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 GNU LGPL license. % % Modified: % % 19 September 2004 % % Author: % % John Burkardt % % Reference: % % Daniel Zwillinger, editor, % CRC Standard Mathematical Tables and Formulae, % 30th Edition, % CRC Press, 1996, pages 580. % % Parameters: % % Input, real X, the argument of the PDF. % 0.0 <= X. % % Input, 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. % % Input, 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 ) ... / r8_factorial ( a - 1 ); end return end