function pdf = point_distance_3d_pdf ( x, a, b )

%*****************************************************************************80
%
%% point_distance_3d_pdf() evaluates the point distance PDF in the 3D.
%
%  Discussion:
%
%    It is assumed that a set of points has been generated in 3D
%    according to a Poisson process.  The number of points in a region
%    of size VOLUME is a Poisson variate with mean value B * VOLUME.
%
%    For a point chosen at random, we may now find the nearest
%    Poisson point, the second nearest and so on.  We are interested
%    in the PDF that governs the expected behavior of the distances
%    of rank A = 1, 2, 3, ... with Poisson density B.
%
%  Formula:
%
%    PDF(X)(A,B) = 3 * ( (4/3) * B * PI )^A * X^( 3 * A - 1 )
%      * EXP ( - (4/3) * B * PI * X * X * X ) / ( A - 1 )!
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    11 January 2021
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Daniel Zwillinger, editor,
%    CRC Standard Mathematical Tables and Formulae,
%    30th Edition,
%    CRC Press, 1996, pages 580.
%
%  Input:
%
%    real X, the argument of the PDF.
%    0.0 <= X.
%
%    integer A, indicates the degree of nearness of the point.
%    A = 1 means the nearest point, A = 2 the second nearest, and so on.
%    0 < A.
%
%    real B, the Poisson point density.  0.0 < B.
%
%  Output:
%
%    real PDF, the value of the PDF.
%
  if ( a < 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'POINT_DISTANCE_3D_PDF - Fatal error!\n' );
    fprintf ( 1, '  Input parameter A < 1.\n' );
    error ( 'POINT_DISTANCE_3D_PDF - Fatal error!' );
  end

  if ( b <= 0.0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'POINT_DISTANCE_3D_PDF - Fatal error!\n' );
    fprintf ( 1, '  Input parameter B <= 0.0.\n' );
    error ( 'POINT_DISTANCE_3D_PDF - Fatal error!' );
  end

  if ( x < 0.0 )
    pdf = 0.0;
  else
    pdf = 3.0 * ( ( 4.0 / 3.0 ) * b * pi ) ^ a ...
      * x ^ ( 3 * a - 1 ) * exp ( - ( 4.0 / 3.0 ) * b * pi * x ^ 3 ) ...
      / factorial ( a - 1 );
  end

  return
end
