{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 } {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 "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 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand([1,2]/x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$*&\"\"\"F%%\"xG!\"\",$F$\"\"#" }}} {EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "norme:=proc( V) (simplify@sqrt@convert)(map(u->u^2,V ),`+`); end:\npv:=proc(U,V) [U[2]*V[3]-U[3]*V[2],U[3]*V[1]-U[1]*V[3],U [1]*V[2]-U[2]*V[1]] end:\nscal:=proc(x,U) [x*U[1],x*U[2],x*U[3]] end: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "M:=[cos(t),sin(t),t]: range_t:=0..2*Pi:range_u:=-2..2:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "M:=[cos(t),sin(t),sin(2*t)]:range_t:=0..2*Pi:range_u: =-5..5:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " dM_sur_dt:=map2(diff,M,t):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 " ds_sur_dt:=norme(dM_sur_dt):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 " T:=expand(dM_sur_dt/ds_sur_dt): \n dT_sur_ds:=expand(map2(diff,T,t)/ds_sur_dt):\n N:=scal(1/norme( dT_sur_ds),dT_sur_ds):\n B:=pv(T,N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 " Rc:=1/norme(dT_sur_ds):\n dB_sur_ds:=expand(map2(diff,B, t)/ds_sur_dt):\n Rt:=-N[1]/dB_sur_ds[1]:\n h:=expand(Rt*diff(Rc,t) /ds_sur_dt):\n subs(t=1,h):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 " \+ polaire:=expand(M+Rc*N+(h+u)*B):\n M1:=expand(M+Rc*N+h*B): \n \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 " " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 118 "graf_M:= spacecurve(M,t=range_t,color=blue,thickne ss=3):\ngraf_M1:= spacecurve(M1,t=range_t,color=red,thickness=3):\n \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "graf_pol:=plot3d(polair e,t=range_t,u=range_u,grid=[100,10]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "display([graf_M,graf_M1,graf_pol],scaling=constrained ,lightmodel=light2);\n" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1167 "restart:\n with(plots):\n norme:= proc( V) (simplify@sqrt@convert)(map(u->u^2,V),`+`); end:\n pv:=proc(U ,V) [U[2]*V[3]-U[3]*V[2],U[3]*V[1]-U[1]*V[3],U[1]*V[2]-U[2]*V[1]]\nend :\n polaire:= proc(M,range_t,range_u,param)\n local dM_sur_dt,ds_su r_dt,dT_sur_ds,T,N,B,Rc,\n Rt,h,polaire,graf_M,graf_M1,graf_pol ,eff,dB_sur_ds,M1:\n dM_sur_dt:=map2(diff,M,t);\n ds_sur_dt:=nor me(dM_sur_dt);\n T:=expand(dM_sur_dt/ds_sur_dt);\n dT_sur_ds:=ex pand(map2(diff,T,t)/ds_sur_dt);\n N:=expand(dT_sur_ds/norme(dT_sur_ ds));\n B:=pv(T,N); # print(T,N,B);\n Rc:=1/norme(dT_sur_ds); \n dB_sur_ds:=expand(map2(diff,B,t)/ds_sur_dt):\n if norme(dB_su r_ds)<>0 then\n Rt:=-N[1]/dB_sur_ds[1]:\n h:=simpl ify(Rt*diff(Rc,t)/ds_sur_dt):\n else h:=0 fi;\n M1:=expand(M+Rc *N+h*B);\n polaire:=expand(M+Rc*N+(h+u)*B);\n graf_M:= spacecurv e(M,t=range_t,color=blue,thickness=3,numpoints=200);\n graf_M1:= sp acecurve(M1,t=range_t,color=red,thickness=3,numpoints=200);\n graf_ pol:=plot3d(polaire,t=range_t,u=range_u,grid=[40,10]);\n if nargs() =4 then eff:=param else eff:=NULL fi;\n display([graf_M,graf_M1,gra f_pol],scaling=constrained,eff);\n end:\n " }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 385 "plotsetup(inline);\n polaire([cos( t)*3,sin(t)*2,0],0..2*Pi,-2..2);\n\n\nn:=1/2:display([polaire([cos(t)* cos(n*t),sin(t)*cos(n*t),sin(n*t)],-Pi..Pi,-\n2..1,orientation=[160,60 ]),\n plot3d([1,t,u],t=0..2*Pi,u=0..Pi,coords=spherical,style= LINE)]);\n polaire([cos(t),sin(t),sin(2*t)],Pi/2..3*Pi/2,-8..8,orienta tion=[40,20]);\n polaire([cos(t),sin(t),t],0..2*Pi,-2..3,orientation=[ 40,80]);\n" }}}{EXCHG {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "n:=1/2:display([polaire([cos(t)*cos(n*t) ,sin(t)*cos(n*t),sin(n*t)],-Pi..Pi,-2..1,orientation=[160,60]),\n \+ plot3d([1,t,u],t=0..2*Pi,u=0..Pi,coords=spherical,style=LINE)]);\n# polaire([cos(t)*cos(n*t),sin(t)*cos(n*t),sin(n*t)],-Pi..Pi,-2..1,orien tation=[160,60]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "n:=1/3:\ndisplay([polaire([cos(t)*cos(n*t) ,sin(t)*cos(n*t),sin(n*t)],-3*Pi/2..3*Pi/2,-2..1,orientation=[-120,50] ),\n plot3d([1,t,u],t=0..2*Pi,u=0..Pi,coords=spherical,style=LI NE)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "#polaire([cos(t)*cos(n*t) ,sin(t)*cos(n*t),sin(n*t)],-3*Pi/2..3*Pi/2,-2..1,orientation=[-120,50] );" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2113 "# Animation : polaire de C = enveloppe du plan norm al du pt courant de C\n# with(plots):\n norme:=proc( V) (simplify@sqrt @convert)(map(u->u^2,V),`+`); end:\n pv:=proc(U,V) [U[2]*V[3]-U[3]*V[2 ],U[3]*V[1]-U[1]*V[3],U[1]*V[2]-U[2]*V[1]]\nend:\n polaire_PN:= proc(M ,range_t,range_u,param)\n local dM_sur_dt,ds_sur_dt,dT_sur_ds,T,N,B ,Rc,\n Rt,h,polaire,graf_M,graf_M1,graf_pol,eff,\n dB_su r_ds,M1,nframes, PN, graf_PN:\n nframes:=43;\n dM_sur_dt:=map2(d iff,M,t);\n ds_sur_dt:=norme(dM_sur_dt);\n T:=expand(dM_sur_dt/d s_sur_dt);\n dT_sur_ds:=expand(map2(diff,T,t)/ds_sur_dt);\n N:=e xpand(dT_sur_ds/norme(dT_sur_ds));\n B:=pv(T,N); # print(T,N,B); \n Rc:=1/norme(dT_sur_ds);\n dB_sur_ds:=expand(map2(diff,B,t)/ds _sur_dt):\n if norme(dB_sur_ds)<>0 then\n Rt:=-N[1]/dB_su r_ds[1]:\n h:=simplify(Rt*diff(Rc,t)/ds_sur_dt):\n else \+ h:=0 fi;\n PN:=expand(M+(u+u*Rc)*N+(v+h)*B);#------plan normal\n \+ M1:=expand(M+Rc*N+h*B);\n polaire:=expand(M+Rc*N+(h+u)*B);\n #- -----graphe plan normal\n graf_PN:=[seq(plot3d(\n subs(t=op (1,range_t)+i*(op(2,range_t)-op(1,range_t))/nframes,PN),\n \+ u=0..2,v=-4..4,grid=[2,2],color=yellow),i=0..nframes)];\n\n gra f_M:= spacecurve(M,t=range_t,color=blue,thickness=3,numpoints=200);\n \+ graf_M1:= spacecurve(M1,t=range_t,color=red,thickness=3,numpoints=2 00);\n graf_pol:=plot3d(polaire,t=range_t,u=range_u,grid=[40,7]);\n if nargs()=4 then eff:=param else eff:=NULL fi;\n display([graf _M,graf_M1,graf_pol,display(graf_PN,insequence=true)],\n scalin g=constrained,eff);\n end:\n # -------------- ellipse ----------\n pol aire_PN([cos(t)*3,sin(t)*2,0],0..2*Pi,-2..2);\n# -------------- h\351l ice ----------\n polaire_PN([cos(t),sin(t),t],0..2*Pi,-4..0,orientatio n=[40,80]);\n# -------------- cl\351lie ----------\n n:=1/2:\n display ([polaire_PN([cos(t)*cos(n*t),sin(t)*cos(n*t),sin(n*t)],\n \+ -Pi..Pi,-2..1,orientation=[160,60]),\n plot3d([1,t,u] 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