File:Double torus illustration.png

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Description

Illustration of en:Double torus

Date (UTC)
Source Own work
Author Oleg Alexandrov
Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Source code

% illustration of a double torus, obtained as an isosurface
function main()

   % 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;
  
   h = 0.1; % h is the grid size. Smaller h means prettier picture.
   
   X = (-Kb-h):h:(3*Kb+h); m = length(X);
   Y = (-Kb-h):h:(Kb+h);   n = length(Y);
   Z = (-r-h):h:(r+h);     k = length(Z);
 
   W = zeros(m, n, k); % the zero level set of this function will be the desired shape
 
   for i=1:m
      for j=1:n
         x = X(i); x = my_map(x, Kb, c);   % map from two torii to one torus
         y = Y(j); 
         W(i, j, :) = (sqrt(x^2+y^2)-R)^2 + Z.^2-r^2; % torus eqn, vectorize in Z
      end
   end

   figure(4); clf; hold on; axis equal; axis off;

   H = patch(isosurface(W, 0));
   isonormals(W, H);
      
   light_green=[184, 224, 98]/256;

   % set some propeties
   set(H, 'FaceColor', light_green, '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))
      
% viewing angle
   view(-165, 42);

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

% save as png
  print('-dpng', '-r500', sprintf('Double_torus_illustration.png'));
   
% 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

File history

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Date/TimeThumbnailDimensionsUserComment
current04:32, 12 July 2008Thumbnail for version as of 04:32, 12 July 2008985 × 1,077 (260 KB)Oleg AlexandrovHigher quality version, using isosurface instead of patches. Same license and all that.
05:49, 6 September 2007Thumbnail for version as of 05:49, 6 September 20071,176 × 1,240 (350 KB)Oleg Alexandrov{{Information |Description= |Source=self-made |Date=Illustration of en:Double torus |Author= Oleg Alexandrov }} {{PD-self}} Category:Differential geometry Category:Files by User:Oleg Alexandrov from en.wikipedia

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