Fractal in MATLAB

I'm a new user of MATLAB and I already met some difficulties with my MATLAB program. It consists in plot Brownian and Bachelier cartoon exactly as you can see following this link.

Longer Link Here

Such function is similar to Bolzano's function. Can someone give me some advice concerning how to generate the new points?

[Andrew Newell]

What have you tried and what difficulties have you had?

[Samuel]

I tried something obtaining a partial result. So far my code is the following

function [x,y]=cartoon(n)
H=1/2; %Hurst's exponent
if nargin ~= 1
    n = 4;
end
% end of recursion
if n <= 0
    x = [0 1];
    y = [0 1];
    % recursive method call
else
    [x0,y0] = cartoon(n-1);
    
    % calculate new co-ordinates
    x = (4/9)*[x0 x0+5/4];
    y = 2/3*[y0 y0+H];
end
plot(x,y,'-');
hold off;
end

The code start to divide the straight and rise line in 3 unlike parts creating a generator with a central interval [4/9,5/9] and so on. But the code is not complete because it doesn't apply the generator in the central intervals (like [4/9,5/9] and the others that step by step are generated). How can I settle my code so that I get the intentional result?

Answer 1

[Andrew Newell]

The key point you are missing is that you need to feed in the coordinates of the endpoints as well as n. That will allow you to recursively call cartoon for each subinterval. Here is a code that will do what you want:

function [x,y]=cartoon(x,y,n,H)
if nargin < 3
  n = 4;
end
if nargin < 4
  H = 1/2; % Hurst's exponent
end

% If n <=0, x and y are equal to the input.
if n > 0

    % Create the first division.
    x = x(1) + (x(2)-x(1))*[0 4/9 5/9 1];
    y = y(1) + (y(2)-y(1))*[0 2/3 2/3*H 1];

    % Subdivide each interval recursively.
    [x1,y1] = cartoon(x(1:2),y(1:2),n-1,H);
    [x2,y2] = cartoon(x(2:3),y(2:3),n-1,H);
    [x3,y3] = cartoon(x(3:4),y(3:4),n-1,H);

    % The vectors x1,x2,x3 overlap at their endpoints.
    x = [x1 x2(2:end) x3(2:end)];
    y = [y1 y2(2:end) y3(2:end)];
end

Note that I have also removed the plot command from inside the function. You can now recreate the left plot using a loop like this:

 for i=1:4
    subplot(4,1,i)
    [x,y] = cartoon([0 1],[0 1],i,1/2);
    plot(x,y)
end

For the right hand plots, you'll need to run cartoon on each of the two line segments and piece the results together.

[Samuel]

Perfect! Thank you so much!

[Shin Kelly]

wow.. this is fabulous work,,,

hello I'm highschool student studying fractal and trying to observe(?) it in animals' organ..

Can you give me some advice for generalizing the photo(of organ) and caculate the fractal dimension? Please,,