function x = sphere01_sample_3d ( )
%*****************************************************************************80
%
%% sphere01_sample_3d() picks a random point on the unit sphere in 3D.
%
% Discussion:
%
% The unit sphere in 3D satisfies:
%
% X * X + Y * Y + Z * Z = 1
%
% Licensing:
%
% This code is distributed under the GNU LGPL license.
%
% Modified:
%
% 12 January 2021
%
% Author:
%
% John Burkardt
%
% Output:
%
% real X(3), the sample point.
%
%
% 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 = - 1.0 + 2.0 * rand ( 1, 1 );
phi = acos ( vdot );
%
% Pick a uniformly random rotation between 0 and 2 Pi around the
% axis of the Z vector.
%
theta = 2.0 * pi * rand ( 1, 1 );
x(1) = cos ( theta ) * sin ( phi );
x(2) = sin ( theta ) * sin ( phi );
x(3) = cos ( phi );
return
end