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0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle45 " -1 259 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle46" -1 260 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle47" -1 261 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 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle 48" -1 262 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }} {PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefine d\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm \+ and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 231 " " 0 "" {TEXT 241 28 "Classification des cubiques." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 " Mail de Vidiani" }}{EXCHG {PARA 232 "" 0 "" {TEXT 242 1845 "Dans le tr\350s long article d'Abhyankar, mondialement connu pour sa m\351thode de \"d\351singularisation des vari\351t\351s \"\n(Memoirs of the college of science, University of Kyoto, serie A, \+ Vol XXXII mathematics No 3, 1960 p 455-511\ndonn\351 en bibliographie \+ pour la citation de Vainsencher (voir message pr\351c\351dent)\nil est dit p 457 :\n\nSoit X,Y,Z,T les coordonn\351es projectives dans P^3 \+ (espace projectif de dimension 3 sur k) et \nF_1 =T^3+XYT+X^3 soit 1+x *y+x^3\nF_2 =T^3+X^2Y soit 1+x^2*y\nF_3 = T^3+YXT+X^2Z soit 1 +x*y+x^2*z\nF_4=T^3+YT^2+YXT+X^2Z soit 1+y+x*y+x^2*z\n \nChaque F_j est un polyn\364me du premier degr\351 en Z et ses coefficients sont \+ premiers entrer eux en X Y T et donc F_j est irr\351ductible dans k[X, Y,Z,T] (k corps alg\351briquement clos de caract\351ristique p)\n\nV_j : est la surface d'\351quation F_j=0 ; K : X=T=0 ; P : X=Y=T=0 Alor s V_j est une surface cubique irr\351ductible dans P^3 avec la droite \+ double L\n\nLa proposition I page 457-458 (deux pages pour l'\351nonc \351) d\351montre p 459-465 que V_1 et V_2 sont des cones avec lke poi nt triple P et que tout cone ayant une double ligne est projectivement \351quivalent \340 l'un d'eux\n\nV_3 est une surface r\351gl\351e de \+ directrice L dite non sp\351ciale\n\nV_4 est une surface r\351gl\351e \+ dit'e sp\351ciale\n\nDans la partie V de la proposition sip la caract \351ristiquedu corps est diff\351rente de 2 et 3, il est d\351montr \351 que toute surface cubique ayant une double ligne est projectiveme nt \351quivalente \340 une des V_j\n\nDans la partie VI il est pr\351c is\351 : dans l'espace \340 9 dimensions P^9 de toutes les surfaces cu biques de P^3 ayant L comme droite singuli\350re, tous les points hors d'une certaine sous vari\351t\351 de P^9 repr\351sentent les surfaces cubiques qui sont projectivement \351quivalentes \340 V_4\n\nNulle pa rt il n'est pr\351cis\351 pour V_3 ou V_4 (mais on est dans un espace \+ projectif complexe) les d\351nominations conoide, Whitney ou Cayley" } {TEXT 243 1 "." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 233 "" 0 "" {TEXT 244 47 " Construction de la suite des \+ coeff pour PovRAY" }}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 8 "rest art:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1914 "i:='i':j:='j':k:= 'k':\n#S:=81*(x^3+y^3+z^3)-189*(x^2*y+x^2*z+y^2*x+y^2*z+z^2*x+z^2*y)+5 4*x*y*z+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1 ;\nS:=64*x^3+48* x^2*z-192*y^2*x+48*y^2*z-31*z^3-54*z^2+72*z:\n#a:=2; b:=1 ; c:=5; d:=- 2;\n#S:=(x^2+y^2+z^2+b^2-d^2)^2 -4*((a*x-c*d)^2+b^2*y^2) ;\n#S := x*(x ^2-y^2)+z^2*(1.+z)+.4000000000*x*y+.4000000000*y*z;\n#S:=-18*x^3-36*x^ 2*z+18*y^2*x-18*x*z^2+18*z^3-12*x^2-12*x*z-3*z^2-2*x-z:\n#S:=(2*x^2+y^ 2+z^2-1)^3-x^2*z^3/10.-y^2*z^3;# coeur de Tore Norstrand\n#S:=x^4+y^4+ z^4-2*(x^2+y^2+z^2)-1;#cube arroundi\n#=============================== ===============\n#================= cubiques r\351gl\351es =========== ======\n#S:=z*x^2-y^2;#parapluie de Whitney \n#S:=z*(x^2+y^2)-x*y; # P lucker\n#===============================================\n#S:=z*(x^2-y ^2)-x*y; # Zindler\n#==============================================\n# S:=(1-x-y-z)*x*z-y^3:#non r\351gl\351e\n#S:=z-x*y+x^3/3;#r\351gl\351e \n#S:=x*y*z+1;# hunt\n#S:=1+6*sqrt(2)*x*y*z+2*(y^3*z+z^3*x+x^3*y);# hu nt\n#S:=x*y+z^3;# hunt\n#S:=x^2*y^2+x^2*z^2+y^2*z^2-x*y*z ;#Steiner-Ro man\n#S:=4*x^2*(x^2+y^2+z^2+z)+y^2*(y^2+z^2-1); #cross-cap\n#S:=y^2-2* x*y^2-x*z^2+x^2*y^2+x^2*z^2-z^4;# ligne triple ??\n#S:=x^2+y^2-x^2*z+y ^2*z+z^2-1;# une forme Cayley \n#S:=4*(x^3+y^3+z^3+1)-(x+y+z+1)^3;# u ne forme Cayley ?\n#S:= 64*x^3+48*x^2*z-192*y^2*x+48*y^2*z-31*z^3-54*z ^2-12*z;\nS:=x^2+y^2+z^2-5/2*x*y*z-1;# surf du dble-six\n#============ ==================================\nS:=expand(1.*S): # sinon pb si x^n tout seul\nn:=degree(S,\{x,y,z\});\nS:=sort(S,[x,y,z],plex):\nex:=con vert(S,list):\ng:=sort(normal(sum(sum(sum(x^k*y^(j)*z^(i),i=0..n-k-j), j=0..n-k),k=0..n)),[x,y,z],plex) :\nbase:=convert(g,list):\nsuite:=NUL L: pp:=1:\nfor i from 1 to nops(base)-1 do # -1 car select(has,12,[x,y ,z]) ne renvoie rien\nif (pp<=nops(ex)) and (select(has,ex[pp],\{x,y,z \}) = base[i]) \n then suite:= suite,coeffs(ex[pp]): pp:=pp+1:\n \+ else suite:= suite,0: fi;\nod:\nsuite:=[suite,subs(x=0,y=0,z=0,S)]; " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,,*$)%\"xG\" \"#\"\"\"F**$)%\"yGF)F*F**$)%\"zGF)F*F**&#\"\"&F)F**(F(F*F-F*F0F*F*!\" \"F*F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&suiteG76\"\"!F&F&$\"\"\"F&F&$!+++++D!\"*F&F&F&F &F&F&F'F&F&F&F&F'F&$!\"\"F&" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 56 "Transformations de Ferr \351ol : Clebsch --> formes diverses" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 373 "# ----- mail de Ferrar\350se\nG:=subs(x=(3+sqrt(3))/ 6*X+(-3+sqrt(3))/6*Y+(sqrt(3))/3*Z/4,\ny=(-3+sqrt(3))/6*X+(3+sqrt(3))/ 6*Y+(sqrt(3))/3*Z/4,\nz=(-1/sqrt(3)*X-1/sqrt(3)*Y+1/sqrt(3)*Z/4),\nt=( 3+sqrt(3))/6*X+(-3+sqrt(3))/6*Y+(sqrt(3))/3*Z/4+(-3+sqrt(3))/6*X+(3+sq rt(\n3))/6*Y+(sqrt(3))/3*Z/4+(-1/sqrt(3)*X-1/sqrt(3)*Y+1/sqrt(3)*Z/4)+ 1,\n(x)^3 + (y)^3 + (z)^3 + (t)^3 - (x+y+z+t)^3);\n" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%\"GG,,*$),(*(\"\"'!\"\",&\"\"$\"\"\"*$F-#F.\"\"#F.F .%\"XGF.F.*(F*F+,&F-F+F/F.F.%\"YGF.F.*(\"#7F+F-F0%\"ZGF.F.F-F.F.*$),(* (F*F+F4F.F2F.F.*(F*F+F,F.F5F.F.*(F7F+F-F0F8F.F.F-F.F.*$),(*(F-F+F-F0F2 F.F+*(F-F+F-F0F5F.F+*(F7F+F-F0F8F.F.F-F.F.*$),2*(F*F+F,F.F2F.F.*(F*F+F 4F.F5F.F.*(\"\"%F+F-F0F8F.F.*(F*F+F4F.F2F.F.*(F*F+F,F.F5F.F.*(F-F+F-F0 F2F.F+*(F-F+F-F0F5F.F+F.F.F-F.F.*$),2*(F-F+F,F.F2F.F.*(F-F+F4F.F5F.F.* (F1F+F-F0F8F.F.*(F-F+F4F.F2F.F.*(F-F+F,F.F5F.F.**F1F.F-F+F-F0F2F.F+**F 1F.F-F+F-F0F5F.F+F.F.F-F.F+" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 249 182 "#Ferr\351ol\n# --------- original --------------\nA:=subs(\{X =(-3*x+y+z+t),Y=(x-3*y+z+t),Z=(x+y-3*z+t),T=2*(x+y+z-t)\},(X^3+Y^3+Z^3 +T^3-(X+Y+Z+T)^3)): B:=subs(t=1,A);B:=expand(B)/6;nops(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,,*$),**&\"\"$\"\"\"%\"xGF+!\"\"%\"yGF +%\"zGF+F+F+F*F+F+*$),*F,F+*&F*F+F.F+F-F/F+F+F+F*F+F+*$),*F,F+F.F+*&F* F+F/F+F-F+F+F*F+F+*$),**&\"\"#F+F,F+F+*&F%\"B G,J\"\"\"!\"\"**\"\"#F&%\"xGF&%\"yGF&%\"zGF&F'*(\"\"(F&F*F&)F+F)F&F&*& \"\"$F&F*F&F&*&F1F&F+F&F&*&F1F&F,F&F&*(F.F&F/F&F,F&F&*(F.F&)F*F)F&F,F& F&*&F1F&)F*F1F&F'*$F6F&F&*$F/F&F&*&F1F&)F,F1F&F'*$)F,F)F&F&*(\"#9F&F+F &F,F&F'*(F.F&F+F&F>F&F&*(F@F&F*F&F+F&F'*(F@F&F*F&F,F&F'*(F.F&F6F&F+F&F &*&F1F&)F+F1F&F'*(F.F&F*F&F>F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 479 "#Ferr\351ol pour r \351duire Todesco\n# --------- original --------------\nS:=288*(1/12*Z +1/3*3^(1/2)*Y+1/3*X)^3+288*(-1/3*3^(1/2)*Y+1/12*Z+1/3*X)^3+288*(-2/3* X+1/12*Z)^3+288*(1/3*T+1/4*Z)^3-288*(1/2*Z+1/3*T)^3;\nf[1]:=(x1,x2,x3, x4,x5)->(x1+x2)/2-x3;\nf[2]:=(x1,x2,x3,x4,x5)->(x1-x2)*sqrt(3)/2;\nf[3 ]:=(x1,x2,x3,x4,x5)->4*(x1+x2+x3);\nf[4]:=(x1,x2,x3,x4,x5)->3*(x4-x1-x 2-x3);\nsol:=solve(\{X=f[1](x1,x2,x3,x4,0),Y=f[2](x1,x2,x3,x4,0),Z=f[3 ](x1,x2,x3,x4,0),T=f[4](x1,x2,x3,x4,0)\},\{x1,x2,x3,x4\});" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,,*&\"$)G\"\"\"),(*&\"#7! \"\"%\"ZGF(F(*(\"\"$F-F0#F(\"\"#%\"YGF(F(*&F0F-%\"XGF(F(F0F(F(*&F'F(), (*(F0F-F0F1F3F(F-*&F,F-F.F(F(*&F0F-F5F(F(F0F(F(*&F'F(),&*(F2F(F0F-F5F( F-*&F,F-F.F(F(F0F(F(*&F'F(),&*&F0F-%\"TGF(F(*&\"\"%F-F.F(F(F0F(F(*&F'F (),&*&F2F-F.F(F(*&F0F-FEF(F(F0F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"fG6#\"\"\"f*6'%#x1G%#x2G%#x3G%#x4G%#x5G6\"6$%)operatorG%&arrowGF/ ,(*&#F'\"\"#F'9$F'F'*&F5F'9%F'F'9&!\"\"F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"#f*6'%#x1G%#x2G%#x3G%#x4G%#x5G6\"6$%)opera torG%&arrowGF/,$*&#\"\"\"F'F6*&,&9$F69%!\"\"F6-%%sqrtG6#\"\"$F6F6F6F/F /F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"$f*6'%#x1G%#x2G%#x 3G%#x4G%#x5G6\"6$%)operatorG%&arrowGF/,(*&\"\"%\"\"\"9$F6F6*&F5F69%F6F 6*&F5F69&F6F6F/F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6#\"\"%f *6'%#x1G%#x2G%#x3G%#x4G%#x5G6\"6$%)operatorG%&arrowGF/,**&\"\"$\"\"\"9 'F6F6*&F5F69$F6!\"\"*&F5F69%F6F:*&F5F69&F6F:F/F/F/" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$solG<&/%#x2G,(*(\"\"$!\"\"F*#\"\"\"\"\"#%\"YGF-F+* &\"#7F+%\"ZGF-F-*&F*F+%\"XGF-F-/%#x4G,&*&F*F+%\"TGF-F-*&\"\"%F+F2F-F-/ %#x3G,&*(F.F-F*F+F4F-F+*&F1F+F2F-F-/%#x1G,(*&F1F+F2F-F-*(F*F+F*F,F/F-F -*&F*F+F4F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "# rajout \351 par moi\nS1:=x1^3+x2^3+x3^3+x4^3-(x1+x2+x3+x4)^3;\nS_hom:=288*exp and(subs(sol,S1));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#S1G,,*$)%#x1G\"\"$\"\"\"F**$)%#x2GF)F*F**$)%#x3GF)F*F**$)%#x4GF)F*F* *$),*F(F*F-F*F0F*F3F*F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&S _homG,0*&\"#k\"\"\")%\"XG\"\"$F(!\"\"*&\"#JF()%\"ZGF+F(F,*(\"#[F(F0F() F*\"\"#F(F(*(\"$#>F()%\"YGF4F(F*F(F(*(\"#CF()%\"TGF4F(F0F(F,*(\"#aF(F< F()F0F4F(F,*(F2F(F0F(F7F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 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x,Y2=y,Z2=z,T2=1,subs(total,S)))*32/3;\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 84 "u:=2*Pi/3;S_rot:=simplify(subs(x=X*cos(u)-Y*sin(u),y=X*sin(u)+Y* cos(u),z=Z,S_simp));" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "War ning, the name arrow has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined an d unprotected\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,,*$)%\"xG\" \"$\"\"\"F**$)%\"yGF)F*F**$)%\"zGF)F*F**$)%\"tGF)F*F**$),*F(F*F-F*F0F* F3F*F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S_homoG,0*(\"$#>\" \"\")%\"YG\"\"#F(%\"XGF(F(*(\"#[F(F)F(%\"ZGF(F(*&\"#kF()F,\"\"$F(!\"\" *&\"#JF()F/F3F(F4*(F.F()F,F+F(F/F(F(*(\"#aF()F/F+F(%\"TGF(F4*(\"#CF(F/ F()F=F+F(F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'S_simpG,0*(\"#[\"\" \")%\"yG\"\"#F(%\"zGF(F(*&\"#CF(F,F(!\"\"*&\"#kF()%\"xG\"\"$F(F/*&\"#J F()F,F4F(F/*&\"#aF()F,F+F(F/*(F'F()F3F+F(F,F(F(*(\"$#>F(F)F(F3F(F(" }} {PARA 11 "" 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FFFC010000000000000408507FFFFFFFFFF400FFFFFFFFFFFFF400FFFFFFFFFFFFFC00 8E38E38E38E39407D31CCBAD1FF4B400FFFFFFFFFFFFF400FFFFFFFFFFFFFC001C71C7 1C71C72407294EA943129CA400FFFFFFFFFFFFF400FFFFFFFFFFFFFBFF555555555555 64063EBDA12F684BE400FFFFFFFFFFFFF400FFFFFFFFFFFFFBFDC71C71C71C72040488 B3CC0705F86400FFFFFFFFFFFFF400FFFFFFFFFFFFF3FDC71C71C71C718C042E09215C 5B4D4400FFFFFFFFFFFFF400FFFFFFFFFFFFF3FF5555555555554C05A57B425ED09794 00FFFFFFFFFFFFF400FFFFFFFFFFFFF4001C71C71C71C71C063BF2A7DB7A8E7400FFFF FFFFFFFFF400FFFFFFFFFFFFF4008E38E38E38E38C068F8EECAF953EC400FFFFFFFFFF FFF400FFFFFFFFFFFFF400FFFFFFFFFFFFFC06D5FFFFFFFFFFE-%+AXESLABELSG6%%\" xG%\"yGQ!F)" 1 2 0 1 10 0 2 1 1 1 2 1.000000 45.000000 45.000000 0 0 " Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sort(expand(subs(z=z+u, B)),[x,y,z],plex);" }{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#, bo*$)%\"xG\"\"$\"\"\"!\"\"**F'F(\"\"#F)F&F+%\"yGF(F(**F'F(F+F)F&F+%\"z GF(F(*&%#PiGF()F&F+F(F(*(F'F(F+F)F&F+F(**F'F(F+F)F&F(F,F+F(**\"\"'F(F& F(F,F(F.F(F)**\"\"%F(F0F(F&F(F,F(F)*(F5F(F&F(F,F(F)**F'F(F+F)F&F(F.F+F (**F+F(F0F(F&F(F.F(F(*(F5F(F&F(F.F(F)*(F'F(F+F)F&F(F)*(F7F(F0F(F&F(F)* *F+F(F'F)F0F+F&F(F(*$)F,F'F(F)**F'F(F+F)F,F+F.F(F(*(F'F(F+F)F,F+F(*&F0 F()F,F+F(F(**F'F(F+F)F,F(F.F+F(**F+F(F0F(F,F(F.F(F(*(F5F(F,F(F.F(F)*(F 'F(F+F)F,F(F)*(F7F(F0F(F,F(F)**F+F(F'F)F0F+F,F(F(*$)F.F'F(F)*(F+F(F0F( )F.F+F(F)*(F'F(F+F)F.F+F(**F7F(F'F)F0F+F.F(F)*(F+F(F0F(F.F(F(*(F'F(F+F )F.F(F)*(F+F(F'F)F0F+F(F0F)F(F)*(\"\")F(\"#FF)F0F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(-1-u^3-3*u-3*u^2,u);" }{TEXT -1 0 " " }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, (in solve) a constant is inval id as a variable, 2/3*Pi\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "s1:=sort(expand(subs(z=z-1,B)),[x,y,z],plex);nops(s1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G,D*$)%\"xG\"\"$\"\"\"!\"\"*&#F)\"\"#F* *&)F(F.F*%\"yGF*F*F**&F-F**&F0F*%\"zGF*F*F**&F-F**&F(F*)F1F.F*F*F***\" \"'F*F(F*F1F*F4F*F+*&F-F**&F(F*)F4F.F*F*F**(\"\"*F*F(F*F4F*F+*&F9F*F(F *F**$)F1F)F*F+*&F-F**&F7F*F4F*F*F**&F-F**&F1F*FF*F1F*F4F*F+ *&F9F*F1F*F**$)F4F)F*F+*&#F>F.F**$F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 249 223 "# \+ --------- modifi\351 --------------\nrestart:\nA:=subs(\{X=(-a*x+y+z+t ),Y=(x-a*y+z+t),Z=(x+y-a*z+t),T=b*(x+y+z-c*t)\},(X^3+Y^3+Z^3+T^3-(X+Y+ Z+T)^3)):\n B:=expand(subs(z=z+b,subs(t=1,a=1,b=1,c=3,A))/6);nops(B); \ncollect(B,[x,y,z]);" }}}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 249 198 "#cst:=subs(x=0,y=0,z=0,B);\n#sol:=solve(subs(x=0,y=0,z=0,B),b);\n sort(subs(b=-1,B),[x,y,z],plex);nops(%);\nsort(27*subs(b=-2/3,B),[x,y, z],plex);nops(%);\n#sort(subs(b=sqrt(5/3),B),[x,y,z],plex);nops(%);" } }}{EXCHG {PARA 236 "> " 0 "" {MPLTEXT 1 249 67 "#B:=collect(expand(sub s(z=z+f,subs(t=1,a=1,b=1,c=3,A))/6),[x,y,z]);" }}}{EXCHG {PARA 236 "> \+ " 0 "" {MPLTEXT 1 249 124 "with(plots):g:=10:\nplotsetup(inline); #wi ndow\nimplicitplot3d(B,x=-g..g,y=-g..g,z=-g..g,style=patchcontour,grid =[30,30,30]);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 242 "" 0 "" {TEXT 257 37 "Ecrire les cylinder \{...\} po ur Povray" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "for i in \{0,1 ,2\} do\nprintf(\"%s %a,%a,%a %s%a,%a,%a %s \\n\",\"cylinder \{<\",op( subs(z=-b,q=i,dr)),\">,<\",op(subs(z=b,q=i,dr)),\"> ,r \}\"); od:\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "restart:\n# ---- droites \+ donn\351es en param\351trique [a*t+a1,b*t+b1,c*t+c1]\n#----- o\371 a,a 1 ...,c1 sont des constantes\nDigits:=4;\ndr:=[]]: \nfor i to nops(dr) do\nprintf(\"/*%d*/ %s %a,%a,%a %s%a,%a,%a %s \\n\",i,\"cylinder\{< \",op(evalf(subs(t=-b,dr[i]),4)),\">,<\", op(evalf(subs(t=b,dr[i]),4)) ,\"> ,r \}\"); od:\n \n" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 298 "with(plots):\na:=0.8:\nimplicitplot3d(36*x*z+36*y*z+114*x*y*z+36* x*y-69*x^2*z-69*x^2*y-69*y^2*z-69*y*z^2-69*x*z^2-18*y^2-18*x^2-18*z^2- 69*x*y^2+37*x^3+37*y^3+37*z^3, x=-a..a,y=-a..a,z=-a..a,scaling=constra ined,grid=[20,20,20],style=patchnogrid,lightmodel=light2,orientation=[ -32,94],grid=[40,40,40]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 244 "" 0 "" {TEXT 260 1 " " }{TEXT 261 15 "Mail de Ferr\351ol" }} {EXCHG {PARA 234 "" 0 "" {TEXT 256 489 "la 1 \350re forme est 1/x + 1/ y +1/z + 6 / (1 - 3(x+y+z)) = 0\nce qui est bien une projection de 1/X + 1/ Y + 1/ Z +1 / T =0 (pourquoi le 6 et le 3 ???)\nEnsuite j'ai e ssay\351 de remplacer X, Y, Z, T dans cette derni\350re \351quation pa r les coordonn\351es t\351tra\351driques :\nhttp://mathworld.wolfram.c om/TetrahedralCoordinates.html\nfactor(subs(\{X=1-z-sqrt(2)*x,Y=1-z+sq rt(2)*x,Z=1+z+sqrt(2)*y,T=1+z-sqrt(2)*y\},1/X+1/Y+1/Z+1/T));\nEt on to mbe sur -1 + z x\262 - y\262 z + z \262 + y \262 + \+ x \262 = 0\n " }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 169 "SH:=n umer(1/X+1/Y+1/Z+1/T);\nS:=numer(subs(X=x,Y=y,Z=z,T=1-(x+y+z),SH));\nw ith(plots):\na:=2;\nimplicitplot3d(S,x=-a..a,y=-a..a,z=-a..a,scaling=c onstrained,grid=[30,30,30]);" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 187 "SH:=numer(1/X+1/Y+1/Z+1/T);\nS:=numer(normal(subs(X=x,Y=y,Z =z,T=1/6-(x+y+z)/2,SH)));\nwith(plots):\na:=1;\nimplicitplot3d(S,x=-a. .2*a,y=-a..2*a,z=-a..2*a,scaling=constrained,grid=[30,30,30]);" }}} {EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 176 "SH:=numer(1/X+1/Y+1/Z+1 /T);\nS:=numer(subs(X=x,Y=y,Z=z,T=1-2*(x+y+z),SH));\nwith(plots):a:=1: \nimplicitplot3d(S,x=-a..2*a,y=-a..2*a,z=-a..2*a,scaling=constrained,g rid=[30,30,30]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "SH:=nu mer(1/X+1/Y+1/Z+1/T);\nS:=numer(subs(X=x,Y=y,Z=z,T=3-(x+y+z),SH));\na: =3: \nimplicitplot3d(S,x=-a..2*a,y=-a..2*a,z=-a..2*a,scaling=constrain ed,grid=[30,30,30]);\n" }}}{EXCHG {PARA 234 "> " 0 "" {MPLTEXT 1 245 160 "SH:=numer(1/X+1/Y+1/Z+1/T);\nS:=numer(subs(X=x,Y=y,Z=z,T=2*x+2*y- z-1,SH));\na:=1.5:\nimplicitplot3d(S,x=-a..a,y=-a..a,z=-a..a,scaling=c onstrained,grid=[30,30,30]);" }}}{EXCHG {PARA 246 "> " 0 "" {MPLTEXT 1 263 153 "# chapeau !tt:=time():\nS:=x^2+y^2+z^3+z^2-3;\nwith(plots): \na:=4;\nimplicitplot3d(S,x=-a..a,y=-a..a,z=-a..a,scaling=constrained, grid=[30,30,30]);\ntime()-tt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 251 "" 0 "" {TEXT 268 1 " " }{TEXT 269 41 "surface de Cayley ave c tetra\350dre r\351gulier" }{TEXT 268 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 391 "solve(27*x*x-6*x-1,x);#ma=mb=mc=sqrt(2)/3\nps:=(u, v) -> add(u[i]*v[i],i=1..3):\nm:=expand(-1*[1,1,1]/9);\nn:=[1,1,1]:\na :=[1/3,0,0];b:=[0,1/3,0];c:=[0,0,1/3];\nprintf(\" v\351rification ma=m b ...=ab=bc : %a \",[ps(m-a,m-a),ps(m-b,m-b),ps(m-c,m-c),ps(b-a,b-a),p s(c-a,c-a),ps(b-c,b-c)]);\nL:=-ps(m-a,n)/sqrt(ps(n,n)):\nL1:=-ps([0,0, 0]-a,n)/sqrt(ps(n,n)):\nrapport:=L/L1;\n#h:=(a+b+c)/3; eq_base:=X+Y+Z- 1/3;\n" }}}{EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 254 8 "restart:" }}} {EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 254 421 "pc:=[x,y,z]:n:=[1,1,1]: \n# k=co tel que pc+k*n appartient \340 la base x+y+z-1/3=0\nco:=solve (x+y+z+3*k -1/3,k):\nproj:=expand(pc+co*n):#projection de m sur base\n m1:=proj+(pc-proj)/2;# affinit\351 de base \"la base\" de rapport 2\nP :=(1-3*X- 3*Y-3*Z)*(X*Y+X*Z+Z*Y)+6*X*Y*Z;# surface de Cayley usuelle. \ns1:=expand( subs(X=m1[1],Y=m1[2],Z=m1[3],P) ):\nS:=subs(X=x,Y=y,Z=z, 72*simplify(subs(x=X-1/9,y=Y-1/9,z=Z-1/9,s1) ));igcd(coeffs(S));" }}} {EXCHG {PARA 241 "> " 0 "" {MPLTEXT 1 254 142 "# Trac\351 de la surfac e de Cayley \"r\351guli\350re\"\nwith(plots):\nd:=1:\nimplicitplot3d(s 1,x=-d..d,y=-d..d,z=-d..d,scaling=constrained,grid=[40,40,40]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "P:=s1:\ndpx:=diff(P,x);dpy: =diff(P,y);dpz:=diff(P,z);\nsol:=[solve(\{dpx,dpy,dpz\},\{x,y,z\})];\n #seq( subs(op(sol[i]),[x,y,z]),i=1..nops(sol));\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 237 "" 0 "" {TEXT 247 56 " Changement d'axes pou r Cayley \"avec t\351tra\350dre r\351gulier\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 475 "#changement d'axes\nwith(linalg):\nK1:=expand([ 1,1,1]/sqrt(3)):\nI1:= expand([2,-1,-1]/sqrt(6)):#expand([1,0,-1]/sqrt (2)):\nJ1:=convert(crossprod(K1,I1),list):\nA:=transpose(matrix(3,3,[o p(I1),op(J1),op(K1)]));\nnewcoord:=evalm(A&*[x,y,z]):\nS:=subs(x=X,y=Y ,z=Z,36*x*z+36*y*z+114*x*y*z+36*x*y-69*x^2*z-69*x^2*y-69*y^2*z-69*y*z^ 2-69*x*z^2-18*y^2-18*x^2-18*z^2-69*x*y^2+37*x^3+37*y^3+37*z^3):# eq de cayley \"r\351guli\350re\"\nS1:=expand(subs(X=newcoord[1],Y=newcoord[ 2],Z=newcoord[3],S)/3);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "restart:\nS1 :=114*x*y*z-69*y*z^2-69*x^2*y-69*x*z^2-69*x*y^2-69*y ^2*z-18*y^2-18*x^2+36*y*z+36*x*z+37*z^3+37*y^3+36*x*y+37*x^3-69*x^2*z- 18*z^2;\nwith(plots):\na:=1;b:=1:\nimplicitplot3d(S1,x=-a..b,y=-a..b,z =-a..b,scaling=constrained,grid=[30,30,30]);\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 " Passages homographiques en tre deux cubiques" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "On aboutit \+ \340 un syst\350me de 20 \351quations de degr\351 inf\351rieur ou \351 gal \340 3 et 16 inconnues ... \347a n'aboutit \351videmment pas touj ours !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1692 "restart:\n#==== ========= mettre en eq1 = \351quation la plus simple ===== \npassage: = proc(eq1,eq2) # equations homog\350nes en (X,Y,Z,T)\n local x,y,a, i, tp1,tp2,tp11,passe,sys,sol,passefin:\n tp1:=numer(normal(expand(s ubs(X=x1,Y=x2,Z=x3,T=x4,eq1))));\n tp2:=numer(normal(eq2));\n pass efin :=\{X= a11*X1+a12*Y1+a13*Z1+a14*T1,Y=a21*X1+a22*Y1+a23*Z1+a24*T1, \n Z=a31*X1+a32*Y1+a33*Z1+a34*T1,T=a41*X1+a42*Y1+a43*Z1+ a44*T1\};\n passe:= seq(x||i=add((a||i||j)*y||j,j=1..4),i=1..4);\n \+ tp11:=subs(y1=X,y2=Y,y3=Z,y4=T,simplify(expand(subs(passe,tp1)))-tp2) :\n tp11:=sort(tp11,[X,Y,Z,T],plex):\n sys:=\{coeff(tp11,X^3),coef f(tp11,Y^3),coeff(tp11,Z^3),coeff(tp11,T^3),#X^3..,T^3 \n \+ coeff(coeff(tp11,X^2),Y),coeff(coeff(tp11,X^2),Z),coeff(coeff(tp11 ,X^2),T),#X^2Y ..\n coeff(coeff(tp11,Y^2),X), coeff(coeff(tp11 ,Y^2),Z), coeff(coeff(tp11,Y^2),T),\n coeff(coeff(tp11,Z^2),X) , coeff(coeff(tp11,Z^2),Z), coeff(coeff(tp11,Z^2),T),\n coeff( coeff(tp11,T^2),X), coeff(coeff(tp11,T^2),Z), coeff(coeff(tp11,T^2),Y) ,\n subs(Z=0,T=0,coeff(coeff(tp11,X),Y)), subs(Y=0,T=0,coeff(c oeff(tp11,X),Z)),#XY..ZT\n subs(Z=0,Y=0, coeff(coeff(tp11,X),T )),subs(X=0,Y=0, coeff(coeff(tp11,Z),T)),\n subs(X=0,T=0, coef f(coeff(tp11,Y),Z)), subs(X=0,Z=0,coeff(coeff(tp11,Y),T)),\n co eff(coeff(coeff(tp11,Y),Z),X),coeff(coeff(coeff(tp11,Y),Z),T),\n \+ coeff(coeff(coeff(tp11,Y),T),X),coeff(coeff(coeff(tp11,T),Z),X)\n \+ \}; \n sol:=[solve(sys,\{seq(seq(a||i||j,j=1..4),i=1..4)\})]: \n if sol=[] then print(\"pas de solution\")\n else prin tf( \"Nombre de solutions = %d\\n\",nops(sol)):\n map( u - >subs(u,passefin),sol): \n fi;\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "plucker:=Z*(X^2+Y^2)-2*X*Y*T;\nzindler:=Z*(X^2-Y^2) -2*X*Y*T;" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "regCayl ey:=2*Z*X^2+4*Z*X+2*Z+Y^3+Y*X^2-Y;\nEqPara:=-X*Z^2+X*T^2+Z*T^2-Z*Y*T; " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "cubique :=-2*z*x^2-x*z^2-z+x*y-x^3+z*y;\ncubhom:=numer(normal(subs(x=X/T,y=Y/T ,z=Z/T,cubique)));\nZindlertocub:=passage(cubhom,EqPara);\n#cub7toZind ler:=allvalues(map( u->solve(u,\{X1,Y1,Z1,T1\}),Zindlertocub));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "subs(a21=1 ,a24=1,\{X1 = -1/2*(X-Y)/a21, T1 = 1/2*(Y+X)/a24, Y1 = 2*a24^2*(Z-T), \+ Z1 = 1/2/a24*(4*a24*a21^2*Z+4*a24*a21^2*T+Y+X)\});\nsimplify(subs(\{Y1 = 2*Z-2*T, X1 = -1/2*X+1/2*Y, Z1 = 2*Z+2*T+1/2*Y+1/2*X, T1 = 1/2*Y+1/ 2*X\},Y1*T1^2+X1^2*Z1-X1^2*T1));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "x77:=expand(y^3-2*y^2*z+y*(z^2+x^2-4)-2*x*z*(x+2)):\nX77:=expand(subs (x=X/T,y=Y/T,z=Z/T,x77)*T^3);\nwhitney:=Z*X^2 -Y^2*T;\nplucker:=Z*(X^2 +Y^2)-2*X*Y*T;\nzindler:=Z*(X^2-Y^2)-2*X*Y*T;\nZindlertoX77:=passage(z indler,X77);\nX77toZindler:=map( u->solve(u,\{X1,Y1,Z1,T1\}),Zindlerto X77);\n#passage(whitney,plucker);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 " matrices po ur LGV" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 354 "ordre:= proc(u,va rs)\n local i, v:\n v:=[0,0,0,0]:\n for i from 1 to 4 do if lhs (u[i])=vars[1] then v[1]:=rhs(u[i]): \n elif l hs(u[i])=vars[2] then v[2]:=rhs(u[i]):\n elif \+ lhs(u[i])=vars[3] then v[3]:=rhs(u[i]):\n eli f lhs(u[i])=vars[4] then v[4]:=rhs(u[i]):\n fi; od:\n v;\nend:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "listecoef:=proc(exp,vars)\n [seq(c oeff(exp,vars[i]),i=1..nops(vars))]\nend:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 304 "depart:=ZindlertoX77:\n# si tu pars de ZindlertoX7 7, tu \351changes varia et varia1\nvaria:=[X1,Y1,Z1,T1]:\nvaria1:=[X,Y ,Z,T]:\nliste:=convert(map(convert,depart,list),list):\nordonne:=map(u -> ordre(u,varia1),liste):\nfor i from 1 to nops(liste) do \nB[i]:=ma trix(4,4,map(u -> listecoef(u,varia),ordonne[i]));\nod;" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "dep art:=X77toZindler:\n# si tu pars de ZindlertoX77, tu \351changes varia et varia1\nvaria1:=[X1,Y1,Z1,T1]:\nvaria:=[X,Y,Z,T]:\nliste:=convert( map(convert,depart,list),list):\nordonne:=map(u -> ordre(u,varia1),lis te):\nfor i from 1 to nops(liste) do \nA[i]:=matrix(4,4,map(u -> liste coef(u,varia),ordonne[i]));\nod;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "seq(simplify(evalm(linalg[multiply](A[i],B[i]) )),i=1..4);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 " V\351rification avec des cas particuli ers" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "#========= Whitney \+ --> Plucker ==================\np1:=[X1=X+Y, Y1=-X+Y, Z1=1/2*Z-1/2*T, T1=-1/2*Z-1/2*T];\np2:=[X1=-X+Y, Y1=X+Y, Z1=1/2*Z+1/2*T, T1=-1/2*Z+1/ 2*T];\nsimplify(subs(p1, subs(X=X1, Y=Y1, Z=Z1, T=T1, whitney)));\nsim plify(subs(p2, subs(X=X1, Y=Y1, Z=Z1, T=T1, whitney)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "#======== Mobius(x77) ---> zindle r ==============\np4:=\{X=-2*Z1, Z=-1/2*Y1-2*T1-1/2*X1, T=-1/4*X1+Z1+ 1/4*Y1, Y=-2*T1\};\nsimplify(subs(X1=X, Y1=Y, Z1=Z, T1=T, subs(op(p4), X77)));\np5:=\{X=-2*Z1, Y=2*T1, Z=1/2*X1+2*T1-1/2*Y1, T=1/4*Y1+Z1+1/4 *X1\};\nsimplify(subs(X1=X, Y1=Y, Z1=Z, T1=T, subs(op(p5), X77)));\np6 :=\{X1=-T, Y1=-Z, Z1=Y-Z, T1=-1/2*X+1/2*T\};\nsimplify(subs(op(p6), su bs(X=X1, Y=Y1, Z=Z1, T=T1, X77)));\np7:=\{X1=T, Y1=-Z, Z1=Y-Z, T1=1/2* X-1/2*T\};\nsimplify(subs(op(p7), subs(X=X1, Y=Y1, Z=Z1, T=T1, X77))); \np8:=\{X1=-2*Z, Y1=-2*T, Z1=-1/2*Y-2*T-1/2*X, T1=Z-1/4*X+1/4*Y\};\nsi mplify(subs(op(p8), subs(X=X1, Y=Y1, Z=Z1, T=T1, X77)));" }}{PARA 11 " " 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 " Solutions complexes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "# ========== que des solutions com plexes ========\nsol:=allvalues(passage(zindler,plucker)):\nrespart:=m ap(u->subs(a11=1,a12=1,a22=1,a21=1,u),sol);\nsimplify(subs(X1=X, Y1=Y, Z1=Z, T1=T, subs(op(respart[1]), zindler)));" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 84 "simplify(subs(X1=X, Y1=Y, Z1=Z, T1=T, subs(Z=- Z1, T=-I*T1, Y=Y1, X=I*X1, zindler)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "# on retrouve les m\352mes ============\np3:=subs(a2 1=0,a22=-1,X77toZindler[2]);# 1 ou 2\np4:=subs(a23=1,X77toZindler[4]); # 3 ou 4" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 " Plucker : trac\351 amusant " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "pluk:=z*(x^2+y^2)-2*x*y; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cyly:=x^2+y^2-2*k*y:\ncylx:=x^2 +y^2-2*k*x:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#factor(simplify(plu k-z*cylx));factor(simplify(pluk-z*cyly));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ellx:=[k+k*cos(t),k*sin(t),sin(t)]:\nelly:=[k*cos(t), k+k*sin(t),cos(t)]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%plukG,&*&%\" zG\"\"\",&*$)%\"xG\"\"#F(F(*$)%\"yGF-F(F(F(F(*(F-F(F,F(F0F(!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "#x:=k+k*cos(t); y:=k*sin(t); " }{TEXT -1 0 "" }{MPLTEXT 1 0 97 "#simplify(cyl);\n#z:=simplify(2*x*y /(x^2+y^2));\npluk_cyl:=[r,t,sin(2*t)]:\nwith(plots):\nn:=10:d:=6:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "g_ellx:=plot3d(ellx,k=-2.. 2,t=0..2*Pi,color=COLOR(RGB,0.5,0.8,15)):\ng_elly:=plot3d(elly,k=-2..2 ,t=0..2*Pi,color=COLOR(RGB,1,1,0.5)):\n# ========== une mani\350re amu sante de tracer Plcker =======\ndisplay([g_ellx,g_elly],scaling=constr ained,orientation=[-20,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 486 486 486 {PLOTDATA 3 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[r*F=$\"+'H!f%f%F:Fde^l7%$\"+]?mGoF=$\"+[G!3q%F:F[f^l7%$\"+TuRIQF=$\"+ \"GO#pZF:Fbf^l7%$\"+9\"yBp(F`v$\"+>pw)z%F:Fif^l7%$!+#o_XI#F=$\"+q)4*)y %F:F`g^l7%$!+%[-0M&F=$\"+))p#)RZF:Fgg^l7%$!+#Qh()G)F=$\"+5UK_YF:F^h^l7 %$!+0>456F:$\"+J$Qy_%F:Feh^l7%$!+$**zIP\"F:$\"+0MToVF:F\\i^l7%$!+X@_8; F:$\"+&=nm<%F:Fci^l7%$!+3.ZF=F:$\"+R\"[d&RF:Fji^l7%$!+f9T6?F:$\"+_PG4P F:Faj^l7%$!+*GDB;#F:$\"+h4KTMF:Fhj^l7%$!+)zLxF#F:$\"+4(fi:$F:F_[_l7%$! +!)>ubBF:$\"+$p!yeGF:Ff[_l7%$!+W*o]R#F:$\"+6&oPb#F:F]\\_l7%$!+T*o]R#F: $\"+a9BYAF:Fd\\_l7%$!+t>ubBF:$\"+s#>7%>F:F[]_l7%$!+'yLxF#F:$\"+d-uV;F: Fb]_l7%$!+u_Ki@F:$\"+2!z'e8F:Fi]_l7%$!+S9T6?F:$\"+=ir!4\"F:F`^_l7%$!+& Gqu#=F:$\"+M$=DW)F=Fg^_l7%$!+>@_8;F:$\"+1zKLiF=F^__l7%$!+j*zIP\"F:$\"+ Wd'eJ%F=Fe__l7%$!+t=456F:$\"+Dlh@FF=F\\`_l7%$!+[5w)G)F=$\"+wxvw9F=Fc`_ l7%$!+P@]S`F=$\"+U/I;(>**F`vFfb_l7%$\"+U0![r*F=$\"+%=(4a?F=F]c_l7%$\"+%>UTC\"F:$\"+nQ dwMF=Fdc_l7%$\"+_bP'\\\"F:$\"+VY/O_F=F[d_l7%$\"+m%QSs\"F:$\"+\\!>OI(F= Fbd_l7%$\"+:FRB>F:$\"+qvMX'*F=Fid_l7%$\"+4\\;\"4#F:$\"+4zFA7F:F`e_l7%$ \"+L-gCAF:$\"+D#f$*\\\"F:Fge_l7%$\"+#o2:K#F:$\"+Q!H7z\"F:F^f_l7%$\"+Rg H!Q#F:$\"+`[4$4#F:Fef_l7%F\\[x$\"+V+++CF:F\\g_lF^g_lFbbl-F/6%7T7%$\"++ +++FF:F`\\`mFfa^l7%$\"+PI$yn#F:$\"+P$o_/$F:F\\b^l7%$\"+Ihp6EF:$\"+xt'[ Q$F:Fcb^l7%$\"+X_n-DF:$\"+84A8PF:Fjb^l7%$\"+,0c_BF:$\"+\"RP\\-%F:Fac^l 7%$\"+zn\"Q;#F:$\"+L%)*[J%F:Fhc^l7%$\"+XKaR>F:$\"+()GMyXF:F_d^l7%$\"+l CU$o\"F:$\"+.]%4\"[F:Ffd^l7%$\"+M*f'*R\"F:$\"+ha))3]F:F]e^l7%$\"+C]\"H 4\"F:$\"+$39*o^F:Fde^l7%$\"+2[C#o(F=$\"+/KS)G&F:F[f^l7%$\"+@r>4VF=$\"+ ;3Rl`F:Fbf^l7%$\"+y`#Rl)F`v$\"+%y7')R&F:Fif^l7%$!+n4VF=$\"+X!>4Y$F` vF_b_l7%$\"+!>XAo(F=$\"+s!of6\"F=Ffb_l7%$\"+h]\"H4\"F:$\"+K$f3J#F=F]c_ l7%$\"+o*f'*R\"F:$\"+,c96RF=Fdc_l7%$\"+'\\AMo\"F:$\"+B-b!*eF=F[d_l7%$ \"+uKaR>F:$\"+I9d;#)F=Fbd_l7%$\"+/o\"Q;#F:$\"+-;5&3\"F:Fid_l7%$\"+A0c_ BF:$\"+[E1v8F:F`e_l7%$\"+i_n-DF:$\"+G\"zno\"F:Fge_l7%$\"+Uhp6EF:$\"+oE 8:?F:F^f_l7%$\"+VI$yn#F:$\"+5 " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 18 "" 0 "" {TEXT -1 42 "Cubiques r\351gl\351es g\351n \351r\351es par des droites " }}{PARA 18 "" 0 "" {TEXT -1 40 "s'appuya nt sur une droite et une conique" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 " Droite des \+ points doubles" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 396 "Principe : soit un point double M de la surface (S) d\"\351quation f(x,y,z) = 0 ; les deux nappes de (S) passant par M ne sont en g\351n\351ral pas tangent es en M donc en ce point, le vecteur normal N : (df/dx, df/dy, df/dz) \+ = 0. Les points de la droite double sont donc \340 rechercher parmi l es points non isol\351s v\351rifiant N = 0. (On peut \351ventuellement obtenir des points d'une ar\352te de rebroussement ?? )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 502 "restart;\ncub1:=-1-x+x^2+x^3+x^2*z +z+2*x*z+y^2*x+y^2-y^2*z:#Droite parall\350le au plan du cercle\ncub2: =2*z*x^2+y*x^2+2*z*x+y*z^2+y^3-y+2*z*y^2 :#cercle c=0\ncub3:=2*z*x^2+4 *z*x+2*z+y^3+y*x^2-y :#cercle c=1\ncub4:=2*z*x^2+x^2*y+6*z*x+y*z^2-y+4 *z+y^3-2*z*y^2 :#cercle c=2\ncub5:= y+x^2*z-x^2:#Droite parall\350le a u plan de la parabole\ncub6:=-2*z*x^2-x*z^2-z+x*y-x^3+z*y :#parabole : droite int\351rieure\ncub7:= -2*z*x^2-x*z^2+z+x*y-x^3+z*y:#parabole : droite ext\351rieure\ncub8:=x-y*z^2-2*y*z*x-y*x^2:# hyperbole et Oz" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow h as been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the prote cted names norm and trace have been redefined and unprotected\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 360 "double:= proc(cubique)\n \+ local n,sol,drd,flag,i,droitedouble,u: \n n:=[diff(cubique,x),di ff(cubique,y),diff(cubique,z)]:\n sol:=[solve(\{op(n)\},\{x,y,z\}) ];\n drd:=\{\}:\n for i from 1 to nops(sol) do u:=sol[i]: \n \+ flag:=(lhs(u[1])=rhs(u[1]))or (lhs(u[2])=rhs(u[2])) or (lhs(u [3])=rhs(u[3])):\n if flag then drd:=sol[i] fi;\n od:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 310 " if drd <>\{\} then droitedou ble:=map(u ->lhs(u)-rhs(u)=0,drd) minus \{0=0\} : \n printf(\" Cubique : %a = 0\\n Droite des points doubles : %a\\n\",cubique,droite double);\n droitedouble;\n else\n printf(\"Cubi que : %a = 0\\n Pas de points doubles\\n\\n\",cubique):\n \{ \};\n fi:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 414 "bonforme:=proc(cubique)\n \+ local sol,vari,varsreste,dr1,dr2,sys,res,neweq;\nsol:=double(cubique );\nif sol=\{\} then print(\"pas de droite de points doubles\")\n el se\n if not(has(sol,x)) then vari:=x:\n elif not(has(sol,y))t hen vari:=y:\n elif not(has(sol,z))then vari:=z: fi;\n varsre ste:=\{x,y,z\} minus \{vari\}:\n dr1:=lhs(sol[1]): dr2:=lhs(sol[2]): \n sys:=\{A-dr1, B-dr2\}:\n res:=solve(sys,varsreste);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 198 " neweq:=subs(A=dr1,B=dr2,collect(expand( subs(res,cubique)),[A,B]));\n printf(\"Equation A*P(x,y,z)+B*Q(x,y, z) =0 avec droite double \{A=0,B=0\}\\n Equation : %a =0\\n\\n\",neweq );\n neweq;\n fi:\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "bonforme(cub1):bonforme(cub2):bonforme(cub3):bonforme(cub4):bonfor me(cub5):" }{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 56 "Cubique : -1- x+x^2+x^3+x^2*z+z+2*x*z+y^2*x+y^2-y^2*z = 0" }}{PARA 6 "" 1 "" {TEXT -1 45 " Droite des points doubles : \{y = 0, x+1 = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 63 "Equation A*P(x,y,z)+B*Q(x,y,z) =0 avec droite doubl e \{A=0,B=0\}" }}{PARA 6 "" 1 "" {TEXT -1 49 " Equation : (x+1-z)*y^2+ (x+1)^3+(-2+z)*(x+1)^2 =0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 " " 1 "" {TEXT -1 53 "Cubique : 2*x^2*z+y*x^2+2*x*z+y*z^2+y^3-y+2*y^2*z \+ = 0" }}{PARA 6 "" 1 "" {TEXT -1 47 " Droite des points doubles : \{y+z = 0, x+1 = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 63 "Equation A*P(x,y,z)+B *Q(x,y,z) =0 avec droite double \{A=0,B=0\}" }}{PARA 6 "" 1 "" {TEXT -1 64 " Equation : (y+z)^3-(y+z)^2*z+((x+1)^2-2*x-2)*(y+z)+z*(x+1)^2 = 0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 "Cubi que : 2*x^2*z+4*x*z+2*z+y^3+y*x^2-y = 0" }}{PARA 6 "" 1 "" {TEXT -1 45 " Droite des points doubles : \{x+1 = 0, y = 0\}" }}{PARA 6 "" 1 " " {TEXT -1 63 "Equation A*P(x,y,z)+B*Q(x,y,z) =0 avec droite double \+ \{A=0,B=0\}" }}{PARA 6 "" 1 "" {TEXT -1 44 " Equation : (2*z+y)*(x+1)^ 2-2*(x+1)*y+y^3 =0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 57 "Cubique : 2*x^2*z+y*x^2+6*x*z+y*z^2-y+4*z+y^3-2*y^2*z = 0 " }}{PARA 6 "" 1 "" {TEXT -1 47 " Droite des points doubles : \{x+1 = \+ 0, y-z = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 63 "Equation A*P(x,y,z)+B*Q( x,y,z) =0 avec droite double \{A=0,B=0\}" }}{PARA 6 "" 1 "" {TEXT -1 62 " Equation : (2*z+y)*(x+1)^2-2*(x+1)*(y-z)+z*(y-z)^2+(y-z)^3 =0" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 25 "Cubique : y+x^2*z-x^2 = 0" }}{PARA 6 "" 1 "" {TEXT -1 22 " Pas de points double s" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q @pas~de~droite~de~points~doubles6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "X77:=expand(y^3-2*y^2*z+y*(z^2+x^2-4)-2*x*z*(x+2));\n bonforme(X77):" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$X7 7G,0*$)%\"yG\"\"$\"\"\"F**(\"\"#F*)F(F,F*%\"zGF*!\"\"*&F(F*)F.F,F*F**& F(F*)%\"xGF,F*F**&\"\"%F*F(F*F/*(F,F*F3F*F.F*F/*(F6F*F4F*F.F*F/" }} {PARA 6 "" 1 "" {TEXT -1 55 "Cubique : y^3-2*y^2*z+y*z^2+y*x^2-4*y-2*x ^2*z-4*x*z = 0" }}{PARA 6 "" 1 "" {TEXT -1 47 " Droite des points doub les : \{x+2 = 0, y-z = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 63 "Equation A *P(x,y,z)+B*Q(x,y,z) =0 avec droite double \{A=0,B=0\}" }}{PARA 6 "" 1 "" {TEXT -1 62 " Equation : (y-2*z)*(x+2)^2-4*(x+2)*(y-z)+z*(y-z)^2+ (y-z)^3 =0" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "for i from 1 to 8 do double(cub||i): printf(\"\\n\" ): od:" }{TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 56 "Cubique : -1-x+x ^2+x^3+x^2*z+z+2*x*z+y^2*x+y^2-y^2*z = 0" }}{PARA 6 "" 1 "" {TEXT -1 45 " Droite des points doubles : \{x+1 = 0, y = 0\}" }}{PARA 6 "" 1 " " {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 53 "Cubique : 2*x^2*z+y*x^2 +2*x*z+y*z^2+y^3-y+2*y^2*z = 0" }}{PARA 6 "" 1 "" {TEXT -1 47 " Droite des points doubles : \{y+z = 0, x+1 = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 "Cubique : 2*x^2*z+4*x*z+2*z+y^3 +y*x^2-y = 0" }}{PARA 6 "" 1 "" {TEXT -1 45 " Droite des points double s : \{x+1 = 0, y = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 57 "Cubique : 2*x^2*z+y*x^2+6*x*z+y*z^2-y+4*z+y^3-2*y^2* z = 0" }}{PARA 6 "" 1 "" {TEXT -1 47 " Droite des points doubles : \{x +1 = 0, y-z = 0\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 25 "Cubique : y+x^2*z-x^2 = 0" }}{PARA 6 "" 1 "" {TEXT -1 22 " Pas de points doubles" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 "Cubique : -2*x^2*z-x *z^2-z+x*y-x^3+z*y = 0" }}{PARA 6 "" 1 "" {TEXT -1 22 " Pas de points \+ doubles" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 42 "Cubique : -2*x^2*z-x*z^2+z+x*y-x^3+z *y = 0" }}{PARA 6 "" 1 "" {TEXT -1 22 " Pas de points doubles" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 35 "Cubique : x-y*z^2-2*y*z*x-y*x^2 = 0" }}{PARA 6 "" 1 "" {TEXT -1 22 " Pas de points doubles" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 52 " Base : une \+ ellipse ( et cas particulier du cercle )" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 " Calculs g\351n\351raux" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart:\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "a:=1:b:=1:c:=2:\nprintf(\"\\#[a = \+ %a,b = %a, c = %a]\",a,b,c);\nbase:=[a*cos(2*t),b*sin(2*t),0];\ndroite :=[0,0,tan(t)];\nM:=expand(base+u*(droite-base));\nsol:=[eliminate(sys ,\{u,t\})]:allvalues(sol[1]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 107 "sol:=simplif y(\{allvalues(solve(sys[1..2],\{u,t\}))\}):\nres:=map(w -> map(numer,s implify(subs(w,sys[3]))),sol);" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "eqA:=subs(sqrt(x^2*(y^2+x^2))=A,res);# remplacer par le radical\nAA:=map(solve,\{op(eqA)\},A);\nmap( u ->factor(numer(norm al( u^2-x^2*(y^2+x^2)))),AA);" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 " Droite parall\350le au plan du cercle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "cubique:=-1-x+ x^2+x^3+x^2*z+z+2*x*z+y^2*x+y^2-y^2*z;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 121 "#[a = 1,b = 1, c = 0];\nbase := [c os(2*t), sin(2*t), 0];\ndroite := [ 0, tan(t),1];\nps:=(U,V)->add(U[i] *V[i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqr t(ps(droite-base,droite-base))):" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 667 "a:=4:n:=10:\n#gbase:=spacecurve(base,t=-Pi/2..Pi/2,t hickness=3,color=red,numpoints=80):\ngbase:=tubeplot(base,t=-Pi/2..Pi/ 2,radius=0.06,color=red,grid=[30,6],style=wireframe):\n#gdroite:=space curve(droite,t=-Pi/2..Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0,numpoin ts=2)):\ngdroite:=tubeplot(droite,t=-Pi/2..Pi/2,radius=0.06,color=COLO R(RGB,0,0.6,0),grid=[30,6],style=wireframe):\ngdouble:=spacecurve([[-1 ,0,-a],[-1,0,3/2*a]],thickness=3,color=COLOR(RGB,1,0.8,0),numpoints=2) :\nwith(plots):\n\ncub:=implicitplot3d(cubique,x=-1.25*a..a,y=-a..a,z= -a/2..3*a/2,grid=[30,30,30],style=patchnogrid):\ngene:=seq(spacecurve( subs(t=i*Pi/n/2,M),u=-2*a..2*a,color=blue,numpoints=2),i=-n..n):" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "display([c ub,gene,gdroite,gdouble,gbase],scaling=constrained,view=[-1.25*a..a,-a ..a,-a/2..3*a/2],orientation=[120,60],axes=boxed);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 675 "# animation\na:=3.5:n:=20:\nmobile:=display([seq(d isplay([spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a..2*a,color=blue,numpo ints=2,thickness=3),\n plot3d(M,t=-Pi/2..-Pi/2+0.0000 1+i*Pi/n,u=-a/2..3*a/2,style=wireframe,grid=[3*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+0.00 1+i*Pi/n,droite)\},\n symbol=circle,symbolsize= 8+4*j,color=blue),j=0..2)\n ]),i=0..n)], insequence=true):\n#p lotsetup(gif,plotoutput=\"c:/cercle_dr_paralleles.gif\",plotoptions=\" width=500, height=500\");\ndisplay([gdroite,gbase,gdouble,mobile],scal ing=constrained,orientation=[-150,60],view=[-a..a,-1.6*a..1.6*a,-a/2.. a*1.5]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 26 " Cas particulier du cercle" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 " cas cercle c=0" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "\ncubique:=2*z*x^2+y*x^2+2*z*x+y*z^2+y^3-y+2*z*y^2;\nchange:=\{Y \+ = -Z1, X = -T1, T = X1+T1, Z = Y1+Z1\}:\nprintf(\"Passage \340 Zindler par :\\n %a\",change);\ncubhom:=numer(normal(subs(x=X/T,y=Y/T,z=Z/T,c ubique)));\nprintf(\" V\351rification : %a -- passage --> %a \\n\",cub hom,simplify(subs(change,cubhom)));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {MPLTEXT 1 0 121 "#[a = 1,b = 1, c = 0];\nbase := [cos(2*t), sin(2*t), 0];\ndroite := [0, 0, tan(t)];\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U) ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqrt(ps(droite-base,d roite-base))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 652 "with(plots):\na: =4:n:=10:\n#gbase:=spacecurve(base,t=-Pi/2..Pi/2,thickness=3,color=red ,numpoints=80):\ngbase:=tubeplot(base,t=-Pi/2..Pi/2,radius=0.06,color= red,grid=[30,6],style=wireframe):\n#gdroite:=spacecurve(droite,t=-Pi/2 ..Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0,numpoints=2)):\ngdroite:=tu beplot(droite,t=-Pi/2..Pi/2,radius=0.06,color=COLOR(RGB,0,0.6,0),grid= [30,6],style=wireframe):\ngdouble:=spacecurve([[-1,-a,a],[-1,a,-a]],th ickness=3,color=COLOR(RGB,1,0.8,0),numpoints=2):\ncub:=implicitplot3d( cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style=patchnogrid):\ng ene:=seq(spacecurve(subs(t=i*Pi/n/2,M),u=-2*a..2*a,color=blue,numpoint s=2),i=-n..n):" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "display([cub,gene,gdouble,gdroite,gbase],scaling=constrained,vi ew=[-a..a,-a..a,-a..a],orientation=[20,70]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 656 "# animation\na:=2:n:=20:\nmobile:=display([seq(dis play([spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a..2*a,color=blue,numpoin ts=2,thickness=3),\n plot3d(M,t=-Pi/2..-Pi/2+0.00001+ i*Pi/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) ,seq(p ointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+0.001+i*Pi/n ,droite)\},\n symbol=circle,symbolsize=8+4*j,co lor=blue),j=0..2)\n ]),i=0..n)], insequence=true):\n#plotsetup (gif,plotoutput=\"c:/cercle_c0.gif\",plotoptions=\"width=500, height=5 00\");\ndisplay([gdroite,gdouble,gbase,mobile],scaling=constrained,ori entation=[-45,60],view=[-1.*a..1.5*a,-1.3*a..1.3*a,-a..a]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 " cas cercle c=1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "re start:\ncubique:=2*z*x^2+4*z*x+2*z+y^3+y*x^2-y;# type RegCayley\n" } {TEXT -1 0 "" }{MPLTEXT 1 0 122 "#[a = 1,b = 1, c = 1];\nbase := [cos( 2*t), sin(2*t), 0];\ndroite := [-1, 0, tan(t)];\nps:=(U,V)->add(U[i]*V [i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqrt(ps(d roite-base,droite-base))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 655 "with (plots):\na:=2:n:=10:\n#gbase:=spacecurve(base,t=-Pi/2..Pi/2,thickness =3,color=red,numpoints=80):\ngbase:=tubeplot(base,t=-Pi/2..Pi/2,radius =0.06,color=red,grid=[30,6],style=wireframe):\n#gdroite:=spacecurve(dr oite,t=-Pi/2..Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0,numpoints=2)): \ngdroite:=tubeplot(droite,t=-Pi/2..Pi/2,radius=0.06,color=COLOR(RGB,0 ,0.6,0),grid=[30,6],style=wireframe):\ngdouble:=spacecurve([[-1,0,a+0. 1],[-1,0,-a]],thickness=3,color=COLOR(RGB,1,0.8,0),numpoints=2):\ncub: =implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style= patchnogrid):\ngene:=seq(spacecurve(subs(t=i*Pi/n/2,M),u=-2*a..2*a,col or=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "display([cub,gene,gdouble,gdroite,gbase],scaling=constrained, view=[-a..a,-a..a,-a..a],orientation=[20,70]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 651 "# animation\nn:=20:\nmobile:=display([seq(display( [spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a..2*a,color=blue,numpoints=2, thickness=3),\n plot3d(M,t=-Pi/2..-Pi/2+0.00001+i*Pi/ n,u=-a..a,style=wireframe,grid=[3*i+2,5]) ,seq(pointp lot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+0.001+i*Pi/n,droi te)\},\n symbol=circle,symbolsize=8+4*j,color=b lue),j=0..2)\n ]),i=0..n)], insequence=true):\n#plotsetup(gif, plotoutput=\"c:/cercle_c1.gif\",plotoptions=\"width=500, height=500\") ;\ndisplay([gdroite,gdouble,gbase,mobile],scaling=constrained,orientat ion=[30,60],view=[-0.7*a..1.5*a,-1.2*a..1.1*a,-a..a]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 " cas cercle \+ c=2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "restart:\ncubique:=2 *z*x^2+x^2*y+6*z*x+y*z^2-y+4*z+y^3-2*z*y^2;\nchange:=\{Y=Z1, X=-1/3*(2 *X1+3*T1)*3^(1/2), Z=Y1+Z1, T=1/3*3^(1/2)*(3*T1+X1)\}:\nprintf(\"Passa ge \340 Plucker par :\\n %a\",change);\ncubhom:=numer(normal(subs(x=X/ T,y=Y/T,z=Z/T,cubique)));\nprintf(\" V\351rification : %a -- passage - -> %a \\n\",cubhom,simplify(subs(change,cubhom)));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 122 "#[a = 1,b = 1, c = 2];\nbase := [cos( 2*t), sin(2*t), 0];\ndroite := [-2, 0, tan(t)];\nps:=(U,V)->add(U[i]*V [i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqrt(ps(d roite-base,droite-base))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 652 "with (plots):\na:=3:n:=10:\n#gbase:=spacecurve(base,t=-Pi/2..Pi/2,thickness =3,color=red,numpoints=80):\ngbase:=tubeplot(base,t=-Pi/2..Pi/2,radius =0.06,color=red,grid=[30,6],style=wireframe):\n#gdroite:=spacecurve(dr oite,t=-Pi/2..Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0,numpoints=2)): \ngdroite:=tubeplot(droite,t=-Pi/2..Pi/2,radius=0.06,color=COLOR(RGB,0 ,0.6,0),grid=[30,6],style=wireframe):\ngdouble:=spacecurve([[-1,-a,-a] ,[-1,a,a]],thickness=3,color=COLOR(RGB,1,0.8,0),numpoints=2):\ncub:=im plicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style=pat chnogrid):\ngene:=seq(spacecurve(subs(t=i*Pi/n/2,M),u=-2*a..2*a,color= blue,numpoints=2),i=-n..n):" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cubiqueG,2*(\"\"#\"\"\"%\"zGF()%\"xGF'F(F(*&F*F(%\"y GF(F(*(\"\"'F(F)F(F+F(F(*&F-F()F)F'F(F(F-!\"\"*&\"\"%F(F)F(F(*$)F-\"\" $F(F(*(F'F(F)F()F-F'F(F2" }}{PARA 6 "" 1 "" {TEXT -1 23 "Passage \340 \+ Plucker par :" }}{PARA 6 "" 1 "" {TEXT -1 77 " \{X = -1/3*(2*X1+3*T1)* 3^(1/2), Z = Y1+Z1, T = 1/3*3^(1/2)*(3*T1+X1), Y = Z1\}" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'cubhomG,2*(\"\"#\"\"\"%\"ZGF()%\"XGF'F(F(*&F* F(%\"YGF(F(**\"\"'F(F)F(F+F(%\"TGF(F(*&F-F()F)F'F(F(*&F-F()F0F'F(!\"\" *(\"\"%F(F)F(F4F(F(*$)F-\"\"$F(F(*(F'F(F)F()F-F'F(F5" }}{PARA 6 "" 1 " " {TEXT -1 113 " V\351rification : 2*Z*X^2+X^2*Y+6*Z*X*T+Y*Z^2-Y*T^2+4 *Z*T^2+Y^3-2*Z*Y^2 -- passage --> -2*Y1*X1*T1+Z1*X1^2+Z1*Y1^2 " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%baseG7%-%$cosG6#,$*&\"\"#\"\"\"%\"t GF,F,-%$sinGF(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'droiteG7%!\" #\"\"!-%$tanG6#%\"tG" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "display([cub,gene,gdroite,gdouble,gbase],scaling=con strained,view=[-a..a,-a..a,-a..a],orientation=[80,60]);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 642 "# animation\nn:=20:\nmobile:=display([seq(dis play([spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a..2*a,color=blue,numpoin ts=2,thickness=3),\n plot3d(M,t=-Pi/2..-Pi/2+0.00001+ i*Pi/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) ,seq(p ointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+0.001+i*Pi/n ,droite)\},\n symbol=circle,symbolsize=8+4*j,co lor=blue),j=0..2)\n ]),i=0..n)], insequence=true):\nplotsetup( gif,plotoutput=\"c:/cercle_c2.gif\",plotoptions=\"width=500, height=50 0\");\ndisplay([gdroite,gbase,gdouble,mobile],scaling=constrained,orie ntation=[60,60],view=[-a*0.9..1.4*a,-a..a,-a..a]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 " Ellipse : cas [a = 2, b = 1, c = -2]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "restart:\ncubique:= \+ 2*z*x^2+8*z*x+8*z-4*y+4*y^3+y*x^2;\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 125 "#[a = 2,b = 1, c = -2];\nbase := [2*cos(2*t), sin(2*t), 0];\ndroi te := [-2, 0, tan(t)];\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqrt(ps(droite-base,droite-bas e))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "gbase:=spacecurve(base,t= -Pi/2..Pi/2,thickness=3,color=red,numpoints=80):\ngdroite:=spacecurve( droite,t=-Pi/2..Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2) :\nwith(plots):\na:=4:n:=10:\ncub:=implicitplot3d(cubique,x=-a..a,y=-a ..a,z=-a..a,grid=[30,30,30],style=patchnogrid):\ngene:=seq(spacecurve( subs(t=i*Pi/n/2,M),u=-2*a..2*a,color=blue,numpoints=2),i=-n..n):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "display([cub,gene,gdroite,gb ase],scaling=constrained,view=[-a..a,-a..a,-a..a],orientation=[20,70]) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "# animation\na:=3:n:=20:\nm obile:=display([seq(display([spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a. .2*a,color=blue,numpoints=2),\n plot3d(M,t=-Pi/2..-Pi /2+0.00001+i*Pi/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+0 .001+i*Pi/n,droite)\},\n symbol=circle,symbolsi ze=6+3*j,color=blue),j=0..2)\n ]),i=0..n)], insequence=true): \ndisplay([gdroite,gbase,mobile],scaling=constrained,orientation=[-10, 80],view=[-1.2*a..1.2*a,-1.2*a..a,-a..a]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 " Ellipse : c as [a = 2,b = 1, c =0]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "re start:\ncubique:=2*z*x^2+y*x^2+4*z*x+4*y*z^2+4*y^3-4*y+8*z*y^2;\n" } {TEXT -1 0 "" }{MPLTEXT 1 0 121 "#[a = 2,b = 1, c =0];\nbase := [2*cos (2*t), sin(2*t), 0];\ndroite := [0,0, tan(t)];\nps:=(U,V)->add(U[i]*V[ i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "M :=expand(base+u*(droite-base)/sqrt(ps( droite-base,droite-base))):\ngbase:=spacecurve(base,t=-Pi/2..Pi/2,thic kness=3,color=red,numpoints=80):\ngdroite:=spacecurve(droite,t=-Pi/2.. Pi/2,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\nwith(plots): \na:=4:n:=10:\n#surfgene:=plot3d(M,t=-Pi/2+0.0001..Pi/2-0.0001,u=-a..a ,style=wireframe):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a.. a,grid=[30,30,30],style=patchnogrid):\ngene:=seq(spacecurve(subs(t=0.0 001+i*Pi/n/2,M),u=-2*a..2*a,color=blue,numpoints=2),i=-n..n):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "display([cub,gene,gdroite,gb ase],scaling=constrained,view=[-a..a,-a..a,-a..a],orientation=[20,70]) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "# anim ation\na:=3:n:=20:\nmobile:=display([seq(display([spacecurve(subs(t=-P i/2+i*Pi/n,M),u=-2*a..2*a,color=blue,numpoints=2),\n \+ plot3d(M,t=-Pi/2..-Pi/2+0.00001+i*Pi/n,u=-a..a,style=wireframe,grid=[3 *i+2,5]) ,seq(pointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n ,base),subs(t=-Pi/2+0.001+i*Pi/n,droite)\},\n s ymbol=circle,symbolsize=6+3*j,color=blue),j=0..2)\n ]),i=0..n) ], insequence=true):\ndisplay([gdroite,gbase,mobile],scaling=constrain ed,orientation=[-10,80],view=[-1.2*a..1.2*a,-1.2*a..a,-a..a]);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 " Ellipse : cas [a = 2,b = 1, c = 4] " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "restart:\ncubique: =4*y^3-8*z*y^2-4*y+y*x^2+4*y*z^2+16*z+12*z*x+2*z*x^2;\n" }{TEXT -1 0 " " }{MPLTEXT 1 0 124 "#[a = 2,b = 1, c = 4]:\nbase := [2*cos(2*t), sin( 2*t), 0];\ndroite := [-4, 0, tan(t)];\nps:=(U,V)->add(U[i]*V[i],i=1..n ops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 506 "M :=expand(base+u*(droite-base)/sqrt(ps(droite-bas e,droite-base))):\ngbase:=spacecurve(base,t=-Pi/2..Pi/2,thickness=3,co lor=red,numpoints=80):\ngdroite:=spacecurve(droite,t=-Pi/2..Pi/2,thick ness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\nwith(plots):\na:=5:n:=1 0:\n#surfgene:=plot3d(M,t=-Pi/2+0.0001..Pi/2-0.0001,u=-a..a,style=wire frame):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30, 30,30],style=patchnogrid):\ngene:=seq(spacecurve(subs(t=i*Pi/n/2,M),u= -2*a..2*a,color=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "display([cub,gene,gdroite,gbase],scaling=constrai ned,view=[-a..a,-a..a,-a..a],orientation=[100,70]);" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "# animation\na:=5:n:=20:\n mobile:=display([seq(display([spacecurve(subs(t=-Pi/2+i*Pi/n,M),u=-2*a ..2*a,color=blue,numpoints=2),\n plot3d(M,t=-Pi/2..-P i/2+0.00001+i*Pi/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=-Pi/2+0.001+i*Pi/n,base),subs(t=-Pi/2+ 0.001+i*Pi/n,droite)\},\n symbol=circle,symbols ize=6+3*j,color=blue),j=0..2)\n ]),i=0..n)], insequence=true): \ndisplay([gdroite,gbase,mobile],scaling=constrained,orientation=[100, 80],view=[-1.2*a..1.2*a,-1.2*a..a,-a..a]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 " Base : la parabole (P) : z = 0, \+ y=x^2" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 " Calculs g\351n\351raux " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 146 "c:=-1:\nprintf(\"\\#c = %a\",c);\nbase:=[t,t* t,0];\ndroite:=[t,0,1];\nM:=expand(base+u*(droite-base));\nsys:=\{x-M[ 1],y-M[2],z-M[3]\};\neliminate(sys,\{u,t\});" }{TEXT -1 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 " Exemple 1 : d\351g\351n\351r\351e en un parabolo\357de e t la droite " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "restart:\n #c = 0:\nbase := [t, t^2, 0];droite := [0, 0, t];\nps:=(U,V)->add(U[i] *V[i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 656 "M :=expand(base+u*(droite-base)/sq rt(ps(droite-base,droite-base)));\n#M := [-t*u+t, -t^2*u+t^2, t*u];\nw ith(plots):\na:=4:\ncubique:= -x*z+y-x^2;\ngbase:=spacecurve(base,t=-a ..a,thickness=3,color=red,numpoints=81):\ngdroite:=spacecurve(droite,t =-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\nwith(plots) :\na:=4:n:=10:\n#surfgene:=plot3d(M,t=-a..a,u=-a..a,style=wireframe): \ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30], style=patchnogrid):\ngene:=seq(spacecurve(subs(t=-a+0.0001+2*i*a/n,M), u=-2*a..2*a,color=blue,numpoints=2),i=0..n):\ndisplay([cub,gene,gdroit e,gbase],scaling=constrained,view=[-a..a,-a..a,-a..a],orientation=[20, 70]);" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 40 " Droite parall\350le au plan de la parabole" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "cubique:=y+ x^2*z-x^2;# type Plucker\npass:=\{Y1 = 2*Z-2*T, X1 = -1/2*X+1/2*Y, Z1 \+ = 2*Z+2*T+1/2*Y+1/2*X, T1 = 1/2*Y+1/2*X\};\nprintf(\"type Plucker :Y*T ^2+X^2*Z-X^2*T --pass--> %a\", simplify(subs(pass,Y1*T1^2+X1^2*Z1-X1^2 *T1)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "base := [t, t^2, 0];\ndr oite := [t, 0, 1];\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "pv:=(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]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "M :=expand(base+u*(droite-base)/sqrt(ps(droite-base,droite-base))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(plots):\na:=4:n:=10:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 513 "#gbase:=spacecurve(base,t=-a..a,th ickness=3,color=red,numpoints=80):\ngbase:=tubeplot(base,t=-a..a,radiu s=0.06,color=red,grid=[30,6],style=wireframe):\n#gdroite:=spacecurve(d roite,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0,numpoints=2)):\ngdro ite:=tubeplot(droite,t=-a..a,radius=0.06,color=COLOR(RGB,0,0.6,0),grid =[30,6],style=wireframe):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a ,z=-a/2..3*a/2,grid=[30,30,30],style=patchnogrid):\ngene:=seq(spacecur ve(subs(t=i*a/n,M),u=-5*a..5*a,color=blue,numpoints=2),i=-n..n):" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cubiqueG,(%\"yG\"\" \"*&%\"zGF')%\"xG\"\"#F'F'*$F*F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%passG<&/%#Y1G,&*&\"\"#\"\"\"%\"ZGF+F+*&F*F+%\"TGF+!\"\"/%#X1G,&* &F*F/%\"XGF+F/*&F*F/%\"YGF+F+/%#Z1G,**&F*F+F,F+F+*&F*F+F.F+F+*&F*F/F6F +F+*&F*F/F4F+F+/%#T1G,&*&F*F/F6F+F+*&F*F/F4F+F+" }}{PARA 6 "" 1 "" {TEXT -1 61 "type Plucker :Y*T^2+X^2*Z-X^2*T --pass--> Z*Y^2+Z*X^2-2*T *Y*X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%baseG7%%\"tG*$)F&\"\"#\"\" \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'droiteG7%%\"tG\"\"!\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "display([cub,gene,gd roite,gbase],scaling=constrained,view=[-a..a,-a..a,-a/2..3*a/2],orient ation=[20,70]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 653 "# animation\na: =3.5:n:=20:\nmobile:=display([seq(display([spacecurve(subs(t=-a+2*i*a/ n,M),u=-5*a..5*a,color=blue,numpoints=2,thickness=3),\n \+ plot3d(M,t=-a-0.0001..-a+2*i*a/n,u=-a/2..3*a/2,style=wireframe,grid =[3*i+2,5]) ,seq(pointplot3d(\{subs(t=-a+2*i*a/n,base ),subs(t=-a+2*i*a/n,droite)\},\n symbol=circle, symbolsize=8+4*j,color=blue),j=0..2)\n ]),i=0..n)], insequence =true):\nplotsetup(inline);\nplotsetup(gif,plotoutput=\"c:/parabole_dr _paralleles.gif\",plotoptions=\"width=500, height=500\");\ndisplay([gd roite,gbase,mobile],scaling=constrained,orientation=[50,70],view=[-a.. a,-a..a,-a/2..a*1.5]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 " Exemple 2 : droite int\351rieur e \340 (P)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 736 "restart:\n#c \+ =-1:\nbase := [t, t^2, 0];droite := [0, 1, t];\n#M := [-t*u+t, -t^2*u+ u+t^2, t*u]:\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand (base+u*(droite-base)/sqrt(ps(droite-base,droite-base))):\nwith(plots) :\na:=4:\ncubique:=-2*z*x^2-x*z^2-z+x*y-x^3+z*y;\ngbase:=spacecurve(ba se,t=-a..a,thickness=3,color=red,numpoints=80):\ngdroite:=spacecurve(d roite,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\nwith (plots):\na:=4:n:=10:\n#surfgene:=plot3d(M,t=-a..a,u=-a..a,style=wiref rame):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,3 0,30],style=patchnogrid):\ngene:=seq(spacecurve(subs(t=i*a/n,M),u=-2*a ..2*a,color=blue,numpoints=2),i=-n..n):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%baseG7%%\"tG*$)F&\"\"#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'droiteG7%\"\"!\"\"\"%\"tG" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cubiqueG,.*(\"\"#\"\"\"%\"zGF()%\"xGF'F(! \"\"*&F+F()F)F'F(F,F)F,*&F+F(%\"yGF(F(*$)F+\"\"$F(F,*&F)F(F0F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "display([cub,gene,gdroite,g base],scaling=constrained,view=[-a..a,-a..a,-a..a],orientation=[-70,70 ]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "# animation\na:=4:n:=20:\nmobile:=displa y([seq(display([spacecurve(subs(t=-a+2*i*a/n,M),u=-2*a..2*a,color=blue ,numpoints=2),\n plot3d(M,t=-a-0.0001..-a+2*i*a/n,u=- a..a,style=wireframe,grid=[3*i+2,5]) ,seq(pointplot3d (\{subs(t=-a+2*i*a/n,base),subs(t=-a+2*i*a/n,droite)\},\n \+ symbol=circle,symbolsize=6+3*j,color=blue),j=0..2)\n \+ ]),i=0..n)], insequence=true):\nplotsetup(inline);\nplotsetup(gif,pl otoutput=\"c:/parabole_c0.gif\",plotoptions=\"width=500, height=500\") ;\n\ndisplay([gdroite,gbase,mobile],scaling=constrained,orientation=[7 0,80],view=[-a..a,-a..3*a/2,-a..a]);" }{TEXT -1 0 "" }{MPLTEXT 1 0 0 " " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 " Exemple 3 : droite ext\351rieure \340 (P)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 724 "#c = 1:\nbase := [t, t^2, 0];droit e := [0, -1, t];\n#M := [-t*u+t, -t^2*u-u+t^2, t*u]:\nps:=(U,V)->add(U [i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(base+u*(droite-base)/sqrt(ps( droite-base,droite-base)));\nwith(plots):\na:=4:\ncubique:= -2*z*x^2-x *z^2+z+x*y-x^3+z*y;\ngbase:=spacecurve(base,t=-a..a,thickness=3,color= red,numpoints=80):\ngdroite:=spacecurve(droite,t=-a..a,thickness=3,col or=COLOR(RGB,0,0.6,0),numpoints=2):\nwith(plots):\nn:=10:\n#surfgene:= plot3d(M,t=-a..a,u=-a..a,style=wireframe):\ncub:=implicitplot3d(cubiqu e,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style=patchnogrid):\ngene:=s eq(spacecurve(subs(t=i*a/n,M),u=-2*a..2*a,color=blue,numpoints=2),i=-n ..n):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%baseG7%%\"tG*$)F&\"\"#\"\" \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'droiteG7%\"\"!!\"\"%\"tG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7%,&*(%\"tG\"\"\"%\"uGF),(*& \"\"%F))F(\"\"#F)F)*$)F(F-F)F)F)F)#!\"\"F/F3F(F),(*(F(F/F*F)F+F2F3*&F* F)F+F2F3*$F.F)F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(cubiqueG,.*( \"\"#\"\"\"%\"zGF()%\"xGF'F(!\"\"*&F+F()F)F'F(F,F)F(*&F+F(%\"yGF(F(*$) F+\"\"$F(F,*&F)F(F0F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "display([cub,gene,gdroite,gbase],scaling=constrained,view=[-a..a,-a.. a,-a..a],orientation=[130,70]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 "# animatio n\nn:=20:\nmobile:=display([seq(display([spacecurve(subs(t=-a+2*i*a/n, M),u=-2*a..2*a,color=blue,numpoints=2),\n plot3d(M,t= -a-0.0001..-a+2*i*a/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=-a+2*i*a/n,base),subs(t=-a+2*i*a/n, droite)\},\n symbol=circle,symbolsize=6+3*j,col or=blue),j=0..2)\n ]),i=0..n)], insequence=true):\nplotsetup(i nline);\n#plotsetup(gif,plotoutput=\"c:/parabole_c1.gif\",plotoptions= \"width=500, height=500\");\n\ndisplay([gdroite,gbase,mobile],scaling= constrained,orientation=[140,80],view=[-a..a,-a..a,-a..a]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Base : l'hy perbole xy = 0, z = 0" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 " Calculs g\351n\351raux" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "restart: \nbase:=[t,1/t,0];\ndroite:=[t,-t,3];\nM:=expand(base+u*(droite-base)) ;\nsys:=\{x-M[1],y-M[2],z-M[3]\}:\neliminate(sys,\{u,t\});" }{TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 " Droite " }{XPPEDIT 18 0 "[t, -t, 3];" "6#7%%\"t G,$F$!\"\"\"\"$" }{TEXT -1 33 " parall\350le au plan de l'hyperbole" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "restart:\ncubique:=3*y*x+z +x^2*z-3;\nbase := [t,1/t, 0];\ndroite := [t, -t, 3];\n# M:=expand(bas e+u*(droite-base));\n#M1:=expand(base1+u*(droite-base1));\nps:=(U,V)-> add(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(base+u*(droite-base)/sqr t(ps(droite-base,droite-base))):\na:=8:n:=10:eps:=0.0001:\nwith(plots) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "gbase:=spacecurve(base,t=-a.. -eps,thickness=3,color=red,numpoints=81):\ngbase1:=spacecurve(base,t=e ps..a,thickness=3,color=red,numpoints=81):\ngdroite:=spacecurve(droite ,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\ncub:=impl icitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style=patch nogrid):\ngene:=seq(spacecurve(subs(t=eps+i*2*a/n,M),u=-a..a,color=blu e,numpoints=2),i=-n..n):\ngene0:=seq(spacecurve(subs(t=10/(i^3+eps),M) ,u=-10*a..10*a,color=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "display([cub,gene,gene0,gdroite,gbase,gbase1 ],scaling=constrained,view=[-a..a,-a..a,-a..a],orientation=[120,70]); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "> " 0 "" {MPLTEXT 1 0 544 "# animation\na:=4:n:=20:\nmobile:= display([seq(display([spacecurve(subs(t=-a+2*i*a/n-0.0001,M),u=-2*a..2 *a,color=blue,numpoints=2),\n plot3d(M,t=-a-0.0001..- a+2*i*a/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) ,se q(pointplot3d(\{subs(t=-a+2*i*a/n-0.0001,base),subs(t=-a+2*i*a/n-0.000 1,droite)\},\n symbol=circle,symbolsize=6+3*j,c olor=blue),j=0..2)\n ]),i=0..n)], insequence=true):\ndisplay([ gdroite,gbase,gbase1,mobile],scaling=constrained,orientation=[-30,50], view=[-a..a,-a..2*a,-a..a]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 " Droite " }{XPPEDIT 18 0 "[t-1/t, 0, 1];" "6#7%,&%\"tG\"\"\"*&F&F&F%!\"\"F(\"\"! F&" }{TEXT -1 33 " parall\350le au plan de l'hyperbole" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "restart:\ncubique:=x*y*z-x*y-z*y^2 +z^2-2*z+1;\nbase := [t,1/t, 0];\ndroite := [t-1/t, 0, 1];\n# M:=expan d(base+u*(droite-base));\n#M1:=expand(base1+u*(droite-base1));\nps:=(U ,V)->add(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(base+u*(droite-base )/sqrt(ps(droite-base,droite-base))):\na:=8:n:=10:eps:=0.0001:\nwith(p lots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "gbase:=spacecurve(base,t =-a..-eps,thickness=3,color=red,numpoints=81):\ngbase1:=spacecurve(bas e,t=eps..a,thickness=3,color=red,numpoints=81):\ngdroite:=spacecurve(d roite,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\ncub: =implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style= patchnogrid):\ngene:=seq(spacecurve(subs(t=eps+i*2*a/n,M),u=-a..a,colo r=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "display([cub,gene,gdroite,gbase,gbase1],scaling=constrained,vie w=[-a..a,-a..a,-a..a],orientation=[120,70]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 544 "# animation\na:=4:n:=20:\nmobile:=display([seq(dis play([spacecurve(subs(t=-a+2*i*a/n-0.0001,M),u=-2*a..2*a,color=blue,nu mpoints=2),\n plot3d(M,t=-a-0.0001..-a+2*i*a/n,u=-a.. a,style=wireframe,grid=[3*i+2,5]) ,seq(pointplot3d(\{ subs(t=-a+2*i*a/n-0.0001,base),subs(t=-a+2*i*a/n-0.0001,droite)\},\n \+ symbol=circle,symbolsize=6+3*j,color=blue),j=0.. 2)\n ]),i=0..n)], insequence=true):\ndisplay([gdroite,gbase,gb ase1,mobile],scaling=constrained,orientation=[-30,50],view=[-a..a,-a.. 2*a,-a..a]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 " Droite [t, t, 1] parall\350le au plan de l'hyperbole" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "restart:\ncubique:=-y*x-z+x ^2*z+1;\nbase := [t, 1/t, 0];\ndroite := [t, t, 1];\n# M:=expand(base+ u*(droite-base));\n#M1:=expand(base1+u*(droite-base1));\nps:=(U,V)->ad d(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(base+u*(droite-base)/sqrt( ps(droite-base,droite-base))):\na:=8:n:=10:eps:=0.0001:\nwith(plots): " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "gbase:=spacecurve(base,t=-a..- eps,thickness=3,color=red,numpoints=81):\ngbase1:=spacecurve(base,t=ep s..a,thickness=3,color=red,numpoints=81):\ngdroite:=spacecurve(droite, t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoints=2):\ncub:=impli citplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,30,30],style=patchn ogrid):\ngene:=seq(spacecurve(subs(t=eps+i*2*a/n,M),u=-a..a,color=blue ,numpoints=2),i=-n..n):\ngene0:=seq(spacecurve(subs(t=30/(i^3+eps),M), u=-10*a..10*a,color=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 128 "display([cub,gene,gene0,gdroite,gbase,gbase1] ,scaling=constrained,\n view=[-a..a,-a..a,-a..a],orientat ion=[120,70]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 544 "# animation\na:=4 :n:=20:\nmobile:=display([seq(display([spacecurve(subs(t=-a+2*i*a/n-0. 0001,M),u=-2*a..2*a,color=blue,numpoints=2),\n plot3d (M,t=-a-0.0001..-a+2*i*a/n,u=-a..a,style=wireframe,grid=[3*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=-a+2*i*a/n-0.0001,base),subs(t =-a+2*i*a/n-0.0001,droite)\},\n symbol=circle,s ymbolsize=6+3*j,color=blue),j=0..2)\n ]),i=0..n)], insequence= true):\ndisplay([gdroite,gbase,gbase1,mobile],scaling=constrained,orie ntation=[-30,50],view=[-a..a,-a..2*a,-a..a]);" }{TEXT -1 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 " Exemple 1 : Oz \"ext\351rieure\" \340 (H)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "restart:\ncubique:=x-y*z^2-2*y*z*x -y*x^2;\n#[a = 2,b = 1, c = 0];\nbase := [t, 1/t, 0];\ndroite := [0, 0 , t];\n# M:=expand(base+u*(droite-base));\n#M1:=expand(base1+u*(droite -base1));\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(ba se+u*(droite-base)/sqrt(ps(droite-base,droite-base))):\na:=8:n:=10:eps :=0.0001:\nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "gbase:= spacecurve(base,t=-a..-eps,thickness=3,color=red,numpoints=81):\ngbase 1:=spacecurve(base,t=eps..a,thickness=3,color=red,numpoints=81):\ngdro ite:=spacecurve(droite,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),nu mpoints=2):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid= [30,30,30],style=patchnogrid):\ngene:=seq(spacecurve(subs(t=eps+i*2*a/ n,M),u=-a..a,color=blue,numpoints=2),i=-n..n):\ngene0:=seq(spacecurve( subs(t=40*sign(i)/(i^2+eps),M),u=-10*a..10*a,color=blue,numpoints=2),i =-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "display([cub,g ene,gene0,gdroite,gbase,gbase1],scaling=constrained,view=[-a..a,-a..a, -a..a],orientation=[120,70]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 558 "# animation\na:=4:n:=20:\nmobile:=display([seq(display([spacecurve(subs (t=-a+2*i*a/n-0.0001,M),u=-2*a..2*a,color=blue,numpoints=2),\n \+ plot3d(M,t=-a-0.0001..-a+2*i*a/n,u=-a..a,style=wireframe,gri d=[3*i+2,5]) ,seq(pointplot3d(\{subs(t=-a+2*i*a/n-0.0 001,base),subs(t=-a+2*i*a/n-0.0001,droite)\},\n \+ symbol=circle,symbolsize=6+4*j,color=blue),j=0..2)\n ]),i=0.. n)], insequence=true):\ndisplay([gdroite,gbase,gbase1,mobile],scaling= constrained,orientation=[-50,70],view=[-1.7*a..1.7*a,-1.7*a..1.7*a,-a. .a]);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 " Exemple 2 : droite [1,1,t] touche (H)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "restart:\ncubique:=z*y^2-3*z*y+z*y*x+2*z-z*x+y*x ^2-x-y*x+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "base := [t, 1/t, 0] ;\ndroite := [1, 1, t];\n# M:=expand(base+u*(droite-base));\n#M1:=expa nd(base1+u*(droite-base1));\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):\n pv:=(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 ]]:\nM:= expand(base+u*(droite-base)/sqrt(ps(droite-base,droite-base)) ):\na:=8:n:=10:eps:=0.0001:\nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 500 "gbase:=spacecurve(base,t=-a..-eps,thickness=3,color= red,numpoints=81):\ngbase1:=spacecurve(base,t=eps..a,thickness=3,color =red,numpoints=81):\ngdroite:=spacecurve(droite,t=-a..a,thickness=3,co lor=COLOR(RGB,0,0.6,0),numpoints=2):\ncub:=implicitplot3d(cubique,x=-a ..a,y=-a..a,z=-a..a,grid=[30,30,30],style=patchnogrid):\ngene:=seq(spa cecurve(subs(t=eps+i*2*a/n,M),u=-a..a,color=blue,numpoints=2),i=-n..n) :\ngene0:=seq(spacecurve(subs(t=10*sign(i)/(i^2+eps),M),u=-10*a..10*a, color=blue,numpoints=2),i=-n..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "display([cub,gene,gene0,gdroite,gbase,gbase1],scalin g=constrained,view=[-a..a,-a..a,-a..a*1.2],orientation=[120,70]);" } {TEXT -1 0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "# animation\na:=8:n:=20:\nmobile:=display([seq( display([spacecurve(subs(t=-a+2*i*a/nc,M),u=-2*a..2*a,color=blue,numpo ints=2),\n plot3d(M,t=-a-0.0001..-a+2*i*a/n,u=-a..a,s tyle=wireframe,grid=[2*i+2,5]) ,seq (pointplot3d(\{subs(t=-a+2*i*a/n-0.0001,base),subs(t=-a+2*i*a/n-0.0001 ,droite)\},\n symbol=circle,symbolsize=6+3*j,co lor=blue),j=0..2)\n ]),i=0..n) ], insequence=true):\ndisplay([ gdroite,gbase,gbase1,mobile],scaling=constrained,orientation=[120,70], view=[-a..a,-a..2*a,-a..a]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 " Exemple 3 : droite [2,2,t] \"int\351rieure\" \340 (H)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "restart:\ncubique:=-y*x^2+2*x*z*y+2*y*x+x+2*z*x-4 *z*y^2-2+6*z*y-6*z^2-8*z+3*y*z^2;\n#[a = 2,b = 1, c = 2];" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 332 "base := [t, 1/t, 0];\ndroite := [2, 2, t]; \n# M:=expand(base+u*(droite-base));\n#M1:=expand(base1+u*(droite-base 1));\nps:=(U,V)->add(U[i]*V[i],i=1..nops(U)):\npv:=(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]]:\nM:= expand(base+u* (droite-base)/sqrt(ps(droite-base,droite-base))):\na:=8:n:=10:eps:=0.0 001:\nwith(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 502 "gbase:=space curve(base,t=-a..-eps,thickness=3,color=red,numpoints=81):\ngbase1:=sp acecurve(base,t=eps..a,thickness=3,color=red,numpoints=81):\ngdroite:= spacecurve(droite,t=-a..a,thickness=3,color=COLOR(RGB,0,0.6,0),numpoin ts=2):\ncub:=implicitplot3d(cubique,x=-a..a,y=-a..a,z=-a..a,grid=[30,3 0,30],style=patchnogrid):\ngene:=seq(spacecurve(subs(t=eps+i*a/n,M),u= -2*a..2*a,color=blue,numpoints=2),i=-n..n):\ngene0:=seq(spacecurve(sub s(t=10*sign(i)/(i^2+eps),M),u=-10*a..10*a,color=blue,numpoints=2),i=-n ..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "display([cub,gene ,gene0,gdroite,gbase,gbase1],scaling=constrained,view=[-a..a,-a..a,-a. .a],orientation=[120,70]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1018 "# a nimation\na:=8:n:=20:\nmobile:=display([seq(display([spacecurve(subs(t =-a+i*a/n-0.0001,M),u=-2*a..2*a,color=blue,numpoints=2),\n \+ plot3d(M,t=-a-0.0001..-a+i*a/n,u=-a..a,style=wireframe,grid=[2*i +2,5]) ,seq(pointplot3d(\{subs(t=-a +i*a/n-0.0001,base),subs(t=-a+i*a/n-0.0001,droite)\},\n \+ symbol=circle,symbolsize=6+3*j,color=blue),j=0..2)\n \+ ]),i=0..n),\n seq(display([spacecurve(subs(t=i*a/n+0.0001,M), u=-2*a..2*a,color=blue,numpoints=2),\n plot3d(M,t=-a. .-0.0001,u=-a..a,style=wireframe,grid=[2*n+2,5]),\n p lot3d(M,t=0.0001..i*a/n,u=-a..a,style=wireframe,grid=[2*i+2,5]) \+ ,seq(pointplot3d(\{subs(t=i*a/n+0.0001,b ase),subs(t=i*a/n+0.0001,droite)\},\n symbol=ci rcle,symbolsize=6+3*j,color=blue),j=0..2)\n ]),i=0..n)\n], ins equence=true):\ndisplay([gdroite,gbase,gbase1,mobile],scaling=constrai ned,orientation=[120,70],view=[-a..a,-a..a,-a..a]);" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "23 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }