Chaos Game and Fractals

[gomgabbar]

Hi All,

I am trying to write a MatLab program that runs a chaos game to generate fractals (or not).

Information found here: http://mathworld.wolfram.com/ChaosGame.html

I've used the n=3 r=1/2 chaos game before to generate the Sierpinski triangle, even on my TI-83. Now I'm working on a MatLab program to run the chaos game for arbitrary n and r values. Can someone take a look at my code and help me find where I went wrong? The code is verified to generate the Sierpinski triangle when n=3 r=1/3, but I can not generate any of the fractals mentioned in the website above.

My m-file:

Code: Select all

clear all;
n=4;
r=1/(n-1);


t=5000;
Ax = 2*pi/n;
Ai = pi-Ax;

Vx = zeros(1,n+1);
Vy = zeros(1,n+1);
Vx(1) = 0;
Vy(1) = 0;
Vx(2) = 1;
Vy(2) = 0;


for i=3:n+1
    Axn = Ax*(i-2);
    Vx(i) = Vx(i-1)+cos(Axn);
    Vy(i) = Vy(i-1)+sin(Axn);
end
Sx=0;
Sy=0;
for i=1:n
    Sx = Sx + Vx(i);
    Sy = Sy + Vy(i);
end
Cx = Sx/n;
Cy = Sy/n;
Nx = zeros(1,t);
Ny = zeros(1,t);
Nx(1) = Cx+Cy*(rand(1)-.5);
Ny(1) = Cy+Cy*(rand(1)-.5);

for i=2:t
    D = rand(1)
    for j=1:n
        if ((j-1)/n < D)&&(D < j/n)
            Nx(i) = Nx(i-1)+r*(Vx(j)-Nx(i-1))
            Ny(i) = Ny(i-1)+r*(Vy(j)-Ny(i-1))
            plot (Vx,Vy,Cx,Cy,Nx,Ny,'.');
            xlim([min(Vx) max(Vx)])
            ylim([min(Vy) max(Vy)])
            drawnow
        end
    end
end

%plot (Vx,Vy,Cx,Cy,Nx,Ny,'.');


[gorcee]

Nothing glaring, but I am not familiar with the algorithm. However, is this intended?

Nx(1) = Cx+Cy*(rand(1)-.5);

[gomgabbar]

Nothing glaring, but I am not familiar with the algorithm. However, is this intended?

Nx(1) = Cx+Cy*(rand(1)-.5);

Ah, glad you asked. I needed to pick a staring place inside an n sided regular polygon. (Cx, Cy) is the Cartesian coordinate for the center, and because the base is always along the x-axis, Cy is also the smallest distance from the edge to the side. Thus, I take the center point, and add (or subtract) at most one half of this distance (in each coordinate) so that the starting point for the algorithm will be a random point, guaranteed to be inside the polygon.


As an update from the original post, I have found the error in my original code. I took r as the distance from the current point toward the selected vertex, when r is actually taken as the distance from the selected vertex to the current point. These two have the same result if r is 1/2; that is why the Sierpinski triangle rendering worked. To update my code, I simply replaced instances of r with (1-r) and it now works for all Sierpinski-like chaos games