Image:Double torus illustration.png

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[edit] Source code

% illustration of a double torus
function main()

   % N = The number of data points. More points means prettier picture.
   N=100;
   
   % big and small radii of the torus
   R = 3; r = 1; 

   % c controls the transition from one ring to the other
   c = 1.3*pi/2;
   
   Kb = R+r;
   Ks = R-r;

   % plot the two rings
   Ttheta=linspace(0, 2*pi, N);
   Pphi=linspace(0, 2*pi, N);
   [Theta, Phi] = meshgrid(Ttheta, Pphi);
   X=(R+r*cos(Theta)).*cos(Phi);
   Y=(R+r*cos(Theta)).*sin(Phi);
   Z=r*sin(Theta);

   figure(4); clf; hold on; axis equal; axis off;
   axis([-Kb 3*Kb -Kb Kb -r r]);
   light_green=[184, 224, 98]/256; % light green
   myplot(X, Y, Z, light_green);
   myplot(X+2*Kb, Y, Z, light_green);
   
% Plot the "waist" joining the two rings. This is the hardest.
% All the code below is devoted to that. It is just a hack
   
%  Dirty tricks to get a grid more adapted to the torus geometry
   N1 = N; N2 = N; N3 = N1;
   N4 = N1;
   small = 0.3*r;
   tiny = 0.001;

   % plot the "waist" of the torus
   XX = linspace(R-r, R+3*r, 4*N);
   YY=linspace(-Kb, Kb, N);
   [X, Y] = meshgrid(XX, YY);

   % Modify Y to make it more adapted to the torus geometry
   M = length(XX);
   for j=1:M
      x = XX(j); x = my_map(x, Kb, c);
      bl = sqrt(Kb^2-x^2);
      bm = sqrt(max(Ks^2-x^2, 0));
      
      k = 10; tiny = 1/N;
      Y1 = linspace(-bl, -bm, N/2-k); % Y1=adaptive_grid(Y1);
      Y2 = linspace(-bm+tiny, bm-tiny, 2*k); %Y2=adaptive_grid(Y2);
      Y3 = linspace(bm, bl, N/2-k);  %  Y3 = adaptive_grid(Y3);
      
      Y(:, j)=[Y1 Y2 Y3]';
   end
   
   [m, n] = size(X);
   for i=1:m
      for j=1:n

         x = X(i, j); x = my_map(x, Kb, c);
         y = Y(i, j);
         
         z = r^2 - (sqrt(x^2+y^2)-R).^2;
         if z < 0
            z = NaN;
         else
            z = sqrt(z);
         end
         
         Z(i, j) = z;
      end
   end
   

   myplot(X, Y, Z, light_green);
   myplot(X, Y, -Z, light_green);


% viewing angle
   view(108, 42);

% add in a source of light
   camlight (-50, 54); lighting phong;

% save as png
%  print('-dpng', '-r300', sprintf('Double_torus_illustration_N%d.png', N));
   
function myplot(X, Y, Z, mycolor)
   
   H=surf(X, Y, Z); 
   
   % set some propeties
   set(H, 'FaceColor', mycolor, 'EdgeColor','none', 'FaceAlpha', 1);
   set(H, 'SpecularColorReflectance', 0.1, 'DiffuseStrength', 0.8);
   set(H, 'FaceLighting', 'phong', 'AmbientStrength', 0.3);
   set(H, 'SpecularExponent', 108);

% This function constructs the second ring in the double torus
% by mapping from the first one.
function y=my_map(x, K, c)

   if x > K
      x = 2*K - x;
   end
   
   if x < K-c
      y = x;
   else
      y = (K-c) + sin((x - (K-c))*(pi/2/c));
   end
   

% take a uniform grid and cluster its points toward the endpoints
function X = adaptive_grid (X)
  
   K = 50;

   n = length(X);
   a = X(1); b = X(n);
   

   if a == b
      return;
   end
   
   X = (X-a)/(b-a);
   X = atan(K*(X-0.5));

   X = (X-X(1))/(X(n)-X(1));
   X = a + (b-a)*X;
   

• Genus (mathematics)
• Euler characteristic
• User:Oleg Alexandrov/Pictures
• Double torus