Clebsch, ses 27 droites, les points de Eckardt

Le programme suivant a été développé afin de tracer les différents éléments explicités dans le paragraphe correspondant du site de Robert Ferréol sur la surface de Clebsch

I. Plans de Sylvester,  points de Eckardt, droites diagonales, droites de Schlafli

restart: # =========== passage équations de droites  en paramétrique [x(t),y(t),z(t)] parametrage:= proc(L)   local n,i,j,k,tamp, uu,vv,ww; global t:   tamp:=NULL: t:='t': n:=nops(L):   for i from 1 to n do        uu:=convert(L[i],list): vv:=uu:        for j from 1 to 3 do             if lhs(uu[j])=rhs(uu[j]) then                     if lhs(uu[j])=x then vv[1]:=t:                             elif lhs(uu[j])=y then vv[2]:=t:                             else vv[3]:=t :                     fi;                     ww:=subs(uu[j]=NULL,uu);                     for k from 1 to 2 do                           if lhs(ww[k])=x then vv[1]:= subs(y=t,z=t,rhs(ww[k])):                                     elif  lhs(ww[k])=y then vv[2]:= subs(x=t,z=t,rhs(ww[k])):                                     else  vv[3]:= subs(x=t,y=t,rhs(ww[k])):                           fi;                     od:             fi;        od: tamp:=tamp,vv: od: [tamp]; end: # ====== PS[1..5] les 5 plans de Sylvester ==========  PS:=[seq([ x[i]=0 ,convert(subsop(i=NULL,[seq(x[j], j= 1..5 )]),`+`)=0],i=1..5)]:

# ===== DE[i] les 10 "droites de Sylvester" DE:='DE': for i from 1 to 4 do     for j from i+1 to 5 do       DE[i,j]:=[ x[i]=0 , x[j] =0 ,convert(subsop(i=NULL,j=NULL,[seq(x[j], j= 1..5 )]),`+`)=0]; od: od: #print(DE);

# ====== Pts_eck les 10 points de Eckardt ========== pt:='pt': for i from 1 to 3 do     for j from i+1 to 4 do        for k from j+1 to 5 do      pt[i,j,k]:=[x[i]=0,op(2,DE[i,j]),op(2,DE[i,k]),    subs(x[i]=0,op(2,DE[i,j]),op(2,DE[i,k]),add(x[l],l=1..5))=0]: od;od:od: Pts_eck:=convert(pt,set): pt:='pt': #nops(Pts_eck); print(Pts_eck);

# ====== droites diagonales ========== # ***** DD[i,j] i=1..5, j=1..3 tamp:=NULL: DD:=NULL: dep:={seq(x[j], j= 1..5 )}: for i from 1 to 5 do    reste:=subs(dep[i]=NULL,dep);    tamp:=dep[i]: un:= reste[1]:    reste:=subs(reste[1]=NULL,reste);    for j from 1 to 3 do      deux:=reste[j]:      reste1:=subs(deux=NULL,reste):      DD:=DD,[tamp,un+reste[j],convert(reste1,`+`)]    od: od: DD:=[DD]:

# ====== droites de Schlafli ========== # ***** Sch1, Sch2 , 1 <= i < = 6 Sch1:=NULL: Sch2:=NULL: fip:=(1+sqrt(5))/2: fim:=(1-sqrt(5))/2: for i from 1 to 3 do      for j from i+1 to 4 do           reste:=subs(x[i]=NULL,x[j]=NULL,[x[1],x[2],x[3],x[4]]):           Sch1:=Sch1,[x[i]+fip*x[j]+reste[1],fip*x[i]+x[j]+reste[2],-fip*(x[i]+x[j])+x[5]]:           Sch2:=Sch1,[x[i]+fim*x[j]+reste[1],fim*x[i]+x[j]+reste[2],-fim*(x[i]+x[j])+x[5]]:      od: od: Sch1:=[Sch1]: Sch2:= [Sch2]:

penta:=[add(x[i]^3, i= 1..5),add(x[i], i= 1..5 )]:

---

Début  

II. Recherche de solutions à 10 points de Eckardt à distance finie

 quelques exemples

#tr:= [x[1] = X, x[2] = Y, x[3] = Z, x[4] = T+X+2*Y+3*Z] ;#10 Eck #tr:= [x[1] = X, x[2] = Y, x[3] = Z, x[4] = T+X-2*Y+2*Z] ;#10 Eck #tr:= [x[1] = X-2*Y+Z, x[2] = X+Y-2*Z, x[3] = Y+Z-2*X, x[4] = T+X] ;#10 Eck #tr:= [x[1] = X, x[2] = Y, x[3] = Z, x[4] =-T+X+2*Y+3*Z] ;#10 Eck #tr:= [x[1] = X, x[2] = Y, x[3] = Z, x[4] =-T-X-Y-Z] ;#4 eck 24 dr #tr:= [x[1]=(-3*X+Y+Z+T)/4,x[2]=(X-3*Y+Z+T)/4,x[3]=(X+Y-3*Z+T)/4,x[4] =(X+Y+Z-T)/2] ;#Ferreol 7 Eck 27 dr #tr:=[x[1] = X+Y, x[2] = Y+Z, x[3] = Z+X, x[4] = T-1*(X+Y+Z)];# forme à 7 pts concise #tr:=[x[1] = -3^(1/2)*Y+X+1/4*Z, x[2] = 3^(1/2)*Y+X+1/4*Z, x[3] = -2*X+1/4*Z, x[4] = 3/4*Z+T];# mienne à 7 termes #tr:= [x[1]=(-9*X+3*Y+3*Z+T)/4,x[2]=(3*X-9*Y+3*Z+T)/4, #      x[3]=(3*X+3*Y-9*Z+T)/4,x[4]=(3*X+3*Y+3*Z-T)/2];#hunt #tr:=[x[1]=-X+Y+Z,x[2] =-Y+Z+T,x[3]=X-Z+T,x[4]=T-(X+Y+Z)];#10 eck ,peu dissymétrique #tr:=[x[1] = -2*X+Y, x[2] = -2*Y+Z, x[3] = -2*Z+X, x[4] = T+Z];#10 eck ,peu dissymétrique #tr:=[x[1]=-X+2*Y+T,x[2] =-Y+2*Z+T,x[3]=2*X-Z+T,x[4]=X+Y];#10 eck

#============= essai de coordonnées ===================== mat:='mat': tr:=[x[1] = -2*X+Y, x[2] = -2*Y+Z, x[3] = -2*Z+X, x[4] = T+Z]; liste:=map(rhs,tr): liste:=map(u ->[coeff(u,X),coeff(u,Y),coeff(u,Z),coeff(u,T)],liste)  : mat:=matrix(4,4,liste): poss:=linalg[det](mat): if poss=0 then printf("%s\n", "Incorrect : Changement de variables non bijectif "):           else  printf("%s\n", "Correct : Changement de variables bijectif "): fi;

eq3:=subs(op(tr),penta[1]): eq4:=subs(op(tr),penta[2]): solx5:=solve(eq4,{x[5]}): pass_dir:= op(tr),op(solx5) ; SH:=subs(solx5,eq3): pts:=subs(pass_dir,Pts_eck): ptsnew:=[seq(solve({op(pts[i])},{X,Y,Z,T}),i=1..nops(pts))]: print("nombre de Eckardt ( dist. finie ou non) = ",nops(pts)): tamp:=NULL: for i from 1 to nops(ptsnew) do       if not(has(ptsnew[i],'T=0')) then tamp:=tamp,ptsnew[i] fi; od: ptsreels:=[tamp]: print("nombre de Eckardt à dist. finie  = ",nops(ptsreels)); if nops(pts)=10 then verif:= [seq(has(ptsnew[i],'T=0'),i=1..nops(ptsnew))]; print(verif); if has(verif,true)then        printf("%a ne  marche pas,%a points de Eckardt à distance finie \n",tr,nops(ptsreels))       else                   printf("%a, marche : 10 pts de Eckardt à distance finie\n",tr):             fi: else  printf("%a ne  marche pas\n",tr) fi: S:= subs(X=x,Y=y,Z=z,T=1,SH); eqE:=subs(X=x,Y=y,Z=z,T=1,ptsreels): Eck:={seq(solve({op(eqE[i])},{x,y,z}),i=1..nops(eqE))}: Eckardt:=map(u -> subs(op(u),[x,y,z]),Eck): #### print(DD); Ps:=subs(X=x,Y=y,Z=z,T=1,subs(pass_dir,PS)): Dd1:=subs(X=x,Y=y,Z=z,T=1,subs(pass_dir,DD)): Sc1:=subs(X=x,Y=y,Z=z,T=1,subs(pass_dir,Sch1)): Sc2:=subs(X=x,Y=y,Z=z,T=1,subs(pass_dir,Sch2)): DD1:=parametrage([seq(solve({op(Dd1[i])},{x,y,z}),i=1..nops(Dd1))]): SC1:=parametrage([seq(solve({op(Sc1[i])},{x,y,z}),i=1..nops(Sc1))]): SC2:=parametrage([seq(solve({op(Sc2[i])},{x,y,z}),i=1..nops(Sc1))]): S:=expand(S): printf("Surface = %a\n eq. développée = %a\n",SH,S/96); printf("Plans de Sylvester : %a\n",Ps); printf("Eckardt (%a) = %a \nDr. diagonales (%a) :\n%a\nDr. de Sclafli (%a) :\n%a\n%a\n" ,nops(Eckardt),Eckardt,nops(DD1),DD1,nops(SC1)+nops(SC2),SC1,SC2):

Vérification éventuelle du double-six

Les droites du double-six  sont mises bout à bout et les indices du double-six sont :

[1,2,3,4,  5,   6  ]
[7,8,9,10,11,12]

La vérification calcule les coordonnées des points d'intersection et les indices des droites correspondantes à ce point sous la forme :

[ [x,y,z], {indice droite 1,indice droite 2} ]

# les droites de Schlaefli sont SC1 et SC2:

>

#### recherche du numéro d'ordre dans l'ensemble "droites" ##### numero:=proc(dr)     local j:     for j from 1 while (j<= nops(droites)) and  evalb(droites[j]<>dr)  do  od:     j; end: #### Numéros d'ordre d'une partie de l'ensemble "droites" ###### ordre_partie:= proc(partie,affichage::{0,1})    local ordre,i:    ordre:=[seq(0,i=1..nops(partie))]:    for i from 1 to 6 do      ordre[i]:=numero(partie[i]);     od:    if affichage = 1 then       printf("Numéros des droites %a",ordre);

>	for i from 1 to nops(ordre) do print(partie[i],"___",droites[ordre[i]]) od;    fi;    ordre; end:

#### 0 pas d'affichage, 1 affichage des points ptscom:=proc(tot,affichage::{0,1})    local i,j,n,sol,pts ,w:    n:=nops(tot):pts:=NULL:    for i from 1 to n-1 do      for j from i+1 to n do        sol:=[solve({op(expand(tot[i]-subs(t=w,tot[j])))}, {t,w})]:                         if sol<>[] then              pts:=pts,[radnormal(subs(op(sol),tot[i]),'rationalized'),{i,j}]        fi;      od;    od:    pts:=[pts]:    if affichage =1 then        printf(" Nombre de droites = %d\n",nops(tot));        printf("Nombre de points d'intersection = %a  %s",nops(pts),            "--- [ point,[n° dr1, n° dr2] dans l'ens. donné]");        print(pts);    fi;   pts; end:

# ====== faire effectuer l'instruction suivante désirée ============

>	#### points communs de toutes les droites pts_dble_six:=ptscom([op(SC1),op(SC2)],1):

>

# =============== tracés =============== with(plots): dx:=-4: fx:=4: dy:=-4: fy:= 4: dz:=-4: fz:=4: dt:=-4: ft:=4: g_surf:=implicitplot3d(S,x=dx..fx,y=dy..fy,z=dz..fz,grid=[40,40,40],style=patchnogrid): g_diag:=spacecurve({op(DD1)},t=dt..ft,numpoints=2,color=blue,thickness=3): g_sc1:=spacecurve({op(SC1)},t=dt..ft,numpoints=2,color=red,thickness=3): g_sc2:=spacecurve({op(SC2)},t=dt..ft,numpoints=2,color=red,thickness=3): plotsetup(inline): display([g_surf,g_diag,g_sc1,g_sc2],scaling=constrained,lightmodel=light2,view=[dx..fx,dy..fy,dz..fz],orientation=[0,75]); #display([g_diag,g_sc1,g_sc2],scaling=constrained,lightmodel=light2,view=[dx..fx,dy..fy,dz..fz],orientation=[0,75]);

---

Début