Dessin de fractals par la methode IFS (iterated functions system) Soient w1,...,wN des transformations contractantes de R^n, A l'unique sous-ensemble compact non vide de R^n qui verifie A=w1(A)U...UwN(A). Pour dessiner A, on va utiliser la methode IFS, qui consiste a partir d'un point P de A, que l'on dessine, puis a tirer au sort parmi les transformations w1,..,wN, prendre le transforme P' de P par la transformation designee par le tirage au sort; on recommence avec P' a la place de P. Pour obtenir de bons resultats, il faudrait beaucoup de pas, dix mille par exemple. Cela pose un probleme de memoire avec Maple. On se contente donc de tirer 1000 points au plus, et les resultats ne seront donc pas tres bons. Remarque . La methode IFS est plus fine que les precedentes, mais Maple est moins bien adapte pour ce type de dessin. D'abord un rappel: homot:=proc(P,lambda,V)
#homothetie de centre P, rapport lambda, appliquee a V
lambda*(V-P)+P;
end: rota:=proc(P,alpha,V);
[cos(alpha)*(V[1]-P[1])-sin(alpha)*(V[2]-P[2])+P[1],sin(alpha)*(V[1]-P[1])+cos(alpha)*(V[2]-P[2])+P[2]];
end: On commence a apprendre a faire les tirage au sort avec Maple : on utilise la fonction "rand" (de random=aleatoire) tire:=rand(1..4): tire();tire();tire(); NiMiIiM= NiMiIiQ= NiMiIiM= Ensuite, il faut apprendre a dessiner des listes de points : with(plots):with(plottools): Warning, the assigned name arrow now has a global binding A:=[0,0]:B:=[2,0]:C:=[1,1]: plot([A,B,C],axes=none,style=point,symbol=point,color=blue); LSUlUExPVEc2KS0lJ0NVUlZFU0c2IzclNyQkIiIhRitGKjckJCIiI0YrRio3JCQiIiJGK0YwLSUrQVhFU0xBQkVMU0c2JFEhNiJGNS0lKkFYRVNTVFlMRUc2IyUlTk9ORUctJSZDT0xPUkc2JiUkUkdCRyRGKyEiIkY/JCIjNUZALSUlVklFV0c2JDskISIlISIjJCIkLyNGSTskRklGSSQiJC0iRkktJSZTVFlMRUc2IyUmUE9JTlRHLSUnU1lNQk9MRzYkRlNGQg== sierpinski:=proc(pas)
local L,P, tire,no,i,A,B,C;
L:=[];
P:=[0,0];
A:=[0,0];
B:=[2,0];
C:=[1,1];
tire:=rand(1..3);
for i from 1 to pas do
no:=tire();
if no=1 then
P:=homot(A,1/2,P)
elif no=2 then
P:=homot(B,1/2,P)
else
P:=homot(C,1/2,P)
fi;
L:=[op(L),P];
od;
L;
end: plot(sierpinski(10000),style=point,symbol=point,axes=none,color=blue,scaling=constrained); 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Si on ne veut pas faire de tirage au sort, mais prendre les 3 transformations alternativement: badsierpinski:=proc(pas)
local L,P,M,no,i,j;
L:=[];
P:=[0,0];
M:=[[0,0],[2,0],[1,1]];
for i from 1 to pas do
for j from 1 to 3 do
P:=homot(M[j],1/2,P);
L:=[op(L),P];
od;
od;
L;
end: p1:=plot(badsierpinski(1000),style=point,symbol=point,color=red,axes=none): p2:=plot([A,B,C,A],axes=none,scaling=constrained,color=black): p3:=[point(1/7*A+2/7*B+4/7*C),point(1/7*B+2/7*C+4/7*A),point(1/7*C+2/7*A+4/7*B)]: display(p1,p2,p3); 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 ce n'est pas fameux! Les tireres s'accumulent en fait vers les points fixes des composees w3w2w1, w1w3w2 et w2w1w3, qui sont respectivement: solve(1/2*( 1/2*(1/2*x+1/2*a)+1/2*b)+1/2*c=x,x); NiMsKEkiYUc2IiMiIiIiIihJImJHRiUjIiIjRihJImNHRiUjIiIlRig= solve(1/2*( 1/2*(1/2*x+1/2*b)+1/2*c)+1/2*1=x,x); NiMsKEkiYkc2IiMiIiIiIihJImNHRiUjIiIjRigjIiIlRihGJw== solve(1/2*( 1/2*(1/2*x+1/2*c)+1/2*a)+1/2*b=x,x); NiMsKEkiY0c2IiMiIiIiIihJImFHRiUjIiIjRihJImJHRiUjIiIlRig= On passe a la courbe de von Koch A:=[0,0];B:=[1,0];C:=[2,0];DE:=[3,0]; NiM+SSJBRzYiNyQiIiFGJw== NiM+SSJCRzYiNyQiIiIiIiE= NiM+SSJDRzYiNyQiIiMiIiE= NiM+SSNERUc2IjckIiIkIiIh w1:=x->homot(A,1/3,x); NiM+SSN3MUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJmhvbW90R0YlNiVJIkFHRiUjIiIiIiIkOSRGJUYlRiU= w2:=x->homot(A,1/3,rota(A,Pi/3,x))+B; NiM+SSN3Mkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwmLUkmaG9tb3RHRiU2JUkiQUdGJSMiIiIiIiQtSSVyb3RhR0YlNiVGMCwkSSNQaUdJKnByb3RlY3RlZEdGOUYxOSRGMkkiQkdGJUYyRiVGJUYl w3:=x->homot(DE,1/3,rota(DE,-Pi/3,x))+C-DE; NiM+SSN3M0c2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoLUkmaG9tb3RHRiU2JUkjREVHRiUjIiIiIiIkLUklcm90YUdGJTYlRjAsJEkjUGlHSSpwcm90ZWN0ZWRHRjkjISIiRjM5JEYySSJDR0YlRjJGMEY7RiVGJUYl w4:=x->homot(DE,1/3,x); NiM+SSN3NEc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJmhvbW90R0YlNiVJI0RFR0YlIyIiIiIiJDkkRiVGJUYl et voici la procedure qui engendre la liste de points de la courbe de Von Koch: vonkoch:=proc(pas)
local L,P, tire,no,i;
L:=[];
P:=A;
tire:=rand(1..4);
for i from 1 to pas do
no:=tire();
if no=1 then
P:=w1(P)
elif no=2 then
P:=w2(P)
elif
no=3 then
P:=w3(P)
else
P:=w4(P)
fi;
L:=[op(L),P];
od;
L;
end: plot(vonkoch(1000),style=point,symbol=point,axes=none,color=green,scaling=constrained); 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