%  Author:   John Burkardt
%  Discussion:
%    In the Buffon-Laplace needle experiment, we suppose that the plane has been
%    tiled into a grid of rectangles of width A and height B, and that a
%    needle of length L is dropped "at random" onto this grid.
%
%    We may assume that one end, the "eye" of the needle falls at the point
%    (X1,Y1), taken uniformly at random in the cell [0,A]x[0,B].
%
%    ANGLE, the angle that the needle makes is taken to be uniformly random.
%    The point of the needle, (X2,Y2), therefore lies at
%
%      (X2,Y2) = ( X1+L*cos(ANGLE), Y1+L*sin(ANGLE) )
%
%    The needle will have crossed at least one grid line if any of the
%    following are true:
%
%      X2 <= 0, A <= X2, Y2 <= 0, B <= Y2.
%
%    This routine simulates the tossing of the needle, and returns the number
%    of times that the needle crossed at least one grid line.
%
%    If L is larger than sqrt ( A*A + B*B ), then the needle will
%    cross every time, and the computation is uninteresting.  However, if
%    L is smaller than this limit, then the probability of a crossing on
%    a single trial is
%
%      P(L,A,B) = ( 2 * L * ( A + B ) - L * L ) / ( PI * A * B )
%
%    and therefore, a record of the number of hits for a given number of
%    trials can be used as a very roundabout way of estimating PI.
%    (Particularly roundabout, since we actually will use a good value of
%    PI in order to pick the random anglesc)
%  Reference:

%    Sudarshan Raghunathan,
%    Making a Supercomputer Do What You Want: High Level Tools for
%    Parallel Programming,
%    Computing in Science and Engineering,
%    Volume 8, Number 5, September/October 2006, pages 70-80.

%  Parameters:

%    Input, double precision A, B, the horizontal and vertical dimensions
%    of each cell of the grid.  0 <= A, 0 <= B.

%    Input, double precision L, the length of the needle.
%    0 <= L <= min ( A, B ).

%    Input, integer TRIAL_NUM, the number of times the needle is
%    to be dropped onto the grid.

%    Input/output, integer SEED, a seed for the random number generator.

%    Output, integer BUFFON_LAPLACE_SIMULATE, the number of times the needle
%    crossed at least one line of the grid of cells.

function   hits = buffon_laplace_simulate(a,b,l,trial_num,seed)

      hits = 0;

      for trial = 1:trial_num

%  Randomly choose the location of the eye of the needle in [0,0]x[A,B],
%  and the angle the needle makes.

%        x1 = a * r8_uniform_01( seed );
%        y1 = b * r8_uniform_01( seed );
%	angle = 2.0 * pi * r8_uniform_01 ( seed );
        x1 = a * rand;%(seed);
        y1 = b * rand;%(seed);
	angle = 2.0 * pi * rand;%(seed);

%  Compute the location of the point of the needle.

        x2 = x1 + l * cos( angle );
        y2 = y1 + l * sin( angle );

%  Count the end locations that lie outside the cell.

%   figure(1),clf % plot the nedle
%   line([0,1],[0,0],'LineWidth',2),hold
%   line([1,1],[0,1],'LineWidth',2)
%   line([0,1],[1,1],'LineWidth',2)
%   line([0,0],[0,1],'LineWidth',2)
%%   axes('position',  [-0.5, -0.5, 2, 2])
%   line([x1,x2],[y1,y2],'LineWidth',2,'Color','r')
%   axis('equal')
        if (x2 <= 0 | a <= x2 |  y2 <= 0 | b <= y2 ),
          hits = hits + 1;
        end 
      end  % for loop

      return