Rotate 1 by N vector

Hello, everyone. Is there a way (function) in Matlab to rotate a row vector of N dimensions by a given angle? I've only been able to use the matrix multiplication for 2D and 3D cases.

Thanks!

[Jan]

To define a rotation you need the angle and the axis to rotate around. How is the axis of rotation defined in your case?

[Peter Popoola]

Thanks, Jan. The axis of rotation in this case I'm not too sure of, but I figure it should be around the first co-ordinate, that is, the x-axis. I don't think it's too important what the axis of rotation is in this algorithm (Imperialist Competitive Algorithm, in case you are interested). We just need to be able to perform a rotation of all variables (dimensions) along one axis. Apologies for the late reply!

[Guillaume]

If you are talking about rotations in N-dimensional space (N>3), then a search with your favourite engine would reveal that:

a) it's far from trivial

b) there's plenty of papers written about it, involving maths that are way beyond me. Clifford algebra is mentioned a lot.

[Peter Popoola]

Thanks for the comment, Guillaume. I didn't know that. I guess I'll have to find a way around it.

[Jan]

I consider this as not complicated:

R = eye(numel(u)) + ...
    (v * u.' - u * v.') * sin(a) + ...
    (u * u.' + v * v.') * (cos(a) - 1);

Here u and v are two orthogonal unit vectors to describe the n-dimensional hyperplane to rotate in. Remember, that in 3D one vector is sufficient to define a plane to rotate in, but in 4D (and N-D), this is not unique anymore. a is the angle in radians to rotate, and the direction is from u to v.

I publish this in the FileExchange currently.

Answer 1

[Rik]

You mean something like this?

v=1:10;
v2=imrotate(v,45);

[Peter Popoola]

Thanks, Rik. However, upon executing this code, I get the following error:

error: Y(2): out of bound 1
error: called from
    interp2 at line 238 column 10
    imremap>grayinterp at line 82 column 12
    imremap at line 67 column 19
    imperspectivewarp at line 124 column 17
    imrotate at line 158 column 20

[Rik]

I don't get an error, so what is the exact code you're using?

[Peter Popoola]

I'm using exactly the code which you presented here. On the Octave shell.

[Rik]

File a bug report with GNU. There are often minor differences between Octave and Matlab, especially in edge cases like this. Apparently Octave assumes both dimensions are larger than 1.

[Peter Popoola]

Okay, Rik. Thank you!

Answer 2

[Jan]

See WikiPedia: Rotation_matrix_from_axis_and_angle

Here the axis of rotation u, the cross-product matrix []_x and the tensor cross-product (x) are not limited to 3D, but work for n dimensions also. So you can create the n-D rotation matrix in a straight-forward way and multiply your vector with.

[EDITED] See FEX: Rotation Matrix

[Peter Popoola]

Thanks once again, Jan. I take it there's no Matlab function that does this? Because I haven't really been able to wrap my head around the formulation given here, and how it can be extended to the N-dimensional case. Thank you for your help!

[Jan]

See FEX: Rotation Matrix .