{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 "" -1 256 "" 0 1 0 0 0 0 0 1 1 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 }{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 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -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 18 "" 0 "" {TEXT -1 70 "Sections de la surface de Mobius par un plan contenant une g\351n\351ratrice" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 10 "Principe :" }{TEXT -1 33 " \nJe prends la g \351n\351ratrice D(u): " }}{PARA 0 "" 0 "" {TEXT -1 193 " \+ r :=2+w*cos(u/2); z:=w*sin(u/2); x:=r*cos(u); y:=r*sin(u); \+ pour w variant dans R.\net un plan P(t) passant par D(u) [ obtenu pa r rotations d'angle t autour de D(u) ]. " }}{PARA 0 "" 0 "" {TEXT -1 56 "J'\351tablis l'\351quation nomm\351e \"eq3\" de la surface dans un " }{TEXT 257 67 "rep\350re dont le plan des YZ est P(t) et l'axe des Z port\351 par D(u)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 243 "On est ramen\351 \340 la section par X=0 de la surface, section qui c ontient la g\351n\351ratrice D(u) obtenue pour (X=0, Y=0 : axe des Z). \nLa conique, reste de l'intersection, est donn\351e par subs(X=0,eq3) /Y : la division par Y enl\350ve la g\351n\351ratrice D(u)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 268 "# ------- initialisations -----------------\nresta rt:\nx77:= expand(y^3-2*y^2*z+y*(z^2+x^2-4)-2*x*z*(x+2));\nu:=2*v:\n# \+ ------- rotation Oz\nrzu:=[x=x1*cos(u)-y1*sin(u),y=x1*sin(u)+y1*cos(u) ,z=z1]:\n#-----translation de 2 sur Ox1 \ntr:=x1=x1-2:\n# ------- rot ation de v % Ay1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 351 "ry1v:=[z1=z2*c os(v)-x2*sin(v),x1=z2*sin(v)+x2*cos(v),y1=y2]:\n# ----------- rotation de t autour de Az2\nrz2t:=[x2=x3*cos(t)-y3*sin(t),y2=x3*sin(t)+y3*cos (t),z2=z3]:\neq1:=simplify(subs(op(rzu),x77)):\neq2:=simplify(subs(op( ry1v),subs(tr,eq1))):\neq3:=simplify(subs(op(rz2t),eq2)):\nsection:=co llect(simplify(expand(subs(x3=0,y3=Y,z3=Z,eq3)/Y),trig),[Y,Z]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 " trac\351 d'une famille de sections pou r une g\351n\351ratrice donn\351e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 321 "graf:=proc(gene,nb_sect)\n global section;\n \+ local i,a,b,sect:\n sect:=evalf(subs(v=gene,section)):\n with (plots):\n display([seq(implicitplot(subs(t=i*2*Pi/nb_sect,sect),Y =-8..8,Z=-8..8,view=[-8..8,-8..8],color=COLOR(HUE,i/nb_sect),grid=[50, 50]),\n i=1..nb_sect)],scaling=constrained);\nend:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "graf(Pi/3,10);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 " trac\351 de quelques sections \+ particuli\350res " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "with(p lots):\n#ellipse\ncas:=subs(t=Pi/4,v=Pi/3,section):\nimplicitplot(cas, Y=-4..4,Z=-2..8,scaling=constrained);\nevalf(cas,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "cas:=subs(t=2*Pi/3,v=0,section):#hyperbo le\nimplicitplot(cas,Y=-6..4,Z=-4..4,scaling=constrained);\nevalf(cas, 4);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "cas: =subs(t=arccos(sqrt(3)/2),v=Pi/4,section):#parabole \nimplicitplot(cas ,Y=-4..4,Z=-2..8,scaling=constrained);\nevalf(cas,4);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "# ellipse bien plac\351e \ncas:=subs(t=arccos(2/9*3^(1/2)*6^(1/2)),v=Pi/3,section):\nwith(plots ):\nimplicitplot(cas,Y=-4..6,Z=0..10,scaling=constrained);\nevalf(cas, 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "# hyperbole bien pl ac\351e\ncas:=subs(t=arccos(-1/3*3^(1/2)*(3-4*cos(1/5*Pi)^2)^(1/2)/sin (Pi/5)),v=Pi/5,section):\nwith(plots):\nimplicitplot(cas,Y=-4..0,Z=-4. .4,scaling=constrained);\nevalf(cas,4);" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 " d\351composition en carr\351s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 229 "#-------------- d\351composition --------\ndecomp:= \+ proc(section,t0,v0)\nlocal a,b,c,d,f,g,fac1,fac2,sec1,sec2,b1,f1, sect ion1;\nsection1:=simplify(expand(subs(t=t0,v=v0,section)),trig);#colle ct(,[Y,Z]);\na:=coeff(section1,Y^2):# en y^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "b:=coeff(section1,Z^2):# en z^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "c:=coeff(coeff(section1,Y),Z):# en yz\nd:=subs(Z=0,co eff(section1,Y)):# en y" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "f:=subs (Y=0,coeff(section1,Z)):# en z\ng:=subs(Y=0,Z=0,section1): # constant \nfac1:=Y+c/2/a*Z+d/2/a:\nsec1:=simplify(expand(section1-a*fac1^2),tri g):\nb1:=coeff(sec1,Z^2):f1:=coeff(sec1,Z):\nfac2:=Z+f1/2/b1:\nsec2:=a *fac1^2+b1*fac2^2 +simplify(expand(section1-a*fac1^2-b1*fac2^2),trig); \n[a,b1,sec2];\nend:\noteplex:=proc(liste)\n local tamp,i;\n tamp: =NULL:\n for i from 1 to nops(liste) do \n if not(has(liste[i] ,I)) then tamp:=tamp,liste[i] fi;\n od:\n [tamp]\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "#verification\n#simplify( expand(subs(t1=t ,v1=v,decomp(section,t1,v1)[3]))-section,trig);\n# ------------------- -----------------------------------#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 80 " R\351soluti on g\351n\351rale : z\351ros du produit des coeff. de la d\351composit ion en carr\351s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "res:=si mplify( expand(subs(t1=t,v1=v,decomp(section,t1,v1))),trig):\nproduit: =simplify(res[1]*res[2]):\nprod:=512*combine(produit,trig):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "sol:=[solve(prod,t)]:nops(sol);\nsol:=co mbine(sol,trig);\n# sol[2] et sol[3] n'ont de solution que pour Pi/4 < v < 3*Pi/4 ( mod Pi )" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 382 "plot(\{sol[1],sol[2],s ol[3]\},v=0..Pi,color=[red,green,blue],view=[0..Pi,-Pi/2..Pi/2]);\ng1: =plot3d(produit,t=0..Pi,v=0..Pi,grid=[80,40],style=patchnogrid):\ng2:= plot3d(0.,t=0..Pi,v=0..Pi,grid=[2,2],color=yellow):\nprint(\"Nature de la section selon la r\351gion du plan des (t,v)\");\nplots[display]([ g1,g2],axes=boxed,orientation=[60,70],title=\"au-dessus : ellipse\\nau -dessous : hyperbole\");" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 637 "#-------- graphe du produit des coeff de --------- ------#\n# ------- la d\351composition en carr\351s de -------------- -# \n# ------- Gauss de la section par un plan Pt -----------#\n# 0 < \+ t < 2*Pi contenant la g\351n\351ratrice Dv0 ------------#\nv0:= Pi/3: \nres:=decomp(section,t,v0):#map(evalf,)\nnulla:=sort(oteplex([solve(r es[1],t)])):\nnullb1:=sort(oteplex([solve(res[2],t)])):\nnullprod:=sor t(oteplex([solve(res[2]*res[1],t)])):\nwith(plots):\nplotsetup(inline) :\nplot(res[2]*res[1],t=0..2*Pi,color=magenta,view=[0..2*Pi,-4..4]);\n print(\"Possibilit\351 de sections paraboliques aux points o\371 annul ation\");\nprint(\"Positif : ellipse ; N\351gatif : hyperbole\");" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "# -------- Valeurs annulant le pro duit des -------------------# \n# -------- coefficients de la d\351com position en carr\351s --------#\n# ------- EXECUTION NON OBLIGATOIRE P OUR LA SUITE ------------#\nprint(\"Points d'annulation \",nullprod); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "#------ ---- trac\351 d'une section par le plan Pt, t=t0 --------#\n#----- es sayer : nullprod[1], nullprod[2]...etc ------------#\n# ---- pour le s sections paraboliques -----------------------# \nwith(plots):\nt0:= nullprod[1]:\nimplicitplot(subs(t=t0,v=v0,section),Y=-10..10,Z=-10..10 ,grid=[50,50],scaling=constrained);\nsecpart:=subs(t=t0,v=v0,section): evalf(secpart);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "decomp(e valf(secpart),t,v)[3];# N\351gliger les \"petits\" termes" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "9 3 3" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }