sample vectors with pre-defined angle

hello,

i want to sample a couple of vectors in a high-dimensional space (using randn), with the constraint that all these vectors must have the same angle between one another. is there an easy way to do that in matlab, e.g. a function that I don't know?

thanks. :)

Answer 1

[José-Luis]

I don't think there is. You could always try this:

 nDim = 5;
 point1 = randn(nDim,1);
 point2 = randn(nDim,1);
 vec = point2 - point1;
 %Rotating, note that this will grow exponentially;
 all_rotations = perms([0 0 ones(1,nDim-2)]);
 rot_vec = vec;
 for ii = all_rotations' %randomly rotating along all possible axes
     rot_angle = 2 * pi() * rand(1);
     sin_pos = find(~ii);
     ii(~ii) = cos(rot_angle);
     rot_mat = diag(ii);
     rot_mat(sin_pos(1),sin_pos(2))=-sin(rot_angle);
     rot_mat(sin_pos(2),sin_pos(1))=sin(rot_angle);
     rot_vec = rot_mat*rot_vec;
 end
 %The coordinates of the rotated point2
 rot_point2 = point1 + rot_vec;

If you always want the same angles, you will need to change the rotation angle to fixed instead of random.

Answer 2

[Jan]

This is not possible for an arbitrary number of vectors. Let's look at the 3D case:

• 2 vectors must be parallel or anti-parallel, but they can point in any direction.
• 3 vectors must be co-planar and include 60 deg angles. The position of the first one can be selected randomly.
• 4 vectors must build a tetraeder and include an angle of 70.53 deg. The tetraeder can be rotated arbitrarily.
• I claim without a proof, that there are not solutions for more than 4 vectors in 3D.

Equivalent restrictions will exist in n-dim spaces.

[José-Luis]

I don't understand what you mean. Rotation is still defined in n-dimensional spaces. From what I understood, the OP want two vectors with the same angle. Given only one angle (and three or more dimensions), there are infinitely many, so I agree it is then impossible to answer.

What I proposed was instead to define rotations along all possible axes (see my post). In that case, the answer will be unique, but you need to specify all those angles.