Summary edit

This is a retouched picture, which means that it has been digitally altered from its original version. Modifications: Vector version. The original can be viewed here: Amoeba4 400.png: . Modifications made by Zerodamage.

Licensing edit

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

Source code edit

% find the amoeba of the polynomial
% p(z, w)=50 z^3+83 z^2 w+24 z w^2+w^3+392 z^2+414 z w+50 w^2-28 z +59 w-100
% See http://en.wikipedia.org/wiki/Amoeba_(mathematics).

function main()

   figure(3); clf; hold on;
   axis equal; axis off;
   axis([-4.5, 5, -3.5, 6]); 
   fs = 20; set(gca, 'fontsize', fs);
   ii=sqrt(-1);
   tiny = 100*eps;
   
   Ntheta = 500; % for Ntheta=500 the code will run very slowly, but will get a good resolution
   NR=      Ntheta; 

   % R is a vector of numbers, exponentiall distributed
   A=-5; B=5;
   LogR  = linspace(A, B, NR);
   R     = exp(LogR);

   % a vector of angles, uniformly distributed
   Theta = linspace(0, 2*pi, Ntheta);

   degree=3;
   Rho = zeros(1, degree*Ntheta); % Rho will store the absolute values of the roots
   One = ones (1, degree*Ntheta);

   % play around with these numbers to get various amoebas
   b1=1;  c1=1; 
   b2=3;  c2=15;
   b3=20; c3=b3/5; 
   d=-80; e=d/4;
   f=0; g=0;
   h=20; k=30; l=60;
   m=0; n = -10; p=0; q=0;
   
%  Draw the 2D figure as union of horizontal slices and then union of vertical slices.
%  The resulting picture achieves much higher resolution than any of the two individually.
   for type=1:2

	  for count_r = 1:NR
		 count_r
		 
		 r = R(count_r);
		 for count_t =1:Ntheta
			
			theta = Theta (count_t);

			if type == 1
			   z=r*exp(ii*theta);

%                         write p(z, w) as a polynomial in w with coefficients polynomials in z 
%                         first comes the coeff of the highest power of w, then of the lower one, etc.
			   Coeffs=[1+m,
				   c1+c2+c3+b1*z+b2*z+b3*z+k+p*z,
				   e+g+(c1+b1*z)*(c2+b2*z)+(c1+c2+b1*z+b2*z)*(c3+b3*z)+l*z+q*z^2,
				   d+f*z+(c3+b3*z)*(e+(c1+b1*z)*(c2+b2*z))+h*z^2+n*z^3];

			else
%                          write p(z, w) as a polynomial in z with coefficients polynomials in w 		
			   w=r*exp(ii*theta);
			   Coeffs=[b1*b2*b3+n,
				   h+b1*b3*(c2+w)+b2*(b3*(c1+w)+b1*(c3+w))+q*w,
				   (b2*c1+b1*c2)*c3+b3*(c1*c2+e)+f+(b1*c2+b3*(c1+c2)+b1*c3+b2*(c1+c3)+l)*w+...
				   (b1+b2+b3)*w^2+p*w^2,
				   d+c3*(c1*c2+e)+(c1*c2+(c1+c2)*c3+e+g)*w+(c1+c2+c3+k)*w^2+w^3+m*w^3];
			end
			
%                       find the roots of the polynomial with given coefficients
			Roots = roots(Coeffs);
			
%                       log |root|. Use max() to avoid log 0.
			Rho((degree*(count_t-1)+1):(degree*count_t))= log (max(abs(Roots), tiny)); 
		 end
		 

%        plot the roots horizontally or vertically
		 if type == 1
		        plot(LogR(count_r)*One, Rho, 'b.');
		 else
		        plot(Rho, LogR(count_r)*One, 'b.');
		 end
		 
	  end

   end
   
   saveas(gcf, sprintf('amoeba4_%d.eps', NR), 'psc2');

Original upload log edit

This image is a derivative work of the following images:

• File:Amoeba4_400.png licensed with PD-self
  • 2007-03-09T03:59:48Z Oleg Alexandrov 1896x1917 (269569 Bytes) Made by myself with Matlab. {{PD-self}}

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