Image:Triple torus illustration.png

function main()
 
  % illustration of a triple torus
 
   % torus radii 
   r = 1;
   R = 3;
 
   Kb = R+r;
   Ks = R-r;
 
   % Km controls the smoothness of the transition from one ring to the others
   Km = 0.5125*Kb;
 
   L = 2.5*(r+R);
 
   h = 0.2; % grid size
 
   X = (-L):h:L; m = length(X);
   Y = (-L):h:L; n = length(Y);
   Z = (-1.1*r):h:(1.1*r); k = length(Z);
 
   W = zeros(m, n, k);
 
   for i=1:length(X)
      for j=1:length(Y)
         x = X(i);
         y = Y(j);
 
         [x, y] = triple_torus_function (x, y, r, R, Kb, Km);
         val = (sqrt(x^2+y^2)-R)^2-r^2;
         W(i, j, :) = val + Z.^2;
 
      end
   end
 
   figure(1); clf; hold on;
   axis equal; axis off;
 
   light_green=[184, 224, 98]/256; % light green
 
   H = patch(isosurface(X, Y, Z, W, 0));
   isonormals(X, Y, Z, W, H);
   mycolor = light_green;
 
   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);
 
   daspect([1 1 1]);
   axis tight;
   colormap(prism(28))
   view(-12, 40);
 
   % add in a source of light
   camlight (-50, 54); lighting phong;
 
   print('-dpng', '-r400',  ...
         sprintf('Triple_torus_illustration.png'));
 
 
function [x, y] = triple_torus_function (x, y, r, R, Kb, Km)
 
% a deformation in the plane, which, when comosed with a torus will give
% a triple torus   
 
% centre of one of the torii
   O = [-Kb, -Kb/sqrt(3)]; 
 
   angle = 2*pi/3;
   Mat = [ cos(angle)  -sin(angle); sin(angle)   cos(angle)  ];
 
   p =[x, y]';
   phi = atan2(y, x);
 
   if phi >= pi/6 & phi <= 5*pi/6
      p = Mat*p; % rotate 120 degree counterclockwise
   elseif phi >= -pi/2 & phi < pi/6
      p = Mat*p; p = Mat*p; % rotate 240 degrees counterclockwise
   end
 
   x=p(1); y = p(2);
 
% reflect against a line, to merge two cases in one
   if y > x/sqrt(3)
 
      p = [x, y];
      v = [cos(2*pi/3), sin(2*pi/3)];
 
      p = p - 2*v*dot(p, v)/dot(v, v);
      x = p(1); y = p(2);
 
   end
 
   if x > O(1)
 
% project to the y axis, to a point B
      if y < O(2)
 
         A = [O(1), y];
         B = [0, y];
      else
 
         A = O;
 
         p = [x, y];
         rho = norm(p-O);
 
         B = O+(Kb/rho)*(p-O);
 
%         t = -O(1)/(x-O(1));
%         B = [0, O(2)+t*(y-O(2))];
 
      end
 
      p = [x, y];
 
      d=norm(p-A);
      q = norm(B-A);
 
      d = my_map(d, q, Km);
      p = (d/q)*B+(1-d/q)*A;
      x=p(1); y=p(2);
 
   end
 
% shift towards the origin
   x = x-O(1);
   y = y-O(2);
 
function y = my_map(x, Kb, Km)
 
   if x > Kb
      y = Km + 1;
   elseif x < Km
      y = x;
   else
      y = Km+sin((pi/2)*(x-Km)/(Kb-Km));
   end