{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 " Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 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 Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Surface de Clebsch - 27 droites s ur une cubique." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 " Re\347u de V idiani - 19/06/03" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Le mail :" }} {PARA 0 "" 0 "" {TEXT -1 800 "Bonjour (:-) 19 Juin 2003 17h27\n\nJe \+ suis en train d'\351crire un nouvel article que je proposerais \340 Qu adrature\n;\n(voir l'\351bauche en tex , ci jointe : si tu n'arrives p as \340 le lire dis le\nmois je t'enverrais un tirage papier)\nIL s'ag it de pr\351ciser certains \351l\351ments du th\351or\350me de Cayley, Salmon, Juel\n: sur une cubique non r\351gl\351e il y a au maximum 27 droites.\nIl se trouve que sur la surface mise sous la forme de Hunt \+ (forme qui d\351pend\ndu choix du plan de l'infini), les 27 droites so nt r\351elles (et non\nhorizontales)\nJ'ai besoin de ton habilet\351 \+ avec Maple, pour obtenir une sortie des 27\nsolutions au besoin num \351rot\351es et sans les % (ce que j'ai fait avec\ninterface(labellin g=false))\n\nJe suis en train de chercher une repr\351sentation param \351trique rationnelle de\nla surface de Clebsch\n@+\nLGV VIDIANI" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 56 "zip maple juel.mws 3 06 03 07 h30 Version 19 06 03 17h36 " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 34 "VIDIANI Dijon (lg_vidiani@ipac.fr)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "restart: #Maple V4" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 69 "L'\351quation en coordonn\351es penta\351drales de l a surface de Clebsch est :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S:=y1^3+y2^3+y3^3+y4^3+y5^3=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 120 "On pose (on remarque que y1+y2+y3+y4+y5=0 comme il est impos\351 aux composan tes penta\351drales pour la surface de Clebsch) :" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 123 "y1:=-1/3*(-2*u0+u1+u2+u3);y2:=-1/3*(u0-2*u1 +u2+u3);y3:=-1/3*(u0+u1-2*u2+u3);y4:=-1/3*(u0+u1+u2-2*u3);y5:=1/3*(u0+ u1+u2+u3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "y1+y2+y3+y4/2 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 48 "On prend comme plan de l'infini u0+u1+u2+4*u3 ; " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 171 "Ce qui revient \340 po ser (lien entre coordonn\351es ordinaires et composantes projectives) \+ x:=u0/(u0+u1+u2+4*u3);y:=u1/(u0+u1+u2+4*u3 );z:=u2/(u0+u1+u2+4*u3);u0+ u1+u2+4*u3:=1; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "#solve( \{x=u0/(u0+u1+u2+4*u3),y=u1/(u0+u1+u2+4*u3 ),z=u2/(u0+u1+u2+4*u3)\},\{ u0,u1,u2,u3\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u0:=x;u1 :=y;u2:=z;u3:=1/4*(1-x-y-z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "S;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(S);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "(288)*\";" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 164 "C'est bien ce qui est donn\351 dans Weisste in, mais le moins qu'on puisse dire, c'est qu'outre le plan de l'infin i, il aurait pu donner le lien entre x,y,z et les yi !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 102 "Calculs de recherche du plan de l'infini et des rel ations entre les coordonn\351es ordinaires et les y1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 120 "solve(\{y1=-1/3*(-2*u0+u1+u2+u3),y2=-1/3*(u0-2*u1+ u2+u3),y3=-1/3*(u0+u1-2*u2+u3),y4=-1/3*(u0+u1+u2-2*u3)\},\{u0,u1,u2,u3 \});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "u0+u1+u2+4*u3= -y2- y3-y4-y1-y3-y4-y2-y1-y4-4*(y3+y2+y1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {MPLTEXT 0 21 113 "Le plan de l'i nfini est y1+y2+y3+y4/2 ! (comme il est bien dit dans le dictionnaire \+ Weisstein \340 l'entr\351e Clebsch)" }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 50 "Donc il fallait poser (ce qui n'\351tait pas \351vid ent)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "x:=(y2+y3+y4)/(6*(y1+y2+y3+y4/2)); y:=(y1+ y3+y4)/(6*(y1+y2+y3+y4/2));z:=(y1+y2+y4)/(6*(y1+y2+y3+y4/2));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 123 "solve(\{x=(y2+y3+y4)/(6*(y1+y2+y3+y4/2)), y =(y1+y3+y4)/(6*(y1+y2+y3+y4/2)),z=(y1+y2+y4)/(6*(y1+y2+y3+y4/2))\},\{y 1,y2,y3,y4\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "y4 := -( 6*x+6*y+6*z-2)/(6^(1/3)); y3 :=simplify( 1/2*y4*(-9*z+3*y+3*x+1)/(3*z+ 3*y+3*x-1)); y1 := simplify(-1/2*y4*(-3*z-1-3*y+9*x)/(3*z+3*y+3*x-1)); y2 := simplify(1/2*(3*z+1-9*y+3*x)*y4/(3*z+3*y+3*x-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "S:=y1^3+y2^3+y3^3+y4^3-(y1+y2+y3+y4 )^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "expand(S); #divise r yi par (-6)^(1/3) n'\351tait pas \351vident....ni indiqu\351 et \351 vite (petite coquetterie de diviser \340 la fin par -6)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 126 " Recherche des droites de la surface de Clebsch sous la forme donn\351e par HUNT. (j'ai fait, ce n'est pas beau, un copier coller)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "H:=(x,y,z)->126*z*x+1-189*z *y^2-189*z*x^2-189*y^2*x+126*z*y-9*x^2-189*z^2*y-189*z^2*x+126*y*x-9*y ^2+81*z^3-189*y*x^2+81*y^3+81*x^3-9*x+54*z*y*x-9*z-9*y-9*z^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "H(a*z+p,b*z+q,z);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "H1:=expand(\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "H2:=collect(H1,z);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 143 "\351criture des coefficients par co pier coller (Esculier fera plus pens\351 avec des listes) (j'ai fait d es copier coller : je sais ce n'est pas fin)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "C3:=81-189*a^2*b+81*a^3+54*a*b-189*b^2+81*b^3-1 89*a^2-189*a-189*b-189*a*b^2;C2:=-9-378*a*p+126*a*b+54*a*q-189*q+126*a -189*p+54*p*b-189*a^2*q-9*b^2-378*b*q+126*b+243*b^2*q-378*a*p*b-9*a^2+ 243*a^2*p-378*a*b*q-189*p*b^2;C1:=126*q-9-18*a*p-9*a+126*p-9*b-189*a*q ^2+126*a*q+243*a*p^2-378*a*p*q+126*p*b-189*p^2*b-378*p*b*q+243*b*q^2+5 4*p*q-18*b*q-189*p^2-189*q^2;C0:=81*p^3-189*p^2*q-189*p*q^2+81*q^3-9*p ^2+126*p*q-9*q^2-9*p-9*q+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "interface(labelling=false): solve(\{C3=0,C2=0,C1=0,C0=0\},\{a,b,p,q\}); #comment se d\351barasser \+ de ces RootOf puisque toutes les racines en sont r\351elles ?" }}} {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 0 {PARA 3 "" 0 "" {TEXT -1 29 " Pg de Vidiani un peu r\351duit " }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 56 "zip maple juel.mws 3 06 0 3 07h30 Version 19 06 03 17h36 " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 34 "VIDIANI Dijon (lg_vidiani@ipac.fr)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "restart: #Maple V4" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 69 "L'\351quation en coordonn\351es penta\351drales de l a surface de Clebsch est :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S:=y1^3+y2^3+y3^3+y4^3+y5^3=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 120 "On pose (on remarque que y1+y2+y3+y4+y5=0 comme il est impos\351 aux composan tes penta\351drales pour la surface de Clebsch) :" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "y1:=-1/3*(-2*u0+u1+u2+u3);y 2:=-1/3*(u0-2*u1+u2+u3);y3:=-1/3*(u0+u1-2*u2+u3);y4:=-1/3*(u0+u1+u2-2* u3);y5:=1/3*(u0+u1+u2+u3);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "y1+y2+y3+y4/2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 48 " On prend comme plan de l'infini u0+u1+u2+4*u3 ; " }}}{EXCHG {PARA 0 " " 0 "" {MPLTEXT 0 21 171 "Ce qui revient \340 poser (lien entre coordo nn\351es ordinaires et composantes projectives) x:=u0/(u0+u1+u2+4*u3); y:=u1/(u0+u1+u2+4*u3 );z:=u2/(u0+u1+u2+4*u3);u0+u1+u2+4*u3:=1; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "#solve(\{x=u0/(u0+u1+u2+4*u3 ),y=u1/(u0+u1+u2+4*u3 ),z=u2/(u0+u1+u2+4*u3)\},\{u0,u1,u2,u3\});" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "u0:=x;u1:=y;u2:=z;u3:=1/4*(1 -x-y-z);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " #S; expand(S);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sort( 288*expand( S),[x,y,z]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 164 "C'est bien ce qui est donn\351 dans Weisstein, mais le moins qu'o n puisse dire, c'est qu'outre le plan de l'infini, il aurait pu donner le lien entre x,y,z et les yi !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 102 "Calculs de recherche du plan de l'infini et des relations entre les coordonn \351es ordinaires et les y1 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "solve(\{y1=-1/3*( -2*u0+u1+u2+u3),y2=-1/3*(u0-2*u1+u2+u3),y3=-1/3*(u0+u1-2*u2+u3),y4=-1/ 3*(u0+u1+u2-2*u3)\},\{u0,u1,u2,u3\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "u0+u1+u2+4*u3= -y2-y3-y4-y1-y3-y4-y2-y1-y 4-4*(y3+y2+y1);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {MPLTEXT 0 21 113 " Le plan de l'infini est y1+y2+y3+y4/2 ! (comme il est bien dit dans le dictionnaire Weisstein \340 l'entr\351e Clebsch)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "x:=(y2+y3+y4)/(6*(y1+y2+y3+y4/2)); \ny:=(y1+ y3+y4)/(6*(y1+y2+y3+y4/2));\nz:=(y1+y2+y4)/(6*(y1+y2+y3+y4/2));" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 366 "restart:\n eq1:=y1=(3+sqrt(3))/6*x+(-3+sqrt(3))/6*y+(sqrt(3))/3*z/4;\n eq2:=y2=(- 3+sqrt(3))/6*x+(3+sqrt(3))/6*y+(sqrt(3))/3*z/4;\n eq3:=y3=(-1/sqrt(3)* x-1/sqrt(3)*y+1/sqrt(3)*z/4);\neq4:=y4=(3+sqrt(3))/6*x+(-3+sqrt(3))/6* y+(sqrt(3))/3*z/4+(-3+sqrt(3))/6*x+(3+sqrt(3))/6*y+(sqrt(3))/3*z/4+(-1 /sqrt(3)*x-1/sqrt(3)*y+1/sqrt(3)*z/4)+1;\nsolve(\{eq1, eq2, eq3, eq4\} ,\{x,y,z,y4\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "#======================== Calculs et r\351sultats ; tra c\351s ==========================" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "solve( \{x=(y2+y3+y4)/(6*(y1+y2+y3+y4/2)), y=(y1+y3+y4)/(6*(y1+y2+y3+y4/2)), \n z=(y1+y2+y4)/(6*(y1+y2+y3+y4/2))\},\{y1,y2,y3,y4\}):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "y4 := -(6*x+6*y+6*z-2)/(6^(1/3)): \+ \ny3 :=simplify( 1/2*y4*(-9*z+3*y+3*x+1)/(3*z+3*y+3*x-1)): \ny1 := sim plify(-1/2*y4*(-3*z-1-3*y+9*x)/(3*z+3*y+3*x-1)): \ny2 := simplify(1/2* (3*z+1-9*y+3*x)*y4/(3*z+3*y+3*x-1)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "S:=sort( expand(y1^3+y2^3+y3^3+y4^3-(y1+y2+y3+y4)^3),[x,y,z]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 " Pg de Giogio Ferrarese" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 " Giorgio Ferrarese\nDi partimento di Matematica\n Universita' di Torino\n Via Carlo Alberto 1 0\n 10123 Torino\n\n Tel. 011 670 2908\n E-mail: giorgio.ferrarese@uni to.it" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "B1:=spacecurve([sqrt(3)/3+(3-5*sqr t(3))/12*t,sqrt(3)/3+(-3-5*sqrt(3))/12*t,-4/sqrt(3)+1/(2*sqrt(3))*4*t] ,t=-0.7..3.7,color=blue):#XXXXXAB\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "B2:=spacecurve([(-1+sqrt(3))/2*t,(1+sqrt(3))/2*t,0],t=-2.3..2.6,co lor=blue):#XXXXXOE\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "B3:=spacec urve([sqrt(3)/6+(3-2*sqrt(3))/6*t,sqrt(3)/6+(-3-2*sqrt(3))/6*t,-4/(2*s qrt(3))-4/(2*sqrt(3))*t],t=-1..1,color=blue):#XXXXXBC" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "B4:=spacecurve([sqrt(3)/3+(-3-5*sqrt(3))/12*t,s qrt(3)/3+(3-5*sqrt(3))/12*t,-4/(sqrt(3))+4/(2*sqrt(3))*t],t=-0.7..3.7, color=blue):#XXXXXAD2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "B5:=spacec urve([(1+sqrt(3))/2*t,(-1+sqrt(3))/2*t,0],t=-2..2.5,color=blue):#XXXXX OE2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "B6:=spacecurve([sqrt(3)/6+( -3-2*sqrt(3))/6*t,sqrt(3)/6+(3-2*sqrt(3))/6*t,-4/(2*sqrt(3))-4/(2*sqrt (3))*t],t=-2.5..2,color=blue):#XXXXXBC2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "B7:=spacecurve([(-3-sqrt(3))/6+(9+sqrt(3))/12*t,(3-sqrt(3))/6 +(-9+sqrt(3))/12*t,-4/sqrt(3)+4/(2*sqrt(3))*t],t=-2..3,color=blue):#XX XXXB3C2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "B8:=spacecurve([(3-sqrt (3))/6+(-9+sqrt(3))/12*t,(-3-sqrt(3))/6+(9+sqrt(3))/12*t,-4/sqrt(3)+4/ (2*sqrt(3))*t],t=-2..2.5,color=blue):#XXXA3D2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "B9:=spacecurve([t,-t,0],t=-1..1,color=blue):#XXXXXOE3 " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "B10:=spacecurve([(-4*sqrt(3))/ 6+(-3+3*sqrt(3))/6*t,(-4*sqrt(3))/6+(3+3*sqrt(3))/6*t,-4/sqrt(3)],t=-1 .5..2,color=blue):#XXXXP3P4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "B11 :=spacecurve([(3-sqrt(3))/6+((3-sqrt(3))/6+(4*sqrt(3))/6)*t,(-3-sqrt(3 ))/6+((-3-sqrt(3))/6+(4*sqrt(3))/6)*t,-4/sqrt(3)],t=-2.5..2,color=blue ):#XXXXP2P5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "B12:=spacecurve([(3 +sqrt(3))/3+((3+sqrt(3))/3-sqrt(3)/3)*t,(-3+sqrt(3))/3+((-3+sqrt(3))/3 -sqrt(3)/3)*t,-4/sqrt(3)],t=-2.4..0.4,color=blue):#XXXXP1P6" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "B13:=spacecurve([(-3+sqrt(3))/6+((-3+sqr t(3))/6-sqrt(3)/6)*t,(3+sqrt(3))/6+((3+sqrt(3))/6-sqrt(3)/6)*t,-4/(2*s qrt(3))],t=-6.5..4.5,color=blue):#XXXXP1P6" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "B14:=spacecurve([(-3-sqrt(3))/12+((-3-sqrt(3))/12+sq rt(3)/3)*t,(3-sqrt(3))/12+((3-sqrt(3))/12+sqrt(3)/3)*t,-4/(2*sqrt(3))] ,t=-5..5,color=blue):#XXXXP3P4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 143 " B15:=spacecurve([-sqrt(3)/3+((3-sqrt(3))/12+sqrt(3)/3)*t,-sqrt(3)/3+(( -3-sqrt(3))/12+sqrt(3)/3)*t,-4/(2*sqrt(3))],t=-4..6,color=blue):#XXXXP 2P5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "A1:=spacecurve([-7/4*t-1/4- 1/4*sqrt(5)-3/4*sqrt(5)*t+1/4*sqrt(3)*t-1/4*sqrt(3)-1/4*sqrt(5)*sqrt(3 )-3/4*sqrt(5)*sqrt(3)*t,1/4-7/4*t-3/4*sqrt(5)*t+1/4*sqrt(5)-1/4*sqrt(3 )-1/4*sqrt(3)*t+3/4*sqrt(5)*sqrt(3)*t-1/4*sqrt(5)*sqrt(3),4*t],t=-0.8. .0.2,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "A2:=spacecurve ([3/4*sqrt(5)*t+1/2-1/2*sqrt(5)+5/4*t-3/4*sqrt(5)*sqrt(3)*t+3/4*sqrt(3 )*t,5/4*t+3/4*sqrt(5)*t-1/2+1/2*sqrt(5)+3/4*sqrt(5)*sqrt(3)*t-3/4*sqrt (3)*t,4*t],t=-0.8..0.6,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "A3:=spacecurve([1/4*sqrt(5)-7/4*t+3/4*sqrt(5)*t-1/4+1/4*sqrt(5)*s qrt(3)+1/4*sqrt(3)*t+3/4*sqrt(5)*sqrt(3)*t-1/4*sqrt(3),1/4-7/4*t+3/4*s qrt(5)*t-1/4*sqrt(5)-1/4*sqrt(3)*t-1/4*sqrt(3)-3/4*sqrt(5)*sqrt(3)*t+1 /4*sqrt(5)*sqrt(3),4*t],t=-0.8..0.7,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "A4:=spacecurve([1/2+5/4*t+1/2*sqrt(5)-3/4*sqrt(5)*t+ 3/4*sqrt(5)*sqrt(3)*t+3/4*sqrt(3)*t,5/4*t-3/4*sqrt(5)*t-1/2-1/2*sqrt(5 )-3/4*sqrt(5)*sqrt(3)*t-3/4*sqrt(3)*t,4*t],t=-0.7..0.3,color=red):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "A5:=spacecurve([5/4*t-1/2-1/2*sqrt (5)-3/4*sqrt(5)*t-3/4*sqrt(3)*t-3/4*sqrt(5)*sqrt(3)*t,5/4*t-3/4*sqrt(5 )*t+1/2+1/2*sqrt(5)+3/4*sqrt(5)*sqrt(3)*t+3/4*sqrt(3)*t,4*t],t=-0.7..0 .3,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "A6:=spacecurve([ 1/2*t-3/2*sqrt(5)*t-1/4*sqrt(5)+1/4+sqrt(3)*t+1/4*sqrt(3)-1/4*sqrt(5)* sqrt(3),-1/4+1/2*t-3/2*sqrt(5)*t+1/4*sqrt(5)-1/4*sqrt(5)*sqrt(3)+1/4*s qrt(3)-sqrt(3)*t,4*t],t=-0.7..0.6,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "A7:=spacecurve([5/4*t+1/2*sqrt(5)-1/2+3/4*sqrt(5)*t+ 3/4*sqrt(5)*sqrt(3)*t-3/4*sqrt(3)*t,5/4*t+1/2-1/2*sqrt(5)+3/4*sqrt(5)* t-3/4*sqrt(5)*sqrt(3)*t+3/4*sqrt(3)*t,4*t],t=-0.7..0.5,color=red):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "A8:=spacecurve([3/2*sqrt(5)*t+1/2* t+1/4*sqrt(5)+1/4+sqrt(3)*t+1/4*sqrt(5)*sqrt(3)+1/4*sqrt(3),-1/4+1/2*t -1/4*sqrt(5)+3/2*sqrt(5)*t-sqrt(3)*t+1/4*sqrt(3)+1/4*sqrt(5)*sqrt(3),4 *t],t=-0.75..0.16,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "A 9:=spacecurve([-1/4+1/2*t-3/2*sqrt(5)*t+1/4*sqrt(5)+1/4*sqrt(3)-sqrt(3 )*t-1/4*sqrt(5)*sqrt(3),1/4+1/2*t-3/2*sqrt(5)*t-1/4*sqrt(5)+1/4*sqrt(3 )+sqrt(3)*t-1/4*sqrt(5)*sqrt(3),4*t],t=-0.7..0.6,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "A10:=spacecurve([-7/4*t+1/4-1/4*sqrt(5)+ 3/4*sqrt(5)*t-1/4*sqrt(3)*t-1/4*sqrt(3)+1/4*sqrt(5)*sqrt(3)-3/4*sqrt(5 )*sqrt(3)*t,-7/4*t+3/4*sqrt(5)*t+1/4*sqrt(5)-1/4+1/4*sqrt(3)*t+1/4*sqr t(5)*sqrt(3)+3/4*sqrt(5)*sqrt(3)*t-1/4*sqrt(3),4*t],t=-0.5..0.6,color= red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "A11:=spacecurve([-1/4+1/2 *t+3/2*sqrt(5)*t-1/4*sqrt(5)-sqrt(3)*t+1/4*sqrt(3)+1/4*sqrt(5)*sqrt(3) ,1/4+1/2*t+3/2*sqrt(5)*t+1/4*sqrt(5)+1/4*sqrt(3)+sqrt(3)*t+1/4*sqrt(5) *sqrt(3),4*t],t=-0.7..0.2,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 255 "A12:=spacecurve([-7/4*t+1/4-3/4*sqrt(5)*t+1/4*sqrt(5)-1/4*sqrt( 3)*t-1/4*sqrt(3)+3/4*sqrt(5)*sqrt(3)*t-1/4*sqrt(5)*sqrt(3),-7/4*t-3/4* sqrt(5)*t-1/4*sqrt(5)-1/4+1/4*sqrt(3)*t-3/4*sqrt(5)*sqrt(3)*t-1/4*sqrt (5)*sqrt(3)-1/4*sqrt(3),4*t],t=-0.7..0.1,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "A:=implicitplot3d( x^3 + y^3 -124*(1/4*z)^3 +6* x^2*(1/4*z)-3*x^2*y-3*x*y^2-18*(1/4*z)-54*sqrt(3)*(1/4*z)^2+6*y^2*(1/4 *z),x=-3..3,y=-3..3,z=-3..3,scaling=CONSTRAINED,grid=[20,20,20]):#,col or=aquamarine(enlev\351 par Esculier)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 180 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xG%\"yGQ!Fhd[l-%(SCALINGG 6#%,CONSTRAINEDG-%+LIGHTMODELG6#%(LIGHT_2G-%&STYLEG6#%,PATCHNOGRIDG" 1 3 0 1 10 0 2 1 4 1 1 1.000000 30.000000 75.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15 " "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "C urve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 58 " repr\351sentations graphiques de surfaces seul es( \340 sauter )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "with(p lots):\n# ======= cubique 1 (Clebsch) =================\n# \351quation la plus fr\351quente sur le net\nS:=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)+54*x*y*z+\n 126*(x*y+x*z+ y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1 ;\n#H4:=sort(expand(subs(z=z+1/3,S)/3 ),[x,y,z]);\nH4 := 27*x^3-63*x^2*y-63*x^2*z-63*x*y^2+18*x*y*z-63*x*z^2 +27*y^3-63*y^2*z\n -63*y*z^2+27*z^3-24*x^2+48*x*y-24*y^ 2+24*z^2+4*x+4*y+4*z;" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "# ======= cubique 2 (Clebsch)=================\nH3:=16*x^3+16*y ^3-31*z^3+24*x^2*z-48*x^2*y-48*x*y^2-72*z-54*sqrt(3)*z^2+24*y^2*z; #To desco\n# ======= cubique 3 (Clebsch)=================\n#Hred:=simplify (expand(sqrt(3)/9*subs(X=X*sqrt(6),Y=Y*sqrt(6),Z=Z*sqrt(3),16*sqrt(2)* X^3+24*X^2*Z-48*sqrt(2)*X*Y^2+24*Y^2*Z-31*Z^3-54*sqrt(3)*Z^2-72*Z))); \nHred := 64*X^3+48*X^2*Z-192*Y^2*X+48*Y^2*Z-31*Z^3-54*Z^2-24*Z;\n# == ===== cubique 4 (Esculier) =================\nS:=-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;" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 183 "with(plots):\na:=2.5:\nimplicitplot3d(Hred, X=-a..a,Y=-a..a,Z=- a..a/2,grid=[40,40,40],\n lightmodel=light2,style=p atchnogrid,scaling=constrained,title=\"vue g\351n\351rale\");" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "implicitplot3d(Hred, X=-0.8..0.8,Y =-0.8..0.8,Z=-2..-1,grid=[20,20,20],\n lightmodel=l ight2,style=patchnogrid,scaling=constrained,title=\"trou central inf \351rieur\");\nimplicitplot3d(Hred, X=-1..3,Y=-0.5..0.5,Z=-3..-0.7,gri d=[20,20,20],\n lightmodel=light2,style=patchnogrid ,scaling=constrained,\n orientation=[90,50],title= \"un trou lat\351ral et le trou central inf\351rieur\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "implicitplot3d(S,x=-1..1,y=-1..1,z =-1..1,scaling=constrained,grid=[40,40,40],style=patchnogrid,axes=box, orientation=[-20,100],lightmodel=light2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 133 "implicitplot3d(H4,x=-1..1.5,y=-1..1.5,z=-1..1.5,sc aling=constrained,grid=[50,50,50],style=patchnogrid,axes=box,orientati on=[50,120]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "implicitp lot3d(H3,x=-3..3,y=-3..3,z=-4..4,scaling=constrained,grid=[50,50,50],s tyle=patchnogrid,axes=box,orientation=[50,70]);" }{TEXT -1 0 "" } {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 " Repr\351sentation graphique de c ubiques contenant 27 droites " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 " # ---------------------------------------------" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "restart:\n#================= Cubique 1 (Cleb sch) =====================\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)+54*x*y*z+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y +z)+1 ;\n" }{TEXT -1 106 "#==================Cubique 2 (Clebsch)====== ================ \n# forme la plus concise tir\351e de Todesco : " } {XPPEDIT 18 0 "64*x^3-192*x*y^2+48*x^2*z+48*y^2*z-31*z^3-54*z^2-24*z; " "6#,0*&\"#k\"\"\"*$%\"xG\"\"$F&F&*(\"$#>F&F(F&%\"yG\"\"#!\"\"*(\"#[F &*$F(F-F&%\"zGF&F&*(F0F&*$F,F-F&F2F&F&*&\"#JF&*$F2F)F&F.*&\"#aF&*$F2F- F&F.*&\"#CF&F2F&F." }{TEXT -1 7 " " }{MPLTEXT 1 0 693 " \n#S:=64 *x^3+48*x^2*z-192*y^2*x+48*y^2*z-31*z^3-54*z^2-24*z;\n#=============== == Cubique 3 (Clebsch) ======================\n# \351quation (9 termes ) du site de Todesco ( les calculs bloquent pour les horizontales)\n# \+ non utilisable== S:=-72*z-31*z^3+24*x^2*z+24*y^2*z+16*y^3-48*x^2*y+16* x^3-54*sqrt(3)*z^2-48*x*y^2;\n#=============== Cubique 4 ( Esculier ) \+ ====================\nS := -18*x^3-36*x^2*z+18*x*y^2-18*x*z^2+18*z^3-1 2*x^2-12*x*z-3*z^2-2*x-z;\n#========================================== =====\nresz:=subs(x=a*z+p,y=b*z+q,S):# droites \340 z non cst\nresx:=s ubs(y=a*x+p,z=b*x+q,S):# droites \340 x non cst \nresx0:=subs(x=p,z=q, S):# droites \340 x cst et z cst\nHx:=collect(resx,x):\nHx0:=collect(r esx0,y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Hz:=collect(resz,z):" } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "Cz:=[seq(coeff(Hz ,z,i),i=0..3)]:\nCx:=[seq(coeff(Hx,x,i),i=0..3)]:\nCx0:=[seq(coeff(Hx0 ,y,i),i=0..3)]:\nsolutionsz :=map(allvalues,\{solve(\{seq(Cz[i]=0,i=1. .4)\},\{a,b,p,q\})\}):\nsolz:=map(u -> subs(op(u),[a*z+p,b*z+q,z]),sol utionsz);\nprintf(\"%s %a\\n\",\"nbre droites non horizontales = \",no ps(solz));\nsolutionsx0 :=map(allvalues,\{solve(\{seq(Cx0[i]=0,i=1..4) \},\{p,q\})\}):\nsolx0:=map(u -> subs(op(u),[p,y,q]),solutionsx0);\npr intf(\"%s %a\\n\",\"nbre droites horizontales avec x cste = \",nops(so lx0));\nsolutionsx :=map(allvalues,\{solve(\{seq(Cx[i]=0,i=1..4)\},\{a ,b,p,q\})\}):\n# ==== j'\351limine les solutions qui sont \340 z non \+ constant =========\nsolutionsxbis:=NULL:\nfor i to nops(solutionsx) do \n if member(b=0 , solutionsx[i]) then solutionsxbis:=solutionsxbis ,solutionsx[i] fi; od:\nsolutionsxbis:=[solutionsxbis]:\nsolx:=map(u - > subs(op(u),[x,a*x+p,b*x+q]),solutionsxbis);\nprintf(\"%s %a\\n\", \" nbre droites horizontales avec x non cste = \",nops(solx));\nprintf(\" %s %a\\n\",\"nombre de droites contenues dans la surface = \",nops(sol x)+nops(solz)+nops(solx0));" }{TEXT -1 0 "" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,6*&\"#=\"\"\")%\"xG\"\"$F(!\"\"*(\"#O F()F*\"\"#F(%\"zGF(F,*(F'F(F*F()%\"yGF0F(F(*(F'F(F*F()F1F0F(F,*&F'F()F 1F+F(F(*&\"#7F(F/F(F,*(F:F(F*F(F1F(F,*&F+F(F6F(F,*&F0F(F*F(F,F1F," }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solzG<47%,(*&,&#\"\"(\"\"'!\"\"*&F, F-\"#P#\"\"\"\"\"#F1F1%\"zGF1F1#F+\"#OF-*&F5F-F/F0F1,(*&,&#F2\"\"$F-*& F;F-F/F0F-F1F3F1F1#\"\"&F5F1*&F5F-F/F0F1F37%,(*&,&#F+F,F-*&F,F-F/F0F-F 1F3F1F1#F+F5F-*&F5F-F/F0F-,(*&,&#F2F;F1*&F;F-F/F0F-F1F3F1F1#F>F5F-*&F5 F-F/F0F1F37%*&,&#F1\"\"%F-*&FSF-F/F0F1F1F3F1,(*&,&#\"#>\"#7F1*&FZF-F/F 0F1F1F3F1F1#F1\"#=F1*&FgnF-F/F0F1F37%*&,&#F1FSF-*&FSF-F/F0F-F1F3F1,(*& ,&FXF1*&FZF-F/F0F-F1F3F1F1FfnF1*&FgnF-F/F0F-F37%FP,(*&,&#FYFZF-*&FZF-F /F0F-F1F3F1F1#F1FgnF-*&FgnF-F/F0F-F37%Fjn,(*&,&#FYFZF-*&FZF-F/F0F1F1F3 F1F1#F1FgnF-*&FgnF-F/F0F1F37%,$*&F2F-F3F1F-,$*(F;F1F2F-F3F1F1F37%Fdp,$ *(F;F1F2F-F3F1F-F37%,&F3F-#F1F,F-,&F3F1#F1F,F-F37%,&*&F2F-F3F1F1#F1F,F -,&*&F2F-F3F1F-#F1F,F-F37%F\\q,&F3F-#F1F,F1F37%Faq,&*&F2F-F3F1F1FiqF1F 37%,(*&,&#F>FZF1*&FZF-F/F0F-F1F3F1F1#F>F5F-*&F5F-F/F0F1,(*&,&#\"#6FZF1 *(F>F1FZF-F/F0F1F1F3F1F1#F+F5F1*&F5F-F/F0F1F37%F',(*&,&FKF1*&F;F-F/F0F 1F1F3F1F1#F>F5F-*&F5F-F/F0F-F37%FA,(*&,&#F2F;F-*&F;F-F/F0F1F1F3F1F1F=F 1*&F5F-F/F0F-F37%F^r,(*&,&#FirFZF-*(F>F1FZF-F/F0F-F1F3F1F1#F+F5F-*&F5F -F/F0F-F37%,(*&,&FarF1*&FZF-F/F0F1F1F3F1F1#F>F5F-*&F5F-F/F0F-,(*&,&#Fi rFZF-*(F>F1FZF-F/F0F1F1F3F1F1#F+F5F-*&F5F-F/F0F1F37%Fdt,(*&,&FhrF1*(F> F1FZF-F/F0F-F1F3F1F1F[sF1*&F5F-F/F0F-F3" }}{PARA 6 "" 1 "" {TEXT -1 35 "nbre droites non horizontales = 18" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solx0G<%7%\"\"!%\"yGF'7%F'F(#!\"\"\"\"'7%F'F(#\"\"\"\"\"$" }} {PARA 6 "" 1 "" {TEXT -1 42 "nbre droites horizontales avec x cste = \+ 3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solxG7(7%%\"xG,&F'\"\"\"#\"\"# \"\"$F)#F)F,7%F',&F'!\"\"#F+F,F0F-7%F',&F'F)#F)\"\"'F)#F0F57%F',&F'F0# F)F5F0F67%F',&F'F0#F)F,F0\"\"!7%F',&F'F)F-F)F=" }}{PARA 6 "" 1 "" {TEXT -1 46 "nbre droites horizontales avec x non cste = 6" }}{PARA 6 "" 1 "" {TEXT -1 49 "nombre de droites contenues dans la surface = \+ 27" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 599 "with(plots):\n#deb:= -1.5: fin :=1.5:debz:=-3:finz:=2: #pour la 2-i\350me forme\ndeb:=-1: f in :=1:debz:=-1:finz:=1:# pour la 4-i\350me\n#deb:=-2: fin := 2.5:debz :=-2:finz:=2.5: pour la 1-\351re forme\nr:=0.005*(finz-debz);\ng_droit es:= tubeplot(\{op(solz)\},radius=r,z=debz..finz,numpoints=2, color=ma genta,style=patchnogrid),\n tubeplot(\{op(solx)\},radius=r, x=deb..fin,numpoints=2, color=black,style=patchnogrid), \n \+ tubeplot(\{op(solx0)\},radius=r,y=deb..fin,numpoints=2, color=black,st yle=patchnogrid):\nsurf:=implicitplot3d(S,x=deb..fin,y=deb..fin,z=debz ..finz,style=patchnogrid,grid=[50,50,50]):" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"#5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "plotsetup(window);\n#plotsetup(jpeg,plotoutput=\"c:/ clebsch.jpg\",plotoptions=\"height=600, width=600\"):\ndisplay([surf,g _droites],scaling=constrained,view=[deb..fin,deb..fin,debz..finz],\n \+ orientation=[60,70],light=[30,40,0.9,0.9,0],light=[120,20,0.8,0. 6,0]);\n#orientation=[50,120] pour la 1-\351re forme\n#lightmodel=ligh t2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 63 " E ssai de repr\351sentation param\351trique et explication de l'\351chec " }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 " Mails de Vidiani " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 938 "Mail 1 de LGV- 20/06/03\nrereBonjour (:- ) 20 Juin 2003 11h06\ninterpr\351tation des solutions de parasites d e la recherche des droites\nhorizontales :\nelles donnent une repr\351 sentation param\351trique que fonction des param\350tres\np,q de la su rface de clebsch (reste \340 savoir si c'est tout ou une partie)\nPour obtenir une repr\351sentation param\351trique rationnelle, voici comm ent (ce\nn'est pas encore fait) je compte faire :\nChoisir deux droite s D1 D2 de la cubique, non horizontales\n(parmi la liste des 22 soluti ons je prendrais les plus simples)\nun plan horizontal par exemple z=0 et y repr\351rer une point P(x=u;z=v)\nPar P faire passer la droite q ui rencontre D1 et D2 (il y en a une seule)\n\nElle rencontre la cubiq ue en trois points dont deux sont sur D1 et D2\nrespectivement, et le \+ troisi\350me est M(u,v) ce qui donnera une\nrepr\351sentation param \351trique rationnelle de la surface de Clebsch\n(originalit\351 mondi ale de LGV pour quadrature)\n@+\nLGV VIDIANI" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 23 "mail2 de LGV - 23/06/03" }}{PARA 0 "" 0 "" {TEXT -1 4150 "Bonjour (:-) Juin 2003\n@+\nLGV VIDIANI\nhttp://www.webmaster -jsc.com/vidiani/\n64 rue de l'Europe 21121\nFontaine Les Dijon\ntel : 03 80 56 65 53 + 03 85 45 37 08\n----- Message d'origine -----\nDe : \+ \"Giorgio Ferrarese\" \n\300 : \"VIDIANI\" \nEnvoy\351 : lundi 23 juin 2003 13:53\nObjet : R e: Clesbsch-Juel fichier gif ; Representation parametrique ?\n\n\nHave a good time, I received your attaches. Many thanks to Alain, who has \nclarified the problems with parametric plotting very well.\nA presto \ngiorgio\n\n\n\nAt 13.38 23/06/2003 +0200, you wrote:\n\nBonjour (:-) 23 Juin 2003 13h30\nMerci je vais analyser tout cela (\340 l'ombre ) car je pars quelques jours\ndans la famille pr\350s de Saint Etienne , via La saone et Loire (ma maisond e\nCampagne dans le payx originair e de la souche de ma famille) et Charlieu\n(pr\350s de Roanne)\nJ'empo rte les tirages papiers pour r\351fl\351chir ;\nEn attendant voici ce \+ que m'a envoy\351 mon coll\350gue et ami (tr\350s fort en\nMaple)\nAla in Esculier avec lequel je travaille souvent (il m'a souvent indiqu \351 de\nbelles astuces)\n\njuelpara.mws est mon programme avec des ta s de v\351rifications annexes et\ninterm\351diaires de v\351rification \n0000clebsch.mws est celui d'Esculier qui explique pourquoi le param \351trage\nne peut marcher (h\351las) : voici ce qu'il m'\351crivais c e matin :\n----------------------------------------------------------- ----------------\n\nJ'ai r\351fl\351chi \340 ta repr\351sentation para m\351trique : \340 part les points des deux\nplans passant par l'une d es droites et parall\350les \340 l'autre, tous les\npoints\nde la surf ace admettent bien un couple (u,v) de repr\351sentant et un seul mais \ncette repr\351sentation param\351trique est inexploitable pour trace r la surface\ncar par exemple pour u fix\351, quand v varie, on passe \+ d'une nappe de la\nsurface \340 une autre d'o\371 les parties de plans parasites.\nSi l'on fait tracer la surface en mode points alors on n' a plus les arcs\nparasites mais il n'est pas ais\351 d'avoir des point s r\351partis sur toute la\nsurface et le r\351sultat n'est pas terrib le du tout.\nOn obtient le m\352me ph\351nom\350ne avec par exemple un hyperbolo\357de ( voir\nl'exemple trait\351 \340 la fin du fichier jo int) ; c'est plus simple de\nvisualiser la cause des plans parasites. \n\nJ'ai am\351lior\351 la repr\351sentation de la surface et ses 27 d roites : je suis\nassez content du r\351sultat !\n\nJ'ai regard\351 au ssi : on peut exprimer changer d'axes et construire la\nsurface comme \+ Y=f(X,Z) mais \347a pose un certain nombre de probl\350mes pour la\nre pr\351sentation !\n\nTu trouveras tout cela dans le fichier joint.\n-- ------------------------------------------------\n\n\n@+\nLGV VIDIANI \nhttp://www.webmaster-jsc.com/vidiani/\n64 rue de l'Europe 21121\nFon taine Les Dijon\ntel : 03 80 56 65 53 + 03 85 45 37 08\n----- Message \+ d'origine -----\nDe : \"Giorgio Ferrarese\" \n\300 : \"VIDIANI\" \nEnvoy\351 : lundi 23 juin 2003 12:06\nObjet : Re: Clesbsch-Juel fichier gif ; Representation pa rametrique ?\n\n\nAt 07.09 23/06/2003 +0200, you wrote:\n>\n>Bonjour ( :-) 23 Juin 2003 06h56\n>\n>Quel est \340 votre avis le choix des m eilleurs droites pour appliquer votre\n>m\351thode pour obtenir une re pr\351sentation param\351trique donnant un trac\351\n>correct (avec Ma ple) de la surface de Clebsch.\n>\n>J'ai obtenu quelque chose, mais au trac\351 il y a des tas de morceaux de\nplans\n>parasites qui sugg \350rent des facteurs communs non simplifi\351s ou des termes\n>nuls a u d\351nominateur comme u+v+1/3 ? (voir les calculs sur le fichier *.p s\n>joint.\n\nI had the same problems, some years ago, when I tried to use parametric\nequations to plot the Diagonal surface. At the time I gave up, maybe now\nI'll try again a little bit. I'll let you know, i f something good will come\nout.\nFor your convenience, I attach a Map le file with the equations of the 27\nlines of Todesco's Clebsch.\n\n Bye\ngiorgio\n\n ----------------------------------------------------- --------------------\n Giorgio Ferrarese\n Dipartimento di Matematica \n Universita' di Torino\n Via Carlo Alberto 10\n 10123 Torino\n\n Tel . 011 670 2908\n E-mail: giorgio.ferrarese@unito.it\n\nAttachment Conv erted: \"C:\\Posta\\Ferrarese\\Attachments\\0000Clebsch.mws\"\n\nAttac hment Converted: \"C:\\Posta\\Ferrarese\\Attachments\\juelpara.mws\"\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "juelpara.mws zip 20 6 0 3 19h50 Vidiani Dijon version 20 6 03 20h10" }}}{EXCHG {PARA 0 "" 0 " " {MPLTEXT 0 21 616 "Je prend comme droites D1 x+y=2/3 z=1/3 et D2 x=- 1/3 y=-z ; simples ne se rencontrant pas et non contenues dans le plan z=0 ; je prend un point P(u,v,0) dans ce plan ; je d\351termine les p lans (par faisceaux) PD1 et PD2, ils d\351terminent une droite que je \+ param\350tre (par z) ; elle rencontre la surface cubique en deux point s d\351ja connus (ceux sur D1 et D2) : il reste un troisi\350me point, ce qui fournit le param\351trage de la surface de Clebsch sous la for me de Hunt, en prem\350re mondiale. Esculier trouvera certainement des m\351thodes beaucoup plus fut\351es que mes copier coller pour report er les x et les y dans les solve !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "solve( \{x+y-2/3+(3*(u+v)-2)*(z-1/3)=0,v*(x+1/3)-(u+1/3)*(y+z)\},\{x,y\});" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "solve(\{-( 9*v*u*z-3*v*u+3*v*z-6*u*z-3*z+9*u^2*z-3*u^2-u)/(3*v+3*u+1)-(9*v*u*z-3* v*u+9*v^2*z-3*v^2-6*v*z-v+3*u*z+z)/(3*v+3*u+1)=2/3\},\{z\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "solve(\{-1/3=-(9 *v*u*z-3*v*u+3*v*z-6*u*z-3*z+9*u^2*z-3*u^2-u)/(3*v+3*u+1),-(9*v*u*z-3* v*u+9*v^2*z-3*v^2-6*v*z-v+3*u*z+z)/(3*v+3*u+1)=-z\},\{z\});" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 37 "d onc elle rencontre les deux droites " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "H:=(x, y,z)->126*z*x+1-189*z*y^2-189*z*x^2-189*y^2*x+126*z*y-9*x^2-189*z^2*y- 189*z^2*x+126*y*x-9*y^2+81*z^3-189*y*x^2+81*y^3+81*x^3-9*x+54*z*y*x-9* z-9*y-9*z^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "F:=simplify(H(-(9*v*u*z-3*v*u+3*v*z-6*u*z-3*z+9*u^2*z-3*u^2-u)/(3 *v+3*u+1),-(9*v*u*z-3*v*u+9*v^2*z-3*v^2-6*v*z-v+3*u*z+z)/(3*v+3*u+1),z )):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(\{F=0\},\{z\}) ;" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 72 "Les deux premi\350res correspondent aux rencontre av ec les droites D1 et D2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "zc:=simplify( 1/3*(1-6*u-6*v+72*v*u-36*v^2-36*u^2-324*v^3*u-1134*v^2* u^2+162*v^2*u-324*v*u^3+162*v*u^2+243*u^4+54*u^3+54*v^3+243*v^4)/(243* u^4-324*v*u^3-324*u^3-1134*v^2*u^2+756*v*u^2-198*u^2-324*v^3*u+756*v^2 *u-252*v*u+12*u+378*v^2+11+243*v^4-180*v-324*v^3)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "interface(labelling=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "xc:=simplify(-(9*v*u*zc-3*v*u+3*v*z c-6*u*zc-3*zc+9*u^2*zc-3*u^2-u)/(3*v+3*u+1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "yc :=simplify(-(9*v*u*zc-3*v*u+9*v^2*zc-3*v^2-6*v*zc-v+3*u*zc+zc)/(3*v+3* u+1)):" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "a:=5;b:=5;\nplot3d([xc, yc,zc],u=-a..a,v=-b..b,grid=[200,200],scaling=constrained,view=[-1..1, -1..1,-1..1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 273 "#les repr\351sentations obtenues \+ ne sont pas \"continues\" : on passe d'une courbe d'une nappe \340 une courbe d'une autre nappe\n# adaptation du pg de LGV (plus simple de p rendre les points courants M(u) et N(v) sur chaque droite et chercher \+ l'intersection de MN avec la surface) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "droite:=NULL:\nfor i from 1 to nops(solutions1bis) do \n \+ droite:=droite,subs(op(solutions1bis[i]),[x,a*x+p,b*x+q]); od:\ndroite :=[droite];\ndroite:=NULL:\nfor i from 1 to nops(solutions) do \n \+ droite:=droite,subs(op(solutions[i]),[a*z+p,b*z+q,z]);\nod:\ndroite:=[ droite];" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "# " }{TEXT -1 0 "" }{MPLTEXT 1 0 257 "d1:=[u, -u, -1/3]: d2:= [0, -v+1/3 , v]:\nd1:= [u, -u, -1/3]: d2:=[-v+1/3, 0, v]:\nduv:=expand(d1+t*(d2-d 1)) :\nH:=81*x^3-189*x^2*y-189*x^2*z-189*x*y^2+54*x*y*z-189*x*z^2+81*y ^3-189*y^2*z-189*y*z^2+81*z^3-9*x^2+126*x*y+126*x*z-9*y^2+126*y*z-9*z^ 2-9*x-9*y-9*z+1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "res:=[solve(sub s(x=duv[1],y=duv[2],z=duv[3],H),t)]:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "para:=simplify(subs(t=res[3],duv));" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "#para1:=map2(collect,si mplify(subs(u=a*U+b*v+1/6,v=c*U+d*v,para)),[U,V]);" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " # r\351ciproque\nsol:=[solve(\{x=para[1],z=para[3]\},\{u,v\})];nops(so l);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "nops( sol);\nsimplify(sol[1]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 479 "w:=0.8:\nplotsetup(inline):\n#======= ===============================================\n# avec \"style=point \", on obtient que la surface ( pas tr\350s belle et ... \n# moins les points dans les plans // passant par chaque droite)\n# avec \"style=p atch\", on obtient une multitude de bouts de plans parasites\n#======= =============================================== \nplot3d(para,u=-2..1. 5,v=-0.8..1.2,scaling=constrained,grid=[200,200],view=[-w..w,-w..w,-w. .w],style=point,orientation=[180,0]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot3d(para,u=-20..20,v=-20..20,scaling= constrained,grid=[500,500],view=[-w..w,-w..w,-w..w],style=point,orient ation=[180,0]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 77 "# ================m\352me probl\350me avec l'hyper bolo\357de (H) ====================" }}{PARA 0 "" 0 "" {TEXT -1 162 " # on comprend plus ais\351ment d'o\371 vient le probl\350me car plus s imple de voir que l'on passe \n# pour certains u fix\351s, d'un c\364t \351 de la nappe \340 l'autre quand v varie" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "restart:\ns:=x^2+y^2-z^2-1; # (H)\nd1:=[1,u,u]; d2:=[2,v,0];\nduv:=expand(d1+t*(d2-d1)):\nsol:=[solve(subs(x=duv[1],y =duv[2],z=duv[3],s),t)]:\npara:=simplify(subs(t=sol[2],duv));" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "plotsetup(inline):\nplot 3d(para,u=0..1,v=-4..4,scaling=constrained,grid=[10,80],view=[-2..2,-2 ..2,-2..2]);\nplot3d(para,u=0..1.2,v=-4..4,scaling=constrained,grid=[1 0,80],view=[-2..2,-2..2,-2..2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 512 "#=========R\351ciproque======\n# tout point de l'hyp erbolo\357de sauf ceux tels x=1 et x=2\n# admet un couple unique (u,v) de repr\351sentant\n# g\351om\351triquement \351vident : soit un tel \+ point M de (H),\n# le plan d\351fini par d1 et M coupe toujours d2 et \+ en un\n# seul point P d'o\371 v, Q=MP(inter)d1 d'o\371 u et le couple (u,v)\nsol:=[allvalues(solve(\{X=para[1],Y=para[2]\},\{u,v\}))]:\nZ:= subs(op(sol[1]),para[3]);\n'X^2+Y^2-Z^2-1'=simplify(subs(x=X,y=Y,z=Z,s ));\nZ:=subs(op(sol[2]),para[3]);\n'X^2+Y^2-Z^2-1'=simplify(subs(x=X,y =Y,z=Z,s));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 " Rotations et dilatation : simplification des \+ \351quations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "rotOz := [co s(t)*X-sin(t)*Y, sin(t)*X+cos(t)*Y, Z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rotOzG7%,&*&-%$cosG6#%\"tG\"\"\"%\"XGF,F,*&-%$sinGF*F,%\"YGF, !\"\",&*&F/F,F-F,F,*&F(F,F1F,F,%\"ZG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "H:=81*x^3-189*x^2*y-189*x^2*z-189*x*y^2+54*x*y*z-189 *x*z^2+81*y^3-189*y^2*z-189*y*z^2+81*z^3-9*x^2+126*x*y+126*x*z-9*y^2+1 26*y*z-9*z^2-9*x-9*y-9*z+1:\n#H4:=sort(expand(subs(z=z+1/3,H)/3),[x,y, z]);\nH4:=18*z^3-z-y-x-9*y*z^2-9*y^2*z-9*x*z^2-9*x*y^2-9*x^2*z-9*x^2*y -3*z^2-6*y*z-3*y^2-6*x*z-6*x*y-3*x^2-18*x*y*z;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 327 "Hbis:=(sort@expand)(subs(x=rotOz[1],y =rotOz[2],z=rotOz[3],H)):\nH42:=(sort@expand)(subs(x=rotOz[1],y=rotOz[ 2],z=rotOz[3],H4)):\nH43:=map(simplify,collect(H42,[Z,Y,X]));\nnulX3da nsH43:=solve(coeff(H43,X,3),t);\nnulY3dansH43:=solve(coeff(H43,Y,3),t) ;\nH44:=sort(expand(map(simplify,subs(t=Pi/4,X=X*sqrt(2),Y=Y*sqrt(2),H 43))),[X,Y,Z]);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "H3:= -72*z-31*z^3+24*x^2*z+ 24*y^2*z+16*y^3-48*x^2*y+16*x^3-54*sqrt(3)*z^2-48*x*y^2;\nH3bis:=(sort @expand)(subs(x=rotOz[1],y=rotOz[2],z=rotOz[3],H3)):\nH3ter:=map(simpl ify,collect(H3bis,[Z,Y,X]));\nHter:=map(simplify,collect(Hbis,[Z,Y,X]) ):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 140 " nulX3dansH3ter:=map(radnormal,simplify([solve(coeff(H3ter,X,3),t )]));\nnulY3dansH3ter:=map(radnormal,simplify([solve(coeff(H3ter,Y,3), t)]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "# --------r\351d uite \340 7 termes ----------\nH34:=sort(simplify(subs(t=Pi/4,X=X*sqrt (6),Y=Y*sqrt(6),Z=Z*sqrt(3),H3ter)/sqrt(3)/3),[X,Y,Z]);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "sol:=[solve(H34,Y)];" }{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 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 61 " Repr\351sentation Y=f(X,Z) [ p\351nible ! ... et pas te rrible ! ]" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 437 "restart: \nwi th(plots):\n#Hred:=16*sqrt(2)*X^3+24*X^2*Z-48*sqrt(2)*X*Y^2+24*Y^2*Z-3 1*Z^3-54*sqrt(3)*Z^2-72*Z;\nrot:=(M,u) -> [M[1]*cos(u)-M[2]*sin(u),M[1 ]*sin(u)+M[2]*cos(u),M[3]]:\n#YY:=1/12*6^(1/2)*((-16*sqrt(2)*X^3-24*X^ 2*Z+72*Z+31*Z^3+54*sqrt(3)*Z^2)/(Z-2*sqrt(2)*X))^(1/2);\nYY:= -1/12*3^ (1/2)*((64*X^3+48*X^2*Z-31*Z^3-54*Z^2-24*Z)/(-Z+4*X))^(1/2);\n#simplif y(4/(48*z-192*x)*3^(1/2)*((z-4*x)*(-64*x^3-48*x^2*z+31*z^3+54*z^2+24*z ))^(1/2));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 596 "a:=2;b:=1;#a:=4;b:=4; # pour la 1-\351re\np1:=plot3d(\{[X,YY,Z],[ X,-YY,Z],rot([X,YY,Z],-2*Pi/3),rot([X,YY,Z],2*Pi/3),\n \+ rot([X,-YY,Z],-2*Pi/3),rot([X,-YY,Z],2*Pi/3)\},\n \+ X=-a..0,Z=0..b,scaling=constrained,grid=[30,30]):\np2:=plo t3d(\{[X,YY,Z],[X,-YY,Z],rot([X,YY,Z],-2*Pi/3),rot([X,-YY,Z],2*Pi/3)\n ,rot([X,YY,Z],+2*Pi/3),rot([X,-YY,Z],-2* Pi/3)\},\n X=0..a,Z=-a..0,scaling=constrained,grid=[40 ,40]):\nplotsetup(window):\ndisplay([p1,p2],view=[-a..a,-a..a,-a..b],l ightmodel=light2,style=patchnogrid,orientation=[60,60]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"bG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "a:=2.5:\nim plicitplot3d(Hred, X=-a..a,Y=-a..a,Z=-3*a/2..a/2,grid=[40,40,40],\n \+ lightmodel=light2,style=patchnogrid,scaling=constrain ed,title=\"vue g\351n\351rale\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "#plot3d(\{[X,YY,Z],[X,-YY,Z]\},X=-a..a,Z=-a..a,scali ng=constrained,grid=[40,40],view=[-a..a,-a..a,-a..a],axes=box);" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 73 " G\351n\351ralisation : des cubiques et les droites conte nues dans ces cubiques" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1379 " restart:\n#==================Cubique 1 (Clebsch)====================== \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)+54*x* y*z+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1 ;# \351quation Clebs ch la plus fr\351quente sur le net\n#vue:=[50,120]: w:=2:\n#========== ========Cubique 2 (Clebsch)======================\n# -------------- To desco------------------------\n# --------blocage pour les horizontales \n# ---- S:=16*x^3+16*y^3-31*z^3+24*x^2*z-48*x^2*y-48*x*y^2-72*z-54*s qrt(3)*z^2+24*y^2*z; \n##============= Cubique 3 (Clebsch pour k=1)== ====================\n#----------obtenue \340 partir de Todesco ------ -----\n# k<0(15dr) ,k=0(6dr), k<1(15dr),**** k =1(27dr)***** k>1 (3dr ) ?\n#S:=64*x^3+48*x^2*z-192*y^2*x+48*y^2*z-31*z^3-54*z^2-24*k*z: S:= subs(k=1,S);\n#vue:=[-50,70]: w:=2:\n#================== Cubique 4 === ===================\n#----- sur le net : ne donne pas Clebsch(24 droit es) ----------\n#------c'est la m\352me pour tout a (homoth\351tie)--- ------\nS:=expand(subs(a=1,x^3+y^3+z^3+a^3 -(x+y+z+a)^3));\nvue:=[55,1 20]: w:=2:\n#==================Cubique 5 ======================\n#---- ----- Esculier - autre cubique \340 27 droites --------\n# rotation Oz de Pi/4 \340 x^3+y^3+7*z^3+1/3^3 -(x+y+z+1/3)^3))*3;\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#vue:=[170, -100]:w:=1:\n#---------------------------------------------------\nS:= x^2+y^2+z^2-5/2*x*y*z-1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "resz:= subs(x=a*z+p,y=b*z+q,S):# droites \340 z non cst\nresx:=subs(y=a*x+p,z =b*x+q,S):# droites \340 x non cst\nresx0:=subs(x=p,z=q,S):# droites \+ \340 x cst\nHx:=collect(resx,x):\nHx0:=collect(resx0,y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Hz:=collect(resz,z):" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1705 "Cz:=[seq(coeff(Hz,z,i),i=0..3)]:\nCx:=[ seq(coeff(Hx,x,i),i=0..3)]:\nCx0:=[seq(coeff(Hx0,y,i),i=0..3)]:\nsolut ionsz :=map(allvalues,\{solve(\{seq(Cz[i]=0,i=1..4)\},\{a,b,p,q\})\}): \n# evalf(solutionsz); # affichage de tout\nprintf(\" R\351solution : \+ nombre de sol. r\351elles ou non = %a\\n\",nops(solutionsz));\n#====== ====\351limination des solutions complexes ===========\ntamp:=NULL:\nf or i to nops(solutionsz) do\nif not(has(solutionsz[i],I)) then tamp:=t amp,solutionsz[i] fi; od:\nsolutionsz:=[tamp]:\nsolz:=radnormal(map(u \+ -> subs(op(u),[a*z+p,b*z+q,z]),solutionsz)); \n# (;) affichage des dro ites \340 z non cst--------------------------------\nprintf(\"%s %a\\n \",\"Nombre de droites non horizontales = \",nops(solz));\nsolutionsx0 :=map(allvalues,\{solve(\{seq(Cx0[i]=0,i=1..4)\},\{p,q\})\}):\ntamp:= NULL:\nfor i to nops(solutionsx0) do\nif not(has(solutionsx0[i],I)) th en tamp:=tamp,solutionsx0[i] fi; od:\nsolutionsx0:=[tamp]: \nsolx0:=ma p(u -> subs(op(u),[p,y,q]),solutionsx0);\n\nprintf(\"%s %a\\n\",\"Nomb re de droites horizontales avec x cste = \",nops(solx0));\nsolutionsx \+ :=map(allvalues,\{solve(\{seq(Cx[i]=0,i=1..4)\},\{a,b,p,q\})\}):\ntamp :=NULL:\nfor i to nops(solutionsx) do\nif not(has(solutionsx[i],I)) th en tamp:=tamp,solutionsx[i] fi; od:\nsolutionsx:=[tamp]:\n\n# ==== j' \351limine les solutions qui sont \340 z non constant =========\nsolut ionsxbis:=NULL:\nfor i to nops(solutionsx) do\n if member(b=0 , solu tionsx[i]) then solutionsxbis:=solutionsxbis,solutionsx[i] fi; od:\ns olutionsxbis:=[solutionsxbis]:\nsolx:=map(u -> subs(op(u),[x,a*x+p,b*x +q]),solutionsxbis);\nprintf(\"%s %a\\n\", \"Nombre droites horizontal es avec x non cste = \",nops(solx));\nprintf(\"%s %a\\n\",\"Nombre tot al de droites contenues dans la surface = \",nops(solx)+nops(solz)+nop s(solx0));" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoo rds has been redefined\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG,B*& \"\"$\"\"\"%\"yGF(!\"\"*&F'F(%\"xGF(F**&F'F(%\"zGF(F**(F'F(F,F()F)\"\" #F(F**(F'F()F,F1F(F.F(F**(F'F(F3F(F)F(F**&F'F()F.F1F(F**(\"\"'F(F)F(F. F(F**&F'F(F0F(F**(F8F(F,F(F.F(F**(F8F(F,F(F)F(F**&F'F(F3F(F**(F'F(F)F( F6F(F**(F'F(F0F(F.F(F**(F'F(F,F(F6F(F***F8F(F,F(F)F(F.F(F*" }}{PARA 6 "" 1 "" {TEXT -1 48 " R\351solution : nombre de sol. r\351elles ou non = 16" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solzG727%,$*&\"\"#!\"\"%\" zG\"\"\"F,F,F+7%,$*&F)F,F+F,F,F,F+7%,$*&F)F,F+F,F*F*F+7%,$*&F)F*F+F,F* F*F+7%,&F+F,#\"\"$\"\"&F,#\"\"%F:F+7%,&F+F,#F9F:F*F;F+7%,&F+F*F8F,#!\" %F:F+7%,&F+F*#F9F:F*FBF+7%F,F'F+7%F,F.F+7%F*F1F+7%F*F4F+7%F;F7F+7%F;F> F+7%FBFAF+7%FBFEF+" }}{PARA 6 "" 1 "" {TEXT -1 40 "Nombre de droites n on horizontales = 16" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&solx0G7\" " }}{PARA 6 "" 1 "" {TEXT -1 47 "Nombre de droites horizontales avec x cste = 0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solxG7*7%%\"xG,$*&\" \"#!\"\"F'\"\"\"F,F,7%F',$*&F*F,F'F,F,F,7%F',$*&F*F,F'F,F+F+7%F',$*&F* F+F'F,F+F+7%F',&F'F,#\"\"$\"\"&F,#\"\"%F:7%F',&F'F,#F9F:F+F;7%F',&F'F+ F8F,#!\"%F:7%F',&F'F+#F9F:F+FB" }}{PARA 6 "" 1 "" {TEXT -1 48 "Nombre \+ droites horizontales avec x non cste = 8" }}{PARA 6 "" 1 "" {TEXT -1 55 "Nombre total de droites contenues dans la surface = 24" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "S;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"#\"\"\"F(*$)%\"yGF'F(F(*$)%\"zGF'F(F(*&# \"\"&F'F(*(F&F(F+F(F.F(F(!\"\"F(F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 12 "with(plots):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 487 "deb:=-w: fin :=w:debz:=-w:finz:=w: r:=0.004*(fin-deb):\n#deb: =-2: fin := 2.5: pour la 1-\351re forme\ng_droites:= tubeplot(\{op(sol z)\},radius=r,z=debz..finz,numpoints=2, color=magenta,style=patchnogri d),\n tubeplot(\{op(solx)\},radius=r,x=deb..fin,numpoints=2 , color=black,style=patchnogrid), \n tubeplot(\{op(solx0)\} ,radius=r,y=deb..fin,numpoints=2, color=black,style=patchnogrid):\nsur f:=implicitplot3d(S,x=deb..fin,y=deb..fin,z=debz..finz,style=patchnogr id,grid=[20,20,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "pl otsetup( inline):\n 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