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!set slib_header_coxaff=\
\\ Polynome minimal de 2*cos(Pi/n)\
real_cyclo(n)={\
if(n==1,return(x+2));\
my(p=polcyclo(2*n),nn=eulerphi(2*n)/2,res=0,c);\
forstep(j=nn,0,-1,\
c = polcoeff(p,nn+j);\
res += c*x^j;\
p -= c*x^(nn-j)*(x^2+1)^j);\
res\
};\
\
\\ p(2*cos(t)) = 2*cos(n*t)\
cheby(n)=2*subst(polchebyshev(n),x,x/2);\
\
\\ m est une matrice de Coxeter\
\\ renvoie les generateurs\
cox_base(m)={\
my(n=#m,nn=1);\
for(i=1,n,for(j=i+1,n,if(m[i,j],nn=lcm(nn,m[i,j]))));\
xx=Mod(x,real_cyclo(nn));\
module=xx.mod;\
racine=2*cos(Pi/nn);\
gram=matrix(n,n,i,j,if(m[i,j],-cos(Pi/m[i,j]),-1));\
v=vector(n,k,matrix(n,n,i,j,\
(i==j)+(i==k)*lift(subst(if(m[k,j],cheby(nn/m[k,j]),2),x,xx))));\
Set(v)\
};\
\
longer(shorter,current,cb)={\
my(l=List());\
foreach(current,w,foreach(cb,s,\
listput(l,simplify((w*s)%module))));;\
setminus(Set(l),shorter)\
};\
\
\\ Sous l'hypothese 1/p+1/q+1/r=1\
\\ imprime le source tikz d'un pavage d'une partie\
\\ du plan avec des triangles d'angles Pi/p, etc.\
aff3(p,q,r,limite)={\
my(g,t=matrix(3,3),u,v,acc,cb=cox_base([1,p,q;p,1,r;q,r,1]),\
sh=Set(),cr=Set(matid(3)),tmp,h,res,xmin,xmax,ymin,ymax,xm,ym);\
for(i=1,3,\
g=subst(cb[i]*cb[1+(i%3)],x,racine);\
\\point fixe de g, moyenne des iteres : intersection de la droite de points fixes, puis z=1\
u=if(i==1,p,if(i==2,r,q));\
v=[0,0,1];\
acc=vector(3);\
for(i=1,u,acc+=v;v=v*g);\
t[i,]=acc/u);\
\\printf("p: %d, q: %d, r: %d\n",p, q, r);\
res=matrix(0,7);\
\\printf("\\begin{tikzpicture}[scale=%.3f]\n",7/limite);\
for(i=1,limite, tmp=longer(sh,cr,cb); sh=cr; cr=tmp;\
foreach(sh,g,h=t*subst(g,x,racine)*[1,0;cos(Pi/p),sin(Pi/p);0,0];\
\\printf("\\%s(%.4f,%.4f)--(%.4f,%.4f)--(%.4f,%.4f)--cycle;\n",\
\\if(i%2,"fill","draw"),h[1,1],h[1,2],h[2,1],h[2,2],h[3,1],h[3,2]);\
res=matconcat([res,[h[1,1],h[1,2],h[2,1],h[2,2],h[3,1],h[3,2],i%2]]~)\
));\
\\print("\\end{tikzpicture}\n");\
for(i=1,matsize(res)[1],for(j=1,3,xm=res[i,2*j-1];ym=res[i,2*j];\
if(xm<xmin,xmin=xm);\
if(xm>xmax,xmax=xm);\
if(ym<ymin,ymin=ym);\
if(ym>ymax,ymax=ym)));\
[[xmin,xmax,ymin,ymax],res];\
};\
/*\
\p3\
print("\nAffines\n");\
aff3(2,4,4,20);\
aff3(3,3,3,20);\
aff3(2,3,6,20);\
*/\
\
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