=constrained,\n view=[-ceil(d)..ceil(d),-ceil(d)..c eil(d)])\n else\n P:=plot([[0,0]],axes=Axes,op(Opts))\nfi;\nif Mode=an ime then\n Animation:=seq(plots[display]\n ([plots[polarplot](Re (r(t)),t=a..k,color=navy,thickness=2,\n numpoints=ceil(4 9+20*k),op(Opts)),\n plottools[line]([0,0],[-1*d*cos(k),-1*d*sin (k)], \n color=grey,linestyle=2),\n plottools[line ]([0,0],[d*cos(k),d*sin(k)],color=orange,linestyle=2),\n plottoo ls[point]([Re(r(k))*cos(k),Re(r(k))*sin(k)],color=orange,\n \+ symbol=circle),P]),k=Pas);\n plots[display](Animation,insequence=t rue,scaling=constrained);\n else\n Courbe:=plots[polarplot](Re(r(t)),t =a..b,color=navy,\n thickness=2,op(Opts)):\n plots[display]([Cou rbe,P],scaling=constrained)\nfi\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Droites" }} {PARA 0 "" 0 "" {TEXT -1 40 " Avec une \351quation lin\351aire de la f orme " }{XPPEDIT 18 0 "a*x+b*y+c = 0;" "6#/,(*&%\"aG\"\"\"%\"xGF'F'*&% \"bGF'%\"yGF'F'%\"cGF'\"\"!" }{TEXT -1 16 ", les \351quations " } {XPPEDIT 18 0 "x=r*cos(theta)" "6#/%\"xG*&%\"rG\"\"\"-%$cosG6#%&thetaG F'" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "y=r*sin(theta)" "6#/%\"yG*&%\"r G\"\"\"-%$sinG6#%&thetaGF'" }{TEXT -1 29 " nous conduisent \340 l'\351 quation" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "r = -c/(a*cos(theta)+b*sin (theta));" "6#/%\"rG,$*&%\"cG\"\"\",&*&%\"aGF(-%$cosG6#%&thetaGF(F(*&% \"bGF(-%$sinG6#F/F(F(!\"\"F5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Obtenez l'\351quation polaire de la droite d' \351quation cart\351sienne " }{XPPEDIT 18 0 "2*x+3*y+2 = 0;" "6#/,(*& \"\"#\"\"\"%\"xGF'F'*&\"\"$F'%\"yGF'F'\"\"#F'\"\"!" }{TEXT -1 40 ". Tr acez ensuite cette \351quation polaire." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Eq:=2*x+3*y+2 = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "readlib(isolate):\nEq_polaire:=isolate(subs(x=r*cos(t heta),y=r*sin(theta),Eq),r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "animepolaireplot(Eq_polaire,theta=-Pi/2..Pi/2,12,fixe,polaire,co lor=orange);" }}}{PARA 0 "" 0 "" {TEXT -1 117 "En animant le trac\351 \+ de cette droite, vous allez mieux voir la mani\350re dont sont engendr \351s les points de cette droite." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "animepolaireplot(Eq_polaire,theta=-Pi/2..Pi/2,12,anim e,polaire,\n style=point,symbol=circle);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 162 "L'animation su ivante illustre une s\351quence de droites parall\350les \340 la droit e pr\351c\351dente. Observez ce qui se passe lorsque la droite parall \350le passe par l'origine." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "A:=seq(animepolaireplot(r=k/(2*cos(theta)+3*sin(theta)),theta=-Pi /2..Pi/2,12,fixe,normal,\n color=orange,view=[-10..10,-10..10]), k=-8..8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(plots,dis play);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(A,inseque nce=true,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 8 "Lorsque " }{XPPEDIT 18 0 "c=0 " "6#/%\"cG\"\"!" }{TEXT -1 107 ", la droite passe alors par l'origine . Alors, puisque de telles droites ne peuvent s'exprimer par la forme \+ " }{XPPEDIT 18 0 "r = (-c)/(a*cos(theta)+b*sin(theta));" "6#/%\"rG*&,$ %\"cG!\"\"\"\"\",&*&%\"aGF)-%$cosG6#%&thetaGF)F)*&%\"bGF)-%$sinG6#F0F) F)F(" }{TEXT -1 55 ", il n'y a donc pas eu de trac\351 passsant par l' origine." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Les droites passant par l'origine sont de la forme " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " = " }{TEXT 301 9 "constante" } {TEXT -1 67 ". Obtenons alors l'inclinaison de la droite d'\351quation cart\351sienne " }{XPPEDIT 18 0 "2*x+3*y+2 = 0;" "6#/,(*&\"\"#\"\"\"% \"xGF'F'*&\"\"$F'%\"yGF'F'\"\"#F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "isolate (Eq,y);" }}}{PARA 0 "" 0 "" {TEXT -1 7 "Alors, " }{XPPEDIT 18 0 "tan(t heta" "6#-%$tanG6#%&thetaG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-2/3;" " 6#,$*&\"\"#\"\"\"\"\"$!\"\"F(" }{TEXT -1 36 ". Utilisons alors la macr o-commande " }{TEXT 303 9 "polarplot" }{TEXT -1 16 " de l'extension " }{TEXT 304 5 "plots" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "with(plots,polarplot):\npolarplot([t,-2/3, t=-5..5],c olor=orange,thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Cercles" }}{PARA 0 "" 0 "" {TEXT -1 110 "Les coordonn\351es polaires d'un cercle ont plusieurs fo rmes. Cela d\351pend de la situation du centre:\n " } {XPPEDIT 18 0 "r=a" "6#/%\"rG%\"aG" }{TEXT -1 63 " \+ centre: (0,0); rayon: a\n " }{XPPEDIT 18 0 "r=2*a*co s(theta) " "6#/%\"rG*(\"\"#\"\"\"%\"aGF'-%$cosG6#%&thetaGF'" }{TEXT -1 48 " centre: (a,0); rayon: a\n " }{XPPEDIT 18 0 "r=2*a*sin(theta)" "6#/%\"rG*(\"\"#\"\"\"%\"aGF'-%$sinG6#%&thetaG F'" }{TEXT -1 37 " centre: (0,a); rayon: a" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "animepo laireplot(r=2.5,theta=0..2*Pi,24,fixe,polaire);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 55 "animepolaireplot(r=2.5,theta=0..2*Pi,24,anime, polaire);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Rempla\347ons " }{XPPEDIT 18 0 "r=2.5" "6#/%\"rG$\"#D!\"\"" }{TEXT -1 5 " par " }{XPPEDIT 18 0 " r=-2.5" "6#/%\"rG,$$\"#D!\"\"!\"\"" }{TEXT -1 76 " pour constater que \+ ces deux \351quations polaires repr\351sentent le m\352me cercle." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "animepolaireplot(r=-2.5,thet a=0..2*Pi,24,anime,polaire);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 305 8 "REMARQUE" }{TEXT -1 168 ": Lorsque l'on \+ fait afficher aucun syst\350me de rep\350re, on voit tr\350s bien le b ras traceur du lieu g\351om\351trique. La partie orange de ce bras nou s indique que la valeur de " }{TEXT 306 1 "r" }{TEXT -1 67 " est posit ive tandis dans la partie grise indique que la valeur de " }{TEXT 307 1 "r" }{TEXT -1 32 " est n\351gative. Rappelons que si " }{TEXT 315 2 "r " }{TEXT -1 30 "< 0, les coordonn\351es polaires " }{XPPEDIT 18 0 " [r,theta]" "6#7$%\"rG%&thetaG" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "[-r, theta+Pi]" "6#7$,$%\"rG!\"\",&%&thetaG\"\"\"%#PiGF)" }{TEXT -1 42 " d \351signe le m\352me point. La macro-commande " }{TEXT 316 16 "animepo laireplot" }{TEXT -1 21 " trace donc le point " }{XPPEDIT 18 0 "[-r, t heta+Pi]" "6#7$,$%\"rG!\"\",&%&thetaG\"\"\"%#PiGF)" }{TEXT -1 4 " si \+ " }{TEXT 317 1 "r" }{TEXT -1 73 " < 0 et, dans ce cas, la partie gris e du bras traceur nous rappelle que " }{TEXT 318 1 "r" }{TEXT -1 13 " \+ est n\351gatif." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Soit le cercle d'\351quation polaire " }{XPPEDIT 18 0 "r= 2*sin(theta)" "6#/%\"rG*&\"\"#\"\"\"-%$sinG6#%&thetaGF'" }{TEXT -1 39 ". Ce cercle est enti\350rement trac\351 avec " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG " }{TEXT -1 5 " [0, " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 16 "[. D ans ce cas, " }{TEXT 309 2 "r " }{TEXT 310 3 "> 0" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "animepolaireplot(r=2*sin(the ta),theta=0..Pi,18,anime,none);" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Ce ce rcle peut aussi \352tre enti\350rement trac\351 avec " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon" "6#%(eps ilonG" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 16 "[. Dan s ce cas, " }{XPPEDIT 308 0 "r<0" "6#2%\"rG\"\"!" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "animepolaireplot(r=2*sin(the ta),theta=Pi..2*Pi,18,anime,none);" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Da ns le cas du trac\351 avec " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 1 " " }{XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 2 " [ " }{XPPEDIT 18 0 "0;" "6#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "2*Pi; " "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 6 "[, le " }{TEXT 311 11 "cercle \+ sera" }{TEXT -1 12 " \351videmment " }{TEXT 312 15 "trac\351 deux fois " }{TEXT -1 16 ": une fois avec " }{TEXT 313 1 "r" }{TEXT -1 22 " > 0 \+ et une fois avec " }{TEXT 314 2 "r " }{TEXT -1 4 "< 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "animepolaireplot(r=2*sin(theta),the ta=0..2*Pi,36,anime,none);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "R appelons que le graphique polaire de " }{XPPEDIT 18 0 " r=f(theta-alph a)" "6#/%\"rG-%\"fG6#,&%&thetaG\"\"\"%&alphaG!\"\"" }{TEXT -1 35 " est celui du graphique polaire de " }{XPPEDIT 18 0 "r=f(theta)" "6#/%\"rG -%\"fG6#%&thetaG" }{TEXT -1 37 " ayant subit une rotation d'un angle \+ " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 11 "Si l'angle " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" } {TEXT -1 95 " est positif, le graphique initial subit une rotation dan s le sens anti-horaire, et si l'angle " }{XPPEDIT 18 0 "alpha" "6#%&al phaG" }{TEXT -1 52 " est n\351gatif, la rotation est dans le sens hora ire. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "A:=animepolaireplo t(r=2*cos(theta),theta=0..Pi,18,fixe,polaire,color=orange):\nB:=animep olaireplot(r=2*cos(theta-3*Pi/4),theta=0..Pi,18,anime,polaire):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plots[display](A,B);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "a:=0:\nb:=3*Pi/4:\nSequence s:=20:\nPas:=seq(a+((b-a)/Sequences)*k,k=0..Sequences):\nA:=seq(animep olaireplot(r=2*cos(theta-k),theta=0..Pi,18,fixe,polaire),k=Pas):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(A,insequence=true,sc aling=constrained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Cardio\357d es et lima\347ons" }}{PARA 0 "" 0 "" {TEXT -1 21 " Formes g\351n\351ra les: \n " }{XPPEDIT 18 0 "r = a + b*cos(theta)" "6#/%\"rG,&%\"aG\"\"\" *&%\"bGF'-%$cosG6#%&thetaGF'F'" }{TEXT -1 5 " et " }{XPPEDIT 18 0 "r \+ = a + b*sin(theta)" "6#/%\"rG,&%\"aG\"\"\"*&%\"bGF'-%$sinG6#%&thetaGF' F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "En g\351n\351ral:\n " }{TEXT 291 10 "|a| >= |b|" } {TEXT -1 3 " " }{TEXT 289 9 "cardio\357de" }{TEXT -1 4 "\n " } {TEXT 292 12 "|a| < |b| " }{TEXT -1 3 " " }{TEXT 290 7 "lima\347on " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepolaireplot(r=2+2*c os(theta),theta=0..2*Pi,24,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 67 "animepolaireplot(r=-2+2*cos(theta),theta=0..2*Pi,24 ,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepo laireplot(r=1+2*cos(theta),theta=0..2*Pi,24,anime,polaire);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepolaireplot(r=3+2*cos(t heta),theta=0..2*Pi,24,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "b:=1:\na:=2+b:\nSequences:=18:\nPas_1:=[seq(a+((b-a) /Sequences)*k,k=0..Sequences)]:\nPas_2:=[seq(a-((b-a)/Sequences)*k,k=0 ..Sequences)]:\nCritere:=(x,y)->evalb(x>y):\nPas:=sort([op(Pas_1),op(2 ..nops(Pas_2),Pas_2)],Critere):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A:=seq(animepolaireplot(r=a+k*cos(theta),theta=0..2*Pi,24,fixe ,normal),k=Pas):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display (A,insequence=true,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Rosaces" }} {PARA 0 "" 0 "" {TEXT -1 16 "Forme g\351n\351rale: " }{XPPEDIT 18 0 "r = a*cos(n*theta)" "6#/%\"rG*&%\"aG\"\"\"-%$cosG6#*&%\"nGF'%&thetaGF'F '" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "r = a*sin(n*theta);" "6#/%\"rG*& %\"aG\"\"\"-%$sinG6#*&%\"nGF'%&thetaGF'F'" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepolaireplot(r=4*cos(2*theta),theta=0..2*Pi,24,an ime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepolai replot(r=5*sin(4*theta),theta=0..2*Pi,24,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "animepolaireplot(r=4*cos(2*(theta+P i/4)),theta=0..2*Pi,24,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Lemniscate " }}{PARA 0 "" 0 "" {TEXT -1 16 "Forme g\351n\351rale: " }{XPPEDIT 18 0 "r^2=cos(2*theta)" "6#/*$%\"rG\"\"#-%$cosG6#*&\"\"#\"\"\"%&thetaGF, " }{TEXT -1 86 ". Il est n\351cessaire, pour cette \351quation, d'effe ctuer la superposition de deux trac\351s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "A:=animepolaireplot(r=cos(2*theta),theta=-Pi/4..Pi/4 ,12,anime,polaire):\nB:=animepolaireplot(r=-cos(2*theta),theta=-Pi/4.. Pi/4,12,anime,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " plots[display](A,B);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 16 "Courbes div erses" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 21 "Spirale \351quiangulaire" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "animepolaireplot(r=theta/12,theta=0..18*Pi,24,anime,none,thickness=3) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 21 "Spirale logarithmique" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "animepolaireplot(r=4*exp(-t heta/12),theta=0..18*Pi,24,anime,polaire,thickness=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 288 8 "Papillon" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "animepolaireplot(r=exp(cos(theta))-2*cos(4*theta),the ta=0..2*Pi,18,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 49 "Rotation dans le sens anti-hora ire d'un angle de " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\" " }{TEXT -1 8 " radian." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 " animepolaireplot(r=exp(cos(theta-Pi/2))-2*cos(4*(theta-Pi/2)),theta=-P i/2..3*Pi/2,24,anime,polaire);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 28 "Autres courbes int\351res santes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "animepolaireplot(r=2+7*sin(3*theta),theta=0..2*Pi,18, anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepol aireplot(r=cos(7*theta/4),theta=0..8*Pi,36,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "animepolaireplot(r=1+cos(th eta/2),theta=0..4*Pi,18,anime,polaire);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "r^2 = 4*cos(theta);" "6#/*$%\"rG\"\"#*&\"\"%\"\"\"-%$co sG6#%&thetaGF)" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "A:=animepolaireplot(r=sqrt(4*cos(theta)),theta=-Pi/2..Pi/2,18,a nime,polaire):\nB:=animepolaireplot(r=-sqrt(4*cos(theta)),theta=-Pi/2. .Pi/2,18,anime,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 285 0 "" }{TEXT -1 49 "Rotation d ans le sens anti-horaire d'un angle de " }{XPPEDIT 18 0 "Pi/3" "6#*&%# PiG\"\"\"\"\"$!\"\"" }{TEXT -1 8 " radian." }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "r^2 = 4*cos(theta-Pi/3)" "6#/*$%\"rG\"\"#*&\"\"%\"\"\"- %$cosG6#,&%&thetaGF)*&%#PiGF)\"\"$!\"\"F2F)" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 170 "A:=animepolaireplot(r=sqrt(4*cos(theta-Pi/3)),thet a=-Pi/6..5*Pi/6,18,anime,polaire):\nB:=animepolaireplot(r=-sqrt(4*cos( theta-Pi/3)),theta=-Pi/6..5*Pi/6,18,anime,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plots[display](A,B);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "r^2 = 4*sin(2*theta);" "6#/*$%\" rG\"\"#*&\"\"%\"\"\"-%$sinG6#*&\"\"#F)%&thetaGF)F)" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 152 "A:=animepolaireplot(r=sqrt(4*sin(2*theta)), theta=0..Pi/2,18,anime,polaire):\nB:=animepolaireplot(r=-sqrt(4*sin(2* theta)),theta=0..Pi/2,18,anime,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "r^4 = 4*sin(2*theta);" "6#/*$%\"rG\"\"%*&\"\"%\" \"\"-%$sinG6#*&\"\"#F)%&thetaGF)F)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "A:=animepolaireplot(r=surd(4*sin(2*(theta)),4),theta =0..Pi/2,18,anime,polaire):\nB:=animepolaireplot(r=-surd(4*sin(2*(thet a)),4),theta=0..Pi/2,18,anime,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Rotation dans le sens horaire d'un angle de " }{XPPEDIT 18 0 "Pi/4 ;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 8 " radian." }}{PARA 0 "" 0 " " {XPPEDIT 18 0 "r^4 = 4*sin(2*(theta+Pi/4));" "6#/*$%\"rG\"\"%*&\"\"% \"\"\"-%$sinG6#*&\"\"#F),&%&thetaGF)*&%#PiGF)\"\"%!\"\"F)F)F)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "A:=animepolaireplot(r=surd( 4*sin(2*(theta+Pi/4)),4),theta=-Pi/4..Pi/4,18,anime,polaire):\nB:=anim epolaireplot(r=-surd(4*sin(2*(theta+Pi/4)),4),theta=-Pi/4..Pi/4,18,ani me,polaire):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1 2" 0 } {VIEWOPTS 1 1 0 1 1 1803 }