function x = ball01_sample_3d ( )
%*****************************************************************************80
%
%% ball01_sample_3d() picks a random point in the unit ball in 3D.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 12 January 2021
%
% Author:
%
% John Burkardt
%
% Output:
%
% real X(3), the sample point.
%
u = rand ( 3, 1 );
%
% Pick a uniformly random VDOT, which must be between -1 and 1.
% This represents the dot product of the random vector with the Z unit vector.
%
% Note: this works because the surface area of the sphere between
% Z and Z + dZ is independent of Z. So choosing Z uniformly chooses
% a patch of area uniformly.
%
vdot = 2.0 * u(1) - 1.0;
phi = acos ( vdot );
%
% Pick a uniformly random rotation between 0 and 2 Pi around the
% axis of the Z vector.
%
theta = 2.0 * pi * u(2);
%
% Pick a random radius R.
%
r = u(3)^( 1.0 / 3.0 );
x(1) = r * cos ( theta ) * sin ( phi );
x(2) = r * sin ( theta ) * sin ( phi );
x(3) = r * cos ( phi );
return
end