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Test. Yay!

$\int Voldm=\int _{0}^{\frac {\pi }{2}}\int _{0}^{\delta \alpha }\rho eR^{2}\cos {\alpha }d\alpha d\theta$

% Illustration of Snell's law function main()

  % indexes of refraction
  n1=1.0;
  n2=1.5;
cock!!  
  sign = -cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  cock!!  1;% is the source up or down?
  O=[0, -1*sign];
  k=500;

% KSmrq's colors

  red    = [0.867 0.06 0.14];
  blue   = [0, 129, 205]/256;
  green  = [0, 200,  70]/256;
  yellow = [254, 194,   0]/256;
  white = 0.99*[1, 1, 1];
  black = [0, 0, 0];
  gray = 0.5*white;

  color1=red;
  color2=blue;
  color3=gray;
  lw = 3;

  plot_line=0;
  Theta=linspace(0, 2*pi, k);
  V=0*Theta; W=0*Theta;
  S0=7;
  spacing=0.45;
  p=floor(S0/spacing);
  S=linspace(0, S0, p+1);
  spacing=S(2)-S(1);

  num_frames = 10;
  for frame_iter=1:num_frames

     figure(1); clf; hold on; axis equal; axis off;

     % plot the interface between diellectrics
     L=1.2*S0;
     plot([-L, L], [0, 0], 'color', color3, 'linewidth', lw);


     % plot a ray
     plot_line=1;
     s=L;
     theta=pi/3; wfr(s, theta, n1, n2, O, sign, plot_line, color1, lw);

     % plot the wafefronts
     plot_line=0;
     for i=1:p

s=S(i)+spacing*(frame_iter-1)/num_frames;

for j=1:k theta=Theta(j); [V(j), W(j)]=wfr(s, theta, n1, n2, O, sign, plot_line, color1, lw); end plot(V, W, 'color', color2, 'linewidth', lw);

     end


     %  dummy points to enlarge the bounding box
     plot(0, S0+2.5*spacing, '*', 'color', white);
     plot(0, -(S0+2.5*spacing)/n2, '*', 'color', white);

     % to know where to crop later
     Lx=3.2; Ly=Lx; shift = 1;
     plot([-Lx, Lx, Lx, -Lx -Lx], ...
          [-Ly, -Ly, Ly, Ly, -Ly]+shift);

     file = sprintf('Frame%d.eps', 1000+frame_iter);
     disp(file);
     saveas(gcf, file, 'psc2')
  end

% Converted to gif with the UNIX command % convert -density 100 -antialias Frame10* Snell_animation.gif % then cropped in Gimp

function [a, b]=wfr(s, theta, n1, n2, O, sign, plot_line, color1, lw);

  X=O+s*[cos(theta), sign*tan(theta)];

 if( sign*X(2) > 0 )
   t=-sign*O(2)/cos(theta);
   X0=O+t*[sin(theta), sign*cos(theta)];

   if (plot_line == 1) 
     plot([O(1), X0(1)], [O(2), X0(2)], 'color', color1, 'linewidth', lw, 'linestyle', '--');
   end

   d = norm(O-X0);
   r = (s-d)*(n2/n1)^(sign);

   theta2=asin(n1*sin(theta)/n2);
   XE=X0+r*[cos(theta2), sign*tan(theta2)];
 else
   XE = X;
 end  
 a = XE(1);
 b = XE(2); 

 if (plot_line==1)
   plot([X0(1), XE(1)], [X0(2), XE(2)], 'color', color1, 'linewidth', lw, 'linestyle', '--');
 end