function pdf = buffon_pdf ( a, l )
%*****************************************************************************80
%
%% BUFFON_PDF evaluates the Buffon PDF.
%
% Discussion:
%
% In the Buffon needle experiment, we suppose that the plane has been
% ruled by vertical lines with a spacing of A units, and that a
% needle of length L is dropped "at random" onto this grid.
%
% Because of the various symmetries, we may assume that this eye of
% this needle lands in the first infinite strip, and we may further
% assume that its Y coordinate is 0. Thus, we have
% the eye as (X1,Y1) with 0 <= X1 <= A and Y1 = 0.
%
% 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.
%
% The probability of a crossing on a single trial is
%
% P(A,L) = ( 2 * L ) / ( PI * A )
%
% 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.
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 01 March 2007
%
% Author:
%
% John Burkardt
%
% Parameters:
%
% Input, real A, the horizontal spacing between the
% vertical grid lines. 0 <= A.
%
% Input, real L, the length of the needle.
%
% Output, real PDF, the Buffon PDF.
%
if ( a < 0.0 )
fprintf ( 1, '\n' );
fprintf ( 1, 'BUFFON_PDF - Fatal error!\n' );
fprintf ( 1, ' Input A < 0.\n' );
error ( 'BUFFON_PDF - Fatal error!' );
elseif ( a == 0.0 )
pdf = 1.0;
return
end
if ( l < 0.0 )
fprintf ( 1, '\n' );
fprintf ( 1, 'BUFFON_PDF - Fatal error!\n' );
fprintf ( 1, ' Input L < 0.\n' );
error ( 'BUFFON_PDF - Fatal error!' );
elseif ( l == 0.0 )
pdf = 0.0;
return
end
pdf = ( 2.0 * l ) / ( pi * a );
return
end