function [ p, t ] = cvt_ellipse_uniform ( n, sample_num, it_num, ...
  delaunay_display )

%*****************************************************************************80
%
%% CVT_ELLIPSE_UNIFORM demonstrates how a CVT can be computed and displayed.
%
%  Discussion:
%
%    This simple example carries out an iterative CVT calculation in an ellipse. 
%    The only change from the code for the unit square is that we have supplied a
%    routine that returns sample points from the ellipse.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    25 December 2018
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Qiang Du, Vance Faber, Max Gunzburger,
%    Centroidal Voronoi Tessellations: Applications and Algorithms,
%    SIAM Review, 
%    Volume 41, 1999, pages 637-676.
%
%  Parameters:
%
%    Input, integer N, the number of generators.
%
%    Input, integer SAMPLE_NUM, the number of sample points.
%
%    Input, integer IT_NUM, the number of iterations to take.
%
%    Input, logical DELAUNAY_DISPLAY, is TRUE (nonzero) if the Delaunay
%    triangulation is also to be displayed.
%
%    Output, real P(N,2), the location of the generators.
%
%    Output, integer T(NT,3), information defining the Delaunay
%    triangulation of the generators.  NT is the number of triangles,
%    which varies depending on the arrangement of the generators.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'CVT_ELLIPSE_UNIFORM:\n' );
  fprintf ( 1, '  A simple demonstration of a CVT computation\n' );
  fprintf ( 1, '  (Centroidal Voronoi Tessellation)\n' );
  fprintf ( 1, '  in an ellipse, with a uniform density.\n' );
  
  if ( nargin < 1 )
    n = 100;
    fprintf ( 1, '\n' );
    fprintf ( 1, 'CVT_ELLIPSE_UNIFORM - Note:\n' );
    fprintf ( 1, '  No value of N was supplied.\n' );
    fprintf ( 1, '  N is the number of generators.\n' );
    fprintf ( 1, '  A default value N = %d will be used.\n', n );
  else
    fprintf ( 1, '\n' );
    fprintf ( 1, '  User specified number of generators = %d\n',  n );
  end

  if ( nargin < 2 )
    sample_num = 1000 * n;
    fprintf ( 1, '\n' );
    fprintf ( 1, 'CVT_ELLIPSE_UNIFORM - Note:\n' );
    fprintf ( 1, '  No value of SAMPLE_NUM was supplied.\n' );
    fprintf ( 1, '  SAMPLE_NUM is the number of sample points.\n' );
    fprintf ( 1, '  A default value SAMPLE_NUM = %d will be used.\n', ...
      sample_num );
  else
    fprintf ( 1, '\n' );
    fprintf ( 1, '  User specified number of sample points = %d\n', ...
      sample_num );
  end
  
  if ( nargin < 3 )
    it_num = 20;
    fprintf ( 1, '  The default number of iterations IT_NUM = %d.\n', ...
      it_num );
  else
    fprintf ( 1, '  The user number of iterations = %d\n', ...
      it_num );
  end


  if ( nargin < 4 )
    delaunay_display = false;
    fprintf ( 1, '\n' );
    fprintf ( 1, 'CVT_ELLIPSE_UNIFORM - Note:\n' );
    fprintf ( 1, '  No value of DELAUNAY_DISPLAY was supplied.\n' );
    fprintf ( 1, '  DELAUNAY_DISPLAY is TRUE (nonzero) if the\n' );
    fprintf ( 1, '  Delaunay triangulation is also to be displayed.\n' );
    fprintf ( 1, '  A default value DELAUNAY_DISPLAY = %d will be used.\n', ...
      delaunay_display );
  else
    fprintf ( 1, '\n' );
    fprintf ( 1, '  User specified DELAUNAY_DISPLAY = %d\n', ...
      delaunay_display );
  end

%
%  This switch is set to 1 (TRUE) if the ACCUMARRAY command is available.
%  That speeds up the calculation a lot.  If you don't have the ACCUMARRAY
%  command, just set this to 0.
% 
  fprintf ( 1, '\n' );
  fprintf ( 1, 'CVT_ELLIPSE_UNIFORM:\n' );
  fprintf ( 1, '  MATLAB''s ACCUMARRAY command can be used for faster\n' );
  fprintf ( 1, '  computation.  This command is not available in\n' );
  fprintf ( 1, '  some versions of MATLAB.  If ACCUMARRAY is available,\n' );
  fprintf ( 1, '  simply make sure that the ACCUMARAY_AVAILABLE variable\n' );
  fprintf ( 1, '  is set to 1!\n' );
  
  accumarray_available = 1;

  if ( accumarray_available )
    fprintf ( 1, '\n' );
    fprintf ( 1, '  CVT_ELLIPSE_UNIFORM will use the ACCUMARRAY command.\n' );
  else
    fprintf ( 1, '\n' );
    fprintf ( 1, '  CVT_ELLIPSE_UNIFORM will NOT use the ACCUMARRAY command.\n' );
  end
%
%  Clear the figure screen.
%
  clf
%
%  Randomize the initial locations of the generators.
%
  p = ellipse_uniform ( n );
    
  for it = 0 : it_num
%
%  Compute the Delaunay triangle information T for the current nodes.
%
    t = delaunay ( p(:,1), p(:,2) );
%
%  Display the Delaunay triangulation if requested.
%
    if ( delaunay_display )
      figure ( 1 );
      triplot ( t, p(:,1), p(:,2) )
      title_string = sprintf ( 'Delaunay, step %d', it );
      title ( title_string );
      axis equal
      view ( 2 )
    end
%
%  Display the CVT generators and their Voronoi diagram.
%
    figure ( 2 )
    voronoi ( p(:,1), p(:,2) );
    title_string = sprintf ( 'Voronoi, step %d', it );
    title ( title_string );
    axis equal
    drawnow
%
%  Generate sample points.  
%
%  These sample points implicitly define the geometry of the region.  
%  If the region is not a unit square, then the range of the sample 
%  data must be changed.
%
%  The data is sampled uniformly.  If a nonuniform density is desired,
%  then the sampling must be done in a biased way.
%    
    ps = ellipse_uniform ( sample_num );
%
%  For each sample point, find K, the index of the nearest generator.
%  We do this efficiently by using the Delaunay information with
%  Matlab's DSEARCH command, rather than a brute force nearest neighbor
%  computation.
%
%  Because MATLAB can do anything it wants, they have DELETED the
%  DSEARCH command.  Now I have to find out how to fix this.
%  
    k(1:sample_num,1) = dsearchn ( p, t, ps );
%
%  The centroid of the Voronoi region associated with each generator
%  is approximated by the average of the sample points it was closest to.
%
    if ( accumarray_available )

      count(1:n,1) = accumarray ( k, ones(sample_num,1) );
      centroid(1:n,1) = accumarray ( k, ps(:,1) );
      centroid(1:n,2) = accumarray ( k, ps(:,2) );

    else

      count(1:n,1) = 0;
      centroid(1:n,1) = 0.0;
      centroid(1:n,2) = 0.0;

      for i = 1 : sample_num
        j = k(i);
        count(j) = count(j) + 1;
        centroid(j,1) = centroid(j,1) + ps(i,1);
        centroid(j,2) = centroid(j,2) + ps(i,2);
      end

    end
%
%  Replace the generators by the centroids.
%
    p(1:n,1) = ( centroid(1:n,1) ./ count(1:n) )';
    p(1:n,2) = ( centroid(1:n,2) ./ count(1:n) )';
    
  end
%
%  Save the graphics.
%
  figure ( 1 );
  filename = 'delaunay.png';
  print ( '-dpng', filename );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Graphics saved as "%s"\n', filename );

  figure ( 1 );
  filename = 'voronoi.png';
  print ( '-dpng', filename );
  fprintf ( 1, '  Graphics saved as "%s"\n', filename );
%
%  Terminate.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'CVT_ELLIPSE_UNIFORM:\n' );
  fprintf ( 1, '  Normal end of execution.\n' );

  return
end
function p = ellipse_uniform ( n )

%*****************************************************************************80
%
%% ELLIPSE_UNIFORM returns sample points from a particular ellipse.
%
%  Discussion:
%
%    This routine returns N points sampled uniformly at random
%    from within the ellipse defined by
%
%      x' * A * x <= R*R
%
%    with A = ( 9  6 ) and R = 6.
%             ( 6 20 )
%
%    The routine requires the Cholesky factor of A.  For this
%    problem, the formulas are worked out explicitly.  The approach
%    generalizes to other ellipses, and with a little more work,
%    to higher dimensions as well.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    06 June 2016
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer N, the number of points to generate.
%
%    Output, real P(N,2), the sample points.
%

%
%  Step 1: Generate N points in the unit circle.
%  (Generate N normally distributed points.
%   Normalize them (now they lie on the circumference).
%   Assign a radius.)
%
  p = randn ( n, 2 );

  p_norm = sqrt ( dot ( p, p, 2 ) );

  r = rand ( n, 1 );

  for j = 1 : 2
    p(1:n,j) = sqrt ( r(1:n) ) .* p(1:n,j) ./ p_norm(1:n);
  end
%
%  Step 2.  The ellipse is x' * A * x <= R*R.
%  A has the Cholesky factorization A = U' * U.
%  Take each point from the unit ball, and:
%  *) multiply by R;
%  *) multiply by inverse(U).
%
%  For our problem, we have A = ( 9  6 )
%                               ( 6 20 )
%
%  and R = 6
%
%  Hence
%
%  U = ( 3 2 )
%      ( 0 4 )
%
%  inverse ( U ) = ( 1/3 -1/6 )
%                  (  0   1/4 )
%
%  
%  so we can do this in one step.
%
  p(1:n,1) = 2.0 * p(1:n,1) - p(1:n,2);
  p(1:n,2) = 1.5 * p(1:n,2);

  return
end
