function p = tetrahedron_sample ( t, n )
%*****************************************************************************80
%
%% tetrahedron_sample() returns random points in a tetrahedron.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 12 November 2010
%
% Author:
%
% John Burkardt
%
% Input:
%
% real T(3,4), the tetrahedron vertices.
%
% integer N, the number of points to generate.
%
% Output:
%
% real P(3,N), random points in the tetrahedron.
%
for j = 1 : n
r = rand ( 1, 1 );
%
% Interpret R as a percentage of the tetrahedron's volume.
%
% Imagine a plane, parallel to face 1, so that the volume between
% vertex 1 and the plane is R percent of the full tetrahedron volume.
%
% The plane will intersect sides 12, 13, and 14 at a fraction
% ALPHA = R^1/3 of the distance from vertex 1 to vertices 2, 3, and 4.
%
alpha = r .^ ( 1.0 / 3.0 );
%
% Determine the coordinates of the points on sides 12, 13 and 14 intersected
% by the plane, which form a triangle TR.
%
tr(1:3,1) = alpha * t(1:3,1) ...
+ ( 1.0 - alpha ) * t(1:3,2);
tr(1:3,2) = alpha * t(1:3,1) ...
+ ( 1.0 - alpha ) * t(1:3,3);
tr(1:3,3) = alpha * t(1:3,1) ...
+ ( 1.0 - alpha ) * t(1:3,4);
%
% Now choose, uniformly at random, a point in this triangle.
%
r = rand ( 1, 1 );
%
% Interpret R as a percentage of the triangle's area.
%
% Imagine a line L, parallel to side 1, so that the area between
% vertex 1 and line L is R percent of the full triangle's area.
%
% The line L will intersect sides 2 and 3 at a fraction
% ALPHA = SQRT ( R ) of the distance from vertex 1 to vertices 2 and 3.
%
alpha = sqrt ( r );
%
% Determine the coordinates of the points on sides 2 and 3 intersected
% by line L.
%
p12(1:3) = alpha * tr(1:3,1) ...
+ ( 1.0 - alpha ) * tr(1:3,2);
p13(1:3) = alpha * tr(1:3,1) ...
+ ( 1.0 - alpha ) * tr(1:3,3);
%
% Now choose, uniformly at random, a point on the line L.
%
beta = rand ( 1, 1 );
p(1:3,j) = beta * p12(1:3) ...
+ ( 1.0 - beta ) * p13(1:3);
end
return
end