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-137.000000 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 18 "" 0 "" {TEXT -1 62 "Surface d \351veloppable engendr\351e par les tangentes \340 une courbe " }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Esculier juin 2003" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "restart:\nwith(plots):\nnorme:=proc( V) (simplify@sq rt@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:\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 " projection sur le plan osculateur" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 854 "pjdeM_sur_po:= proc(M,range_t)\n local dM_s ur_dt,ds_sur_dt,ds,s,dT_sur_ds,T,N,B,Rc, \n Rt,dB_sur_ds, f,sol_ f,X,Y: \n dM_sur_dt:=map2(diff,M,t);\n ds_sur_dt:=norme(dM_sur_dt );\n T:=expand(dM_sur_dt/ds_sur_dt):\n dT_sur_ds:=expand(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:=e xpand(map2(diff,B,t)/ds_sur_dt):\n if norme(dB_sur_ds)<>0 then \n \+ Rt:=-N[1]/dB_sur_ds[1]: else Rt=infinity fi;\n s:=int(subs(t= u,ds_sur_dt),u=op(1,range_t)..t);\n sol_f:=dsolve(\{diff(f(t),t)=1/R t*ds_sur_dt,f(op(1,range_t))=0\},f(t)):\nf:=op(2,sol_f);\n#print(diff( X(t),t)=cos(f)*ds_sur_dt,diff(Y(t),t)=sin(f)*ds_sur_dt); \ndsolve(\{ diff(X(t),t)=cos(f)*ds_sur_dt,diff(Y(t),t)=sin(f)*ds_sur_dt,\n \+ X(op(1,range_t))=0,Y(op(1,range_t))=0\},\{X(t),Y(t)\});\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "C:=[cos(t),sin(t),t];\nres: =pjdeM_sur_po(C,0..2*Pi):\nif op([1,1],res)=X(t) then X:=op([1,2],res) ;Y:=op([2,2],res)\n else Y:=op([1,2],res);X:=op([2 ,2],res) fi;\nplot([X,Y,t=0..2*Pi*sqrt(2)],title=\"projection\",scalin g=constrained);\n# le r\351sultat Y := 2*sin(1/2*sqrt(2)*t);X := -2*co s(1/2*sqrt(2)*t)+2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"CG7%-%$cosG6#%\"tG-%$sinGF(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"YG,&-%$cosG6#,$*&-%%sqrtG6#\"\"#\"\"\"%\"tGF/#F/F.!\"#F.F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,$-%$sinG6#,$*&-%%sqrtG6#\"\"#\" \"\"%\"tGF/#F/F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3u-VmkjbIF!#=$\"37tl:?Dv s=!#>7$$\"3w!>m$=zdm]F-$\"3r:3Q@!eR_'F07$$\"39\"=8&[1A1wF-$\"3'*)o%yTq #G]\"F-7$$\"3Ts'Rx=%[-5!#<$\"3w79`h\"eQp#F-7$$\"3!4NJQ8<^A\"F>$\"3eFt! 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_=F>$\"3y8en(HcXv#F>7$$!3n6'3)Q(\\2$>F>$\"3&)=ho7$$!392pT)R[K)> F>$\"37!**elN8$eAF>7$$!3!\\$*zBl!****>F>$\"3sxUUoZ61?F>7$$!3a2UAp$[L)> F>$\"3aY'p3NbCu\"F>7$$!311fJEfhL>F>$\"3;@oXK<**)[\"F>7$$!3oU_3Ra,Z=F>$ \"3b7$$!3$z:#)H_p;t\"F>$\"3')Qo(G'))R$***F-7$$!3-W.Yf9O$e \"F>$\"3'QWb^@*H\"y(F-7$$!3)z;;MTW&39F>$\"30[e%*R#*R,eF-7$$!3-*oA=rnhA \"F>$\"3]Cq,CKk*>%F-7$$!3iikxhR0^**F-$\"3Y\"*46x\\K^EF-7$$!3vZyYrr$*>x F-$\"3QZB7,\")**\\:F-7$$!3Ne/n)pn.@&F-$\"3w4fnnEA1pF07$$!3X0v`KY$>s#F- $\"3_qz\\3$*)3'=F07$$!3Su/DT&[#zW!#FF(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F( -%(SCALINGG6#%,CONSTRAINEDG-%&TITLEG6#Q+projection6\"-%+AXESLABELSG6$Q !Ff[lFj[l-%%VIEWG6$%(DEFAULTGF^\\l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 27.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 48 " animation : enveloppe des plans tgts \340 l'h\351lice" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 168 "restart:\nwith(plots):\nnorme:=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:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "helice_PO:= proc(range_t,range_u,param)\n loc al M,T,N,cer,dev,graf_M,graf_dev,eff,dep,larg,PO,nfram,graf_PO,graf_ra y,graf_cer,graf_tg:\n M:=[cos(t),sin(t),t]; \n T:=expand([-sin(t) , cos(t), 1]/sqrt(2));\n N:=[-cos(t), -sin(t), 0];\n cer:=expand(M +(2-2*cos(sqrt(2)/2*(u+t)))*N-2*sin(sqrt(2)/2*(u+t))*T);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1096 "\n dev:=expand(M+u*T);\n PO:=simplify(e xpand(M+u*cos(-sqrt(2)/2*t+v)*T+(2+u*sin(-sqrt(2)/2*t+v))*N));\n dep :=op(1,range_t); larg:=op(2,range_t)-op(1,range_t); \n nfram:=40; \+ \ngraf_PO:= display([seq(\n plot3d(subs(t=dep+i*larg/nfram,PO), u=2..4,v=-Pi..Pi-1,grid=[10,20],color=yellow,style=patchnogrid),\n \+ i=0..nfram)],insequence=true);\ngraf_cer:=display([seq(\n \+ spacecurve([op(subs(t=dep+i*larg/nfram,cer)),u=0..2*Pi],color=blue,thi ckness=2),\n i=0..nfram)],insequence=true);\ngraf_ray:=display ([seq( polygonplot3d([subs(t=dep+i*larg/nfram,u=0,cer),subs(t=d ep+i*larg/nfram,u=Pi*2/sqrt(2),cer)],\n color=blue,t hickness=2,style=line),\n i=0..nfram)],insequence=true);\ngraf _tg:=display([seq(\n polygonplot3d([subs(t=dep+i*larg/nfram,M), subs(t=dep+i*larg/nfram,M-2*T)],\n color=blue,thickn ess=2,style=line),\n i=0..nfram)],insequence=true);\n\n graf _M:= spacecurve(M,t=range_t,color=red,thickness=3);\n if nargs()=4 the n eff:=param else eff:=NULL fi;\n graf_dev:=plot3d(dev,t=range_t,u=r ange_u,eff,grid=[20,5]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 " disp lay([graf_M,graf_dev,graf_PO,graf_ray,graf_cer,graf_tg],scaling=constr ained,eff);\nend:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name ch angecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "# h\351lice circulaire\n#helice_PO(0..2*Pi,-4..0,orientation=[ 10,50]);" }{TEXT 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tangentes" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "developpable:= proc(M,range _t,range_u,param)\n local dM_sur_dt,dev,graf_M,graf_dev,eff: \n d M_sur_dt:=map2(diff,M,t);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 207 " de v:=expand(M+u*dM_sur_dt);\n # print(dev);\n graf_M:= spacecurve(M, t=range_t,color=red,thickness=3);\n if nargs()=4 then eff:=param els e eff:=NULL fi;\n graf_dev:=plot3d(dev,t=range_t,u=range_u,eff);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " display([graf_M,graf_dev],scalin g=constrained,eff);\nend:\nplotsetup(inline);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "n :=1/2:#cl\351lie\ndisplay( [ developpable([cos(t)*cos(n*t),sin(t)*cos( n*t),sin(n*t)],\n -Pi..Pi,-1.5..2,orientation=[-5 5,70]),\n plot3d([1,t,u],t=0..2*Pi,u=0..Pi,coords=spherical, style=patchnogrid)],\n color=COLOR(ZHUE),lightmodel=light2);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6) 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