{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 }{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 "Headi ng 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 "Warning" -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 "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "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 }} {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 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 16 "Hyperbolographes" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 " Hype rbolographe par anti-parall\351logramme" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Tiges articul\351es AB, BC, CD formant un quadrilat\350r e non convexe : A et D fixes, AB=CD, AD=BC ; le point d'intersection E de AB et CD d\351crit une ellipse" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 74 " solution 1 : la 1-\350re branche est parcourue plus rapidement que la 2-i\350me" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1866 "resta rt;\nwith(plots):\nN2:=V->sqrt(V[1]^2+V[2]^2):\n\nhyperboleAP:= proc(a a,bb)\n local a,b, u,U,t,t1,u0,k,A,B,E,C,D,cercle_B,cercle_D,tiges, tigeEA,tigeEC,AB,courbe,arti,artiE,nbf,Cbar:\n Cbar:=0.7,0.2,0. 2:\n \n b:= 2*aa;a:=sqrt(4*bb^2+b^2) ;\n U:=simplify([solve(( b* cos(t)-a-b*cos(w) )^2 + (b*sin(t)-b*sin(w) )^2-a^2,w)][2]):\n A:=[- a/2,0]: D:=[a/2,0]: B:=[-a/2+b*cos(t),b*sin(t)]:\n cercle_B:=plot([ op(B),t=0..2*Pi],color =grey,thickness=2):\n cercle_D:=plot([a/2+b* cos(w),b*sin(w),w=0..2*Pi],color =grey,thickness=2):\n C:=[a/2+b*co s(U),b*sin(U)]:\n AB:=B-A; \n E:=expand(D+u*(C-D)); u0:=solve(( A-E)[1]*AB[2]-(A-E)[2]*AB[1],u);\n E:=simplify(subs(u=u0,E));\n \+ nbf:=20:t1:=-arctan(bb/aa):\n arti:=display([seq(PLOT(POINTS(op (evalf(subs(t=t1+j*2*Pi/nbf,[A,B,C,D]))),\n \+ COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=0..nbf)],insequence=true):\n \+ artiE:=display([seq(PLOT(POINTS(op(evalf(subs(t=t1+j*2*Pi/nbf,[E ]))),\n COLOR(RGB,1,0,0),SYMBOL(BOX,12))),j =0..nbf)],insequence=true):\n tiges:=display([seq(PLOT(CURVES(e valf(subs(t=t1+j*Pi*2/nbf,[A,B,C,D])),\n CO LOR(RGB,Cbar),THICKNESS(3))),j=0..nbf)],insequence=true):\n tig eEA:=display([seq(PLOT(CURVES(evalf(subs(t=t1+j*Pi*2/nbf,[A,E])),\n \+ COLOR(RGB,Cbar),THICKNESS(3), LINESTYLE(3))), j=0..nbf)],inse quence=true):\ntigeEC:=display([seq(PLOT(CURVES(evalf(subs(t=t1+j*Pi*2 /nbf,[C,E])),\n COLOR(RGB,Cbar),THICKNESS(3), LINESTYLE(3)) ), j=0..nbf)],insequence=true):\n courbe:=animate([op(subs(t=t1 +t*w,E)),t=0..2*Pi],w=0..1,frames=nbf+1,color=red):\n display([ cercle_B,cercle_D,arti,artiE,tiges,tigeEA,tigeEC,courbe],scaling=const rained,\n title=cat(\"X^2/\",convert(aa^2,string),\" - Y ^2/\",convert(bb^2,string),\" = 1\"),view=[-a..a,-a..a],axes=none):#\n \n end:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoor ds has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " hyperboleAP(1,1);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 39 " solution 2 : modifications des d\351fauts" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2812 "restart;\nwith(plots):\nN2:=V->sq rt(V[1]^2+V[2]^2):\n\nhyperboleAP:= proc(aa,bb)\n local a,b, u,U,t, t1,t2,u0,k,A,B,E,C,D,cercle_B,cercle_D,ass,\n tiges,tigeEA,tig eEC,AB,courbe,arti,artiE,nbf,Cbar:\n Cbar:=0.7,0.2,0.2:\n b: = 2*aa;a:=sqrt(4*bb^2+b^2) ;\n U:=simplify([solve(( b*cos(t)-a-b*c os(w) )^2 + (b*sin(t)-b*sin(w) )^2-a^2,w)][2]):\n A:=[-a/2,0]: D:=[ a/2,0]: B:=[-a/2+b*cos(t),b*sin(t)]:\n cercle_B:=plot([op(B),t=0..2 *Pi],color =grey,thickness=2):\n cercle_D:=plot([a/2+b*cos(w),b*sin (w),w=0..2*Pi],color =grey,thickness=2):\n ass:=plot([bb/aa*x,-bb/a a*x],x=-4*aa..4*aa,color=red,linestyle=2):\n C:=[a/2+b*cos(U),b*sin (U)]:\n AB:=B-A; \n E:=expand(D+u*(C-D)); u0:=solve((A-E)[1]*AB [2]-(A-E)[2]*AB[1],u);\n E:=simplify(subs(u=u0,E));\n\n nbf:=15: t1:=arctan(bb/aa)-0.15: t2:=arctan(bb/aa)+0.2:\n\n arti:=display([s eq(PLOT(POINTS(op(evalf(subs(t=-t1+2*j*t1/nbf,[A,B,C,D]))),\n \+ COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=0..nbf),\n \+ seq(PLOT(POINTS(op(evalf(subs(t=t2+2*j*(Pi-t2)/nb f,[A,B,C,D]))),\n COLOR(RGB,0,0,0),SYMBOL(C IRCLE,12))),j=1..nbf)],insequence=true):\n artiE:=display([seq(PLOT (POINTS(op(evalf(subs(t=-t1+2*j*t1/nbf,[E]))),\n \+ COLOR(RGB,1,0,0),SYMBOL(BOX,12))),j=0..nbf),\n \+ seq(PLOT(POINTS(op(evalf(subs(t=t2+2*j*(Pi-t2)/nbf,[E]))),\n \+ COLOR(RGB,1,0,0),SYMBOL(BOX,12))),j=1..nbf)],in sequence=true): \n tiges:=display([seq(PLOT(CURVES(evalf(subs(t=-t1 +2*j*t1/nbf,[A,B,C,D])),\n COLOR(RGB,Cbar), THICKNESS(3))),j=0..nbf),\n seq(PLOT(CURVES(eva lf(subs(t=t2+2*j*(Pi-t2)/nbf,[A,B,C,D])),\n \+ COLOR(RGB,Cbar),THICKNESS(3))),j=1..nbf)],insequence=true):\n tige EA:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+2*j*t1/nbf,[A,10*E-9*A]) ),\n COLOR(RGB,Cbar),THICKNESS(3), LINESTYLE(1))), j=0..nbf ),\n seq(PLOT(CURVES(evalf(subs(t=t2+2*j*(Pi-t2)/nbf,[A,10*E -9*A])),\n COLOR(RGB,Cbar),THICKNESS(3), LINESTYLE(1))), j= 1..nbf)],insequence=true):\n tigeEC:=display([seq(PLOT(CURVES(evalf (subs(t=-t1+2*j*t1/nbf,[C,10*E-9*C])),\n COLOR(RGB,Cbar),TH ICKNESS(3), LINESTYLE(1))), j=0..nbf),\n seq(PLOT(CURVES(eva lf(subs(t=t2+2*j*(Pi-t2)/nbf,[C,10*E-9*C])),\n COLOR(RGB,Cb ar),THICKNESS(3), LINESTYLE(1))), j=1..nbf)],insequence=true):\n\n \+ courbe:=display([seq(plot([op(E),t=-t1..-t1+t1/2/nbf+2*t1*k/nbf],color =red),k=0..nbf),\n seq(plot(\{[op(E),t=-t1..t1],[o p(E),t=t2..t2+2*k*(Pi-t2)/nbf]\},\n color=red), k=1..nbf)],insequence=true):\n display([cercle_B,cercle_D,ass,arti, tiges,tigeEA,tigeEC,artiE,courbe],\n view=[-1.5*a..1.5*a,- a..a],axes=none):#\n\n end:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "hyperboleAP(1,1);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 367 247 247 {PLOTDATA 2 "6&-%(ANIMATEG6A7.-%'POINTSG6(7$$ !+iN@99!\"*$\"\"!F/7$$\"*StX&>F-$!+Ng*p=\"F-7$$\"*3[7q#F-$\"+U`WS;F-7$ $\"+iN@99F-F.-%&COLORG6&%$RGBGF/F/F/-%'SYMBOLG6$%'CIRCLEG\"#7-%'CURVES G6%7&F*F0F5F:-F>6&F@$\"\"(!\"\"$\"\"#FNFO-%*THICKNESSG6#\"\"$-FG6&7$F* 7$$\"+&4\"*=o&!\")$!+j(*>%H%FenFJFQ-%*LINESTYLEG6#\"\"\"-FG6&7$F57$$\" +il)f;%Fen$!+r0gqdFenFJFQFhn-F(6%7$$\"+*)))44WF-$FgnF--F>6&F@F[oF/F/-F B6$%$BOXGFE-FG6$7S7$$\"3&*>S*p9*44W!#<$!3;P&*)=>+UH%Fep7$$\"3A56DJ(*z% R%Fep$!3!R=XV(o^zUFep7$$\"3e(3h9**HCQ%Fep$!31S5\"f$H\"oE%Fep7$$\"3[s+ \"zQ)foVFep$!3/*G\"Q/`g_UFep7$$\"3I)))H$H$fZN%Fep$!3rN&3b_(QQUFep7$$\" 3lMcx.#p5M%Fep$!3\"zcn/G?VA%Fep7$$\"3w\"p&)Q<]%GVFep$!3F_(\\[V^8@%Fep7 $$\"3=TiqdxX:VFep$!3]!y3Wi'*z>%Fep7$$\"36Eaeb$*4-VFep$!3/7QWSKE%=%Fep7 $$\"3+(e\"4=/V,U5% Fep7$$\"3.]IM%p6G@%Fep$!3mvoe]aS#4%Fep7$$\"3o%[.'=AE*>%Fep$!3qeY(*zfXy SFep7$$\"3i[.n*>\\y=%Fep$!3[c(zz$RqmSFep7$$\"3AZfs0xkuTFep$!3y%[Lv(y5` SFep7$$\"3#H5Tb]GI;%Fep$!3JGXjP#R6/%Fep7$$\"3%onp')[^.:%Fep$!3ws)GPjy! 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l-Fc`l6$;$!0I>roSEC%!#9$\"0I>roSEC%Fgbdm;$!0?YZ7F%GGFgbdm$\"0?YZ7F%GGF gbdm" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8 " "Curve 9" "Curve 10" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT -1 28 " Hyperbolographe de Delaunay" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "tige AB, A fixe ; parall\351logramme articul\351 BC ED, les points L de BC et M de BD sont tels que BL=BM et d\351crivent \+ une droite fixe" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1284 "with(p lots):\nhyperboleD:= proc(a,b) \n local u,u0,t1,t,w,A,B,E,F,tiges,c ourbe,arti,nbf,d,L,M,Cbar:\n Cbar:=0.7,0.2,0.2:\n M:= [t ,b/a*t] :\n A:=[0,u]: u0:=[solve((A-M)[1]^2+(A-M)[2]^2-(1.5*a)^ 2,u)]:\n B:=evalf(subs(u=u0[1],A)):A:=evalf(subs(u=u0[2],A)):\n \n L:= simplify(expand((A+B)/2)) :\n E:=expand(L+u*(M-L) ): u0:=[solve(expand(A-E)[1]^2+expand(A-E)[2]^2-4*a^2,u)]:\n F: =simplify(subs(u=u0[1],E)):E:=simplify(subs(u=u0[2],E)):\n#print(A,B,E ,F);print(\"M\",M);break; \n nbf:=15:t1:=1.5*a;\n \+ arti:=display([seq(PLOT(POINTS(op(evalf(subs(t=-t1+j*2*t1/nbf,[A,B,E, F,M,B]))),\n COLOR(RGB,0,0,0),SYMBOL(CIRCLE ,12))),j=0..nbf)],insequence=true):\n#print(arti);break:\n tige s:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+j*2*t1/nbf,[A,E,B,F,A,M,B ])),\n COLOR(RGB,Cbar),THICKNESS(3))),j=0.. nbf)],insequence=true):\n courbe:=animate([op(subs(t=-t1+t*w,E) ),t=0..2*t1],w=0..1,frames=nbf+1),\n animate([op(subs(t =-t1+t*w,F)),t=0..2*t1],w=0..1,frames=nbf+1):\n display([plot([ b/a*t,-b/a*t],t=-1.5*t1..1.5*t1,color=grey,thickness=2,linestyle=[1,2] ),\nplot([[0,-2.2],[0,2.2]],color=grey,thickness=2),\n \+ arti,tiges,courbe],axes=none):#\n \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "hyperboleD(1,1);" }}{PARA 13 "" 1 "" {GLPLOT2D 247 256 256 {PLOTDATA 2 "6&-%(ANIMATEG627)-%'POINTSG6*7$$\"\"!F,$!#:! 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7F]r7$$!3?ogFc7NC8FW$!3AA+++++]iFisFc]v7$F^]x$\"3!y(**********\\iFisFc gv7$F`jr$\"3y(**********\\(=F]rF`^v7$$!3Y@K=)>&Gf8FW$\"3y(**********\\ 7$F]r7$Fi]m$\"3y(***********\\PF]r7$$!3'p\\+1bVLR\"FW$\"3y(********** \\P%F]rF\\_v7$$!3y**********\\P9FW$\"3c&**********\\i&F]r7$$!3+`*R))[( 3j9FW$\"3c&***********\\iF]r7$$!3?qXBbv&3\\\"FW$\"3c&**********\\(oF]r 7$$!3haXdK1p?:FW$FbatF]r7$$!3Axna$znCb\"FW$\"3c&**********\\7)F]r7$$!3 !)*=cD>sge\"FW$\"3c&***********\\()F]rF\\[w7$$!3o*px^R7$e;FW$\"2c***** **********FW7$$!3;yA1&*Rt'p\"FW$\"3c**********\\i5FWF\\fw7$$!3G')fmJGo xU!3>FW$\"3c***********\\P\" FW7$$!3es4,^Cc`>FW$\"3b**********\\P9FW7$$!3c**************>FW$\"3c*** **********\\\"FWFX-FB6$7SF[oFbfmF_imF\\\\nFjfnFadoFginFceoFbfo7$FbgoF_ [x7$F[aqFb[x7$FaaqFe[xF_]rFj\\wF\\^r7$FbcqFh[xFi^r7$$\"3=(\\+1bVLR\"FW F]\\x7$Fg_rF`\\x7$$\"3o@K=)>&Gf8FWFe\\x7$F^]uFh\\xFc`s7$Fd]uF[]x7$$\"3 ?ogFc7NC8FWF`]xFcav7$F[dxFc]xFa^w7$Fd`sFf]xFgav7$$\"3Y@K=)>&Gf8FWF[^x7 $Fg_rF^^x7$$\"3'p\\+1bVLR\"FWFc^xF[bv7$$\"3y**********\\P9FWFh^x7$$\"3 +`*R))[(3j9FWF]_x7$$\"3?qXBbv&3\\\"FWFb_x7$$\"3haXdK1p?:FWFg_x7$$\"3Ax na$znCb\"FWF[`x7$$\"3!)*=cD>sge\"FWF``xF``w7$$\"3o*px^R7$e;FWFe`x7$$\" 3;yA1&*Rt'p\"FWFj`xFciw7$$\"3G')fmJGoxU!3>FWF^bx7$$\"3es4,^Cc`>FWFcbx7$$\"3c** ************>FWFhbxFXF]oF`yFcbl-%+AXESLABELSG6%Q\"t6\"Q!Fdgx-%%FONTG6# %(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;$!$D#FT$\"$D#FTFigx" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 44.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 " H yperbolographe par cercle directeur" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "tige AB et parall\351logramme CLBM, E l'intersection de AB et \+ LM : A et C fixes : E d\351crit une ellipse de demi-axes AB/2 et s qrt(AB^2-AC^2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 778 "with(plo ts):\nN2:=V->sqrt(V[1]^2+V[2]^2):\n \nHyperboleCD:= proc(a,b)\n loc al u,t,t1,t2,w,ass,w0,k,A,B,E,C,c,BC,L,tigeAB,para,M,Cbar,courbe,arti, artiE,nbf:\n \n c:=sqrt(a^2+b^2): Cbar:=0.7,0.2,0.2:\n \+ A:=[0,0]; B:=[2*a*cos(t),2*a*sin(t)] ; C:= [2*c,0] ;\n B C:=expand(B-C): BC:=simplify([BC[1]/N2(BC),BC[2]/N2(BC)]);\n L: =simplify(expand((B+C)*0.5+sqrt((a+c)^2-N2(expand((B-C)/2))^2)*[-BC[2] ,BC[1]])):\n M:=simplify(expand((B+C)*0.5-sqrt((a+c)^2-N2(expan d((B-C))/2)^2)*[-BC[2],BC[1]])):\n E:=expand(L+w*(M-L)):w0:=sim plify(solve(E[2]*B[1]-E[1]*B[2],w));\n E:=simplify(subs(w=w0,E) ):\nt1:=arctan(b/a)-0.2; t2:=arctan(b/a)+0.3; nbf:=8:#mettre un nombr e IMPAIR\n ass:=plot([b/a*(t-c),-b/a*(t-c)],t=-c..3*c,color=red ,linestyle=2):" }}{PARA 12 "> " 1 "" {MPLTEXT 1 0 1966 " arti:= display([seq(PLOT(POINTS(op(evalf(subs(t=-t1+j*2*t1/nbf,[A,B,C,L,M]))) ,\n COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))), j=1..nbf),\n seq(PLOT(POINTS(op(evalf(subs(t=t2+j *(Pi-t2)*2/nbf,[A,B,C,L,M]))),\n COLOR(RG B,0,0,0),SYMBOL(CIRCLE,12))),j=1..nbf)],insequence=true):\n arti E:=display([seq(PLOT(POINTS(op(evalf(subs(t=-t1+j*2*t1/nbf,[E]))),\n \+ COLOR(RGB,1,0,0),SYMBOL(CIRCLE,12))),j=1..nbf), \n seq(PLOT(POINTS(op(evalf(subs(t=t2+j*(Pi-t2)* 2/nbf,[E]))),\n COLOR(RGB,1,0,0),SYMBOL(CIRCLE ,12))),j=1..nbf)],insequence=true):\n tigeAB:=display([seq(PLOT (CURVES(evalf(subs(t=-t1+j*2*t1/nbf,[A,10*B-9*A,B,E,L])),\n \+ evalf(subs(t=-t1+j*2*t1/nbf,[E,M])),\n \+ COLOR(RGB,Cbar),THICKNESS(3),LINESTYLE(1))),j=1..nbf) ,\n seq(PLOT(CURVES(evalf(subs(t=t2+j*(Pi-t2)* 2/nbf,[A,B,E,L])),\n evalf(subs (t=t2+j*(Pi-t2)*2/nbf,[M,E])),\n COLOR(RGB, Cbar),THICKNESS(3),LINESTYLE(1))),j=1..nbf)],insequence=true):\n \+ para:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+j*2*t1/nbf,[A,B,L,C, M,B])),\n COLOR(RGB,Cbar),THICKNESS(3))),j= 1..nbf),\n seq(PLOT(CURVES(evalf(subs(t=t2+j*(Pi -t2)*2/nbf,[A,B,L,C,M,B])),\n COLOR(RGB,Cba r),THICKNESS(3))),j=1..nbf)],insequence=true):\n courbe:=displa y([seq(plot([op(E),t=-t1..-t1+j*2*t1/nbf],color=red),j=1..nbf),\n \+ seq(plot(\{[op(E),t=-t1..t1],[op(E),t=t2..t2+j*(Pi-t 2)*2/nbf]\},color=red),j=1..nbf)],\n insequence=true): \n print(display([plot([op(B),t=0..2*Pi],color=grey,thickness=2),\n \+ ass,arti,artiE,tigeAB,para,courbe],\n \+ \nview=[-c..3*c,-3*a..3*a],axes=none))\n#[plot([op(E),t=0..2*Pi],co lor=green),1\n#\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "H yperboleCD(1,1);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 321 321 321 {PLOTDATA 2 "6&-%(ANIMATEG627,-%'POINTSG6)7$$\"\"!F,F+7$$\"+mHJ5=! \"*$!+U-d,&)!#57$$\"+CrUGGF0F+7$$\"+3rB2QF0$!+!z))o?#F07$$\"+;)H]J)F3$ \"+lcRz:_3$4]SKz$f]EF^p7$$\"3?Z)o:'4`0UF^p $!3A_xA+;/1EF^p7$$\"3#*p)H'3'G)pTF^p$!33kQE_WwnDF^p7$$\"3nZ6\"\\-'oITF ^p$!305rG.BrDDF^p7$$\"3?14Mb\"fC4%F^p$!3O&o!*R;_X[#F^p7$$\"3M8@')4>gcS F^p$!3+`t^Tp&eW#F^p7$$\"3_t-E!GW[-%F^p$!3W0B#Ha7:T#F^p7$$\"3#GN9#p8/)) RF^p$!3yq<<>$QF^p$!3Qaq\"3\"3E,AF^p7$$\"38j(*)y( *R=!QF^p$!35H-m*eB\"o@F^p7$$\"3!)ff=r&>[x$F^p$!378ZR&HJ$Q@F^p7$$\"3/@2 3xyMYPF^p$!3>teffz&o5#F^p7$$\"3drNlO%*[F^p7$$\"3w>eh*z=eh$F^p$!3AW][&z$Rh>F^p7$$\"3Kd (zI/W@f$F^p$!3&p1\\h%>yM>F^p7$$\"3)zql5;`kc$F^p$!3wNFRuh\"e!>F^p7$$\"3 UPY=zs)Qa$F^p$!3Mza&pA&H!)=F^p7$$\"30oYpj%=._$F^p$!39u4!4;eN&=F^p7$$\" 3c\"**)G&H;%*\\$F^p$!3\"e'3kHLxH=F^p7$$\"3!z&y>J[/xMF^p$!3!))f**)ouB/= F^p7$$\"3WwOsz*RkX$F^p$!31Em^)yU1y\"F^p7$$\"3KySHn)\\`V$F^p$!3'\\d=$>X TcAkL$F^p$!3'e>8#*z5;k\"F^p7$$\"3rqVK\"e<$>LF^p$! 3db&*o]!\\:i\"F^p7$$\"3T'fd(GV5+LF^p$!3Ei)*\\Q@$*)f\"F^p7$$\"3U$y)\\#* RE$G$F^p$!3[(y].OM!z:F^p7$$\"3'*RLCNsllKF^p$!31k%G;$R:e:F^p7$$\"3_)f#Q xO8\\KF^p$!3.6Jg8O[Q:F^p7$$\"3P=?(*zbsJKF^p$!3O=A\"y7zw^\"F^p-%'COLOUR G6&FD$\"*++++\"FZF+F+-%+AXESLABELSG6$Q!6\"Fi_l-%%VIEWG6$%(DEFAULTGF^`l -FU6%7S7$$F]oF,F+7$$\"3O=AzuCF\")>F^p$\"3iG`6ljbIF!#=7$$\"3lAYw>/wM>F^ p$\"3o\\)*=>zdm]Fi`l7$$\"3+2Ax&H<(\\=F^p$\"3?nEr\\1A1wFi`l7$$\"3/s$fP= 91t\"F^p$\"3$Hq!*y=%[-5F^p7$$\"336]d1/&3e\"F^p$\"3b!f/S8<^A\"F^p7$$\"3 !*pysL[^;9F^p$\"3cs%y,a3>T\"F^p7$$\"3%e'QE\\6HB7F^p$\"3)HC/*4QE#e\"F^p 7$$\"3p&[h7Ja@+\"F^p$\"3gQQvF_!3t\"F^p7$$\"3C[;M*Gz-k(Fi`l$\"3]1?@-JJ[ =F^p7$$\"3mLHW_Dx]]Fi`l$\"39xi*oit^$>F^p7$$\"3qY-BXOF!p#Fi`l$\"3i>([FZ B=)>F^p7$$!3+u%3[&R@%>\"!#?$\"3!fAjMk*****>F^p7$$!3:'*H[=Z'\\s#Fi`l$\" 3#R*Q_d%\\8)>F^p7$$!3u6Vr+VO#H&Fi`l$\"3Z%QO)4lqG>F^p7$$!3!zvG@'y@YvFi` l$\"3'ebkrIt@&=F^p7$$!3\\*\\CZ0j%35F^p$\"3bGdh9q8FF^p$\"3P(QHKl(oV^F i`l7$$!3Q&R$)Q+19)>F^p$\"36'3/,,]3s#Fi`l7$$!3+_m&=Q!****>F^p$\"3^\\W)H ()=E?'F_dl7$$!3]5Yu.wo\")>F^p$!3T&=)R7PE+FFi`l7$$!3S52X$48[$>F^p$!3%32 ^l0mX1&Fi`l7$$!3UImgUCl_=F^p$!3S>qjsqWMvFi`l7$$!3*\\l>kAP[t\"F^p$!3E`q !Hx\"e^**Fi`l7$$!3k%)fDaN@*e\"F^p$!3#)[f!Q&)[U@\"F^p7$$!34l#=u=8@U\"F^ p$!34WOTi&piS\"F^p7$$!3woN-'G<(47F^p$!3=E:c_=F^p7$$!36c\\ei6 L<_Fi`l$!3k)Q%>R(\\2$>F^p7$$!3ABC[]N8$e#Fi`l$!3kyOh)R[K)>F^p7$$!3]edte ow9hF_dl$!3G@ZQ_1****>F^p7$$\"3cgWE2lWvDFi`l$!3M\\p,p$[L)>F^p7$$\"3`ky S\"p#35^Fi`l$!3o'p$*e#fhL>F^p7$$\"3Uod6rnfrwFi`l$!3iSTVQa,Z=F^p7$$\"3Z A&>_6g1+\"F^p$!3&f?6@_p;t\"F^p7$$\"3%)\\w*)z+(=A\"F^p$!3!3zp$e9O$e\"F^ 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!3V2&QOWFg$>F^p7$$\"3mn1H-yNUNF^p$!3%G.bNsi&y=F^p7$$\"3@MssPz4,NF^p$!3 5#\\3C\\*oJ=F^p7$$\"3?oEy.W6bMF^p$!3a%3hVKA\"zFZk8y-+LF^p$!3UIge!*=%))f\"F^p7$$\"3saMnt)))o E$F^p$!3gL%3)*><'f:F^p7$$\"3=qyw#z3DB$F^p$!3:zta%)ph=:F^p7$$\"3!Gh.j&4 A)>$F^p$!3It/6'R!Rx9F^p7$$\"3qp#G,g&\\pJF^p$!3Cs#\\mOsDW\"F^p7$$\"3Yh$ yM9)eRJF^p$!3Ov$oM'*GgS\"F^p7$$\"3bMdfM;&)4JF^p$!3QA$RE@v$p8F^p7$$\"3i v7!R'y%=3$F^p$!3\"Q)z(*zEaM8F^p7$$\"3=jVXY%Hd0$F^p$!3<2()>&>eOwZ< \"F^p7$$\"3Re&=ET*)R$HF^p$!3k=%oKRHW9\"F^p7$$\"30,ALl\\68HF^p$!3()H?KM 'fl6\"F^p7$$\"3iwDzP,/#*GF^p$!3%R,`K\\/\")3\"F^p7$$\"3YmZ#H%H9sGF^p$!3 #*3[b?K#41\"F^p7$$\"3Y%Rs\\@K?&GF^p$!3)GO!*R+8J.\"F^p7$$\"31L4xnCLLGF^ p$!3[UO)[DDp+\"F^p7$$\"3J=(*QVD'[\"GF^p$!3eXFdH)Hs!)*Fi`l7$$\"3_F.UGP< (z#F^p$!39a#>C:**Gb*Fi`l7$$\"37[kO%Q^9y#F^p$!3iKBDfk$QK*Fi`l7$$\"39\") fn\"*f.kFF^p$!3%Hqui7Nl1*Fi`l7$$\"3oBFxxu)*[FF^p$!3\">FF^p$!3Y;<77&)=$Q)F i`l7$$\"3Hgs9#f;Uq#F^p$!3c%[m81\"G\\\")Fi`lFa_lFf_lF[`lF_`lF[`mFe_n7,- 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2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 " Hyperbol ographe de Descartes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "un triang le mobile ABC rectangle en B est tel que CB se d\351place sur la droi te x=-a ; soit Q de CB tel que CQ=b.\nLe point E intersection de OQ et AC d\351crit une hyperbole " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1357 "with(plots):\n#--------------------------------\n# x=-a cot \351 BC du triangle ABC, b = longueur CQ, c=longueur CB\n#------------ ---------------------\nhyperboleDesc:= proc(a,b,c)\n local d,u,u0,t ,OO,t1,triangle,k,A,B,E,C,Q,AC,EC,tigeOQ,courbe,arti,nbf,Cbar,tigeL:\n Cbar:=0.7,0.2,0.2:\n d:=1:\n OO:=[0,0]:B:=[- a,t+c]:\n A:=expand(B+[1,0]): C:= [-a,t] :\n Q:=[-a,t+b ] : E:=u*Q:\n EC:=expand(E-C):AC:=expand(A-C):\n u0: =solve(expand(EC[1]*AC[2]-EC[2]*AC[1]),u):\n\n E:=simplify(expa nd(subs(u=u0,E))):\n nbf:=20:\n t1:=3*a;\n arti:= display([seq(PLOT(POINTS(op(evalf(subs(t=-t1+j*2*t1/nbf,[Q,E,OO]))),\n COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=0.. nbf)],insequence=true):\n triangle:=display([seq(PLOT(CURVES(ev alf(subs(t=-t1+j*2*t1/nbf,[A,B,C,A])),\n CO LOR(RGB,Cbar),THICKNESS(3))),j=0..nbf)],insequence=true):\n tig eOQ:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+j*2*t1/nbf,[OO,Q,E,A])) ,\n COLOR(RGB,Cbar),THICKNESS(3),LINESTYLE( 1))),j=0..nbf)],insequence=true):\n courbe:=animate([op(subs(t= -t1+t*w,E)),t=0..2*t1],w=0..1,frames=nbf+1):\n display([plot([[ -a,-3*a],[-a,3*a]],color=grey,thickness=2),\n arti,tri angle,tigeOQ,courbe],axes=none,view=[-a..1,-3*a..4*a])#\n\n \nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "hyperboleDesc(0.9,1,2); " }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 303 262 262 {PLOTDATA 2 "6&-%(ANIMATEG677'-%'POINTSG6'7$$!\"*!\"\"$!#(*GeF/$\"3%[OBDRp\" 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3Sqig)>*G\"\\'F]p$\"3%\\uy-;U#*G\"F/7$Fgay$\"3U&['['['['Q\"F/7$$!3d\"3 &HiUMRmF]p$\"3O#)4aZ6j%[\"F/7$$!3@^nUpj+2nF]p$\"3-[Y6E()f$e\"F/7$$!3OO t\"G$y*3x'F]p$\"3eIlVL/K$o\"F/Ff\\[l7$$!3?rL&)\\Ic))oF]p$\"3]B$H+R(y%) =F/Fa\\\\l7$$!3yH$f$p*GW*pF]p$\"3kJ\"Gh?9')3#F/7$$!3Tj&p3EyM/(F]p$\"31 %3EyM/8>#F/7$$!3!>d#Rwc=!4(F]p$\"3k#[@Z'GY%H#F/7$$!3!\\t1f-:Z8(F]p$\"3 3]'=[*p0)R#F/7$$!37 " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 " Hyperbolographe de L'Hospital" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 153 "une tige AEB se d\351place de fa\347on que A et B d \351crivent deux droites perpendiculaires ( ou non de L'Hospital ) : E d\351crit une ellipse de demi-axes AE et BE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2207 "with(plots):\nhyperboleH:= proc(a,b)\n loca l A,B,E,F,coef,ass,tigeAF,tigeEB,tigeAE,t1,t2,t3,courbe,arti,nbf,Cbar: \n Cbar:=0.7,0.2,0.2:\n A:=[-a,0]: B:=[t,-b/a*t] :\n \+ E:=expand(A+u*(B-A)):F:=subs(u=solve(E[2]-b/a*E[1],u),E):\n \+ E:=simplify(expand(B+F-A));\n nbf:=8:\nt1:=3*a; t2:=-a/2+0.1: t3:=2*a: coef:=1.95:\n ass:=plot([b/a*t,-b/a*t],t=-2*a..2*a,color =red,linestyle=2):\n\n arti:=display([seq(PLOT(POINTS(op(evalf( subs(t=-t1+j*(-a/coef+t1)/nbf,[A,B,E,F]))),\n CO LOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=1..nbf),\n \+ seq(PLOT(POINTS(op(evalf(subs(t=t2+j*(t3-t2)/nbf,[A,B,E]))),\n \+ COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=1..nbf)],insequ ence=true):\n tigeAE:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+ j*(-a/coef+t1)/nbf,[A,E])),\n COLOR(RGB,Cbar),THI CKNESS(3))),j=1..nbf),\n seq(PLOT(CURVES(evalf( subs(t=t2+j*(t3-t2)/nbf,[F,B])),\n COLOR(RGB,Cbar ),THICKNESS(3))),j=1..nbf)],insequence=true):\n tigeAF:=display( [seq(PLOT(CURVES(evalf(subs(t=-t1+j*(-a/coef+t1)/nbf,[A,F])),\n \+ COLOR(RGB,Cbar),THICKNESS(3),LINESTYLE(1)) ),j=1..n bf),\n seq(PLOT(CURVES(evalf(subs(t=t2+j*(t3-t2) /nbf,[E,B])),\n COLOR(RGB,Cbar),THICKNESS(3), LINESTYLE(1)) ),j=1..nbf)],insequence=true):\n tigeEB:=display([ seq(PLOT(CURVES( evalf(subs(t=-t1+j*(-a/coef+t1)/nbf,[E,B])),\n \+ COLOR(RGB,Cbar),THICKNESS(3),LINESTYLE(1)) ),j=1 ..nbf),\n seq(PLOT(CURVES( evalf(subs(t=t2+j*(t 3-t2)/nbf,[A,F])),\n COLOR(RGB,Cbar),THICK NESS(3),LINESTYLE(1)) ),j=1..nbf)],insequence=true):\n courbe:=d isplay([seq(plot([op(E),t=-t1..-t1+j*(-a/coef+t1)/nbf],color=red),j=1. .nbf)],\n [seq(display([plot([op(E),t=-t1..-a/1 .9],color=red),\n plot([op(E),t=t2 ..t2+j*(t3-t2)/nbf],color=red)])\n ,j=1..nb f)],insequence=true):\n display([ass,arti,courbe,tigeAE,tigeAF, tigeEB],scaling=constrained,\n axes=none,view=[-2*a..2*a ,-2*a..2*a])#\n\n \nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "hyperboleH(1,1);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%(ANIMATEG627+-%'POIN TSG6(7$$!\"\"\"\"!$F-F-7$$!+kD5*o#!\"*$\"+kD5*o#F27$$!+8YI.BF2$\"+90! \\2#F27$$!+)\\??9'!#5F;-%&COLORG6&%$RGBGF-F-F--%'SYMBOLG6$%'CIRCLEG\"# 7-%'CURVESG6$7S7$$!33+++++++E!#<$\"3!**************R#FN7$$!3!*eCgC^\\$ 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45.000000 44.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2663 "hyperboleH:= proc(a,b)\n \+ local A,B,E,F,coef,ass,tigeE1E2,tigeAE,t1,t2,t3,dAB1,dAB2,\n \+ courbe,arti,nbf,Cbar,B1,B2,B3,B4,E1,E2,C,c,curseur,long,larg,cL,cl :\n Cbar:=0.7,0.2,0.2:\n A:=[-a,0]: B:=[t,-b/a*t] :\n cL:=2: cl:=1:\n dAB1:=expand(A+10*(B-A)):dAB2:=expand (A-10*(B-A)):\n c:=sqrt(a^2+b^2):\n larg:=[b/c,-a/c]: \+ long:=[a/c/4,b/c/4]:\n E:=expand(A+u*(B-A)):F:=subs(u=solve(E[ 2]-b/a*E[1],u),E):\n B1:=expand(B+cL*long): B2:=expand(B-cL*lo ng):\n B3:=expand(B1+cl*larg):B4:=expand(B2+cl*larg):\n \+ E:=simplify(expand(B+F-A));\n E1:=expand(E+8*long): E2:=ex pand(E-8*long):\n C:= expand(B-cl/2/larg[1]/larg[2]*[a/c,-b/c ]):\n \n nbf:=20:\n t1:=3*a; t2:=-a/2+0.1:t3:=2*a: coef: =1.95:\n ass:=plot([b/a*t,-b/a*t],t=-3*a..3*a,color=grey,thickne ss=2):\n \n arti:=display([seq(PLOT(POINTS(op(evalf(subs(t=-t1 +j*(-a/coef+t1)/nbf,[A,B,E,F,C]))),\n COLOR(RGB ,0,0,0),SYMBOL(CIRCLE,12))),j=1..nbf),\n seq(P LOT(POINTS(op(evalf(subs(t=t2+j*(t3-t2)/nbf,[A,B,E,C]))),\n \+ COLOR(RGB,0,0,0),SYMBOL(CIRCLE,12))),j=1..nbf)],insequenc e=true):\n tigeAE:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+j* (-a/coef+t1)/nbf,[dAB1,dAB2])),\n COLOR(RGB,Cbar ),THICKNESS(3))),j=1..nbf),\n seq(PLOT(CURVES( evalf(subs(t=t2+j*(t3-t2)/nbf,[dAB1,dAB2])),\n C OLOR(RGB,Cbar),THICKNESS(3))),j=1..nbf)],insequence=true):\n ti geE1E2:=display([seq(PLOT(CURVES(evalf(subs(t=-t1+j*(-a/coef+t1)/nbf,[ E1,E2])),\n COLOR(RGB,Cbar),THICKNESS(3))),j=1.. nbf),\n seq(PLOT(CURVES(evalf(subs(t=t2+j*(t3- t2)/nbf,[E1,E2])),\n COLOR(RGB,Cbar),THICKNESS(3 ))),j=1..nbf)],insequence=true):\n curseur:=display([seq(\n \+ PLOT(CURVES(evalf(subs(t=-t1+j*(-a/coef+t1)/nbf,[B1,B2,B4,B3, B1])),\n COLOR(RGB,Cbar),THICKNESS(3)) ),j=1 ..nbf),\n seq(PLOT(CURVES(evalf(subs(t=t2+j*(t3 -t2)/nbf,[B1,B2,B4,B3,B1])),\n COLOR(RGB,Cba r),THICKNESS(3)) ),j=1..nbf)],insequence=true):\n courbe:=displ ay([seq(plot([op(E),t=-t1..-t1+j*(-a/coef+t1)/nbf],color=red),j=1..nbf )],\n [seq(display([plot([op(E),t=-t1..-a/1.9] ,color=red),\n plot([op(E),t=t2.. t2+j*(t3-t2)/nbf],color=red)])\n ,j=1..nbf )],insequence=true):\n display([ass,arti,tigeAE,tigeE1E2,courb e,curseur],axes=none,view=[-3*a..3*a,-2*a..2*a])#,tigeAE\n \n \n en d:\n hyperboleH(1,1);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 419 281 281 {PLOTDATA 2 "6&-%(ANIMATEG6J7+-%'POINTSG6)7$$!\"\"\"\"!$F-F-7$$!+E5kvG !\"*$\"+E5kvGF27$$!+\"yv3[#F2$\"+siSqAF27$$!+XvM_g!#5$!+WvM_gF=7$$!+XU `o@F2$\"+XU`o@F2-%&COLORG6&%$RGBGF-F-F--%'SYMBOLG6$%'CIRCLEG\"#7-%'CUR VESG6%7$7$$!+E5kv>!\")$F4FU7$$\"+E5kvYo$F27$$!+U$*3&*QF2$\"* 4F>c)F2FenF[o-FO6$7S7$$!33+++++++E!#<$\"3!**************R#Fbp7$$!3[8B$ G(yR(f#Fbp$\"3U86r$y!=(R#Fbp7$$!3Az!*ynR8&f#Fbp$\"3%)\\0!ogFZR#Fbp7$$! 3!RLxP=)e#f#Fbp$\"3?Z5\")Q&o>R#Fbp7$$!351N:ad-!f#Fbp$\"3Qg5V_4>*Q#Fbp7 $$!3a\")*fEtvue#Fbp$\"31\"p%3QjU'Q#Fbp7$$!3QOJ8!e#Fbp$\"3=C$[;9j% 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