Retrieving angle of rotation from transformation matrix

[Phong]

Hi!

How do I calculate the angle of rotation for each axis by a given 4x4 transformation matrix? The thing is that all values are a kind of mixed up in the matrix, so I cannot get discrete values to start calculating with anymore.

Thanks,


Phong

 

[HallsofIvy]

A four by four transformation matrix? Are you rotating in four dimensional space or is this a projective space?

First find the eigenvalues. A rotation matrix, in four dimensions may have two real and two complex-conjugate eigenvalues or two pairs of complex eigenvalues. If there are two real eigenvalues they must be either 1 or negative one. The eigenvectors corresponding to those eigenvalues give the axes of rotation. The complex eigenvalues will have modulus 1 and are of the form $cos(\theta)\pm i sin(\theta)$ where $\theta$ is the angle of rotation.

Two pairs of complex rotation give two simultaneous rotations in four space but are again of the form $cos(\theta)+ i sin(\theta)$. What those mean depends upon how you are writing vectors in four space.

If you are talking about a matrix representing a rotation matrix projectively, then you can renormalize to make the last row [0 0 0 1] and the last column $$\begin{bmatrix}0 \\ 0 \\ 0\\ 1\end{bmatrix}$$. The 3 by 3 matrix made up of the first three rows and columns will have one eigenvalue of 1 (the corresponding eigenvector gives the axis of rotation) and two complex conjugate eigenvalues of modulus 1. They will be of the form $cos(\theta)+ i sin(\theta)$ where $\theta$ is the angle of rotation.

 

[Phong]

Hello!

Thank you very much for your detailed reply. I must admit that I'm pretty new to transformation matrices and have not yet entirely understood the mathematical meaning of eigenvalues and eigenvectors although I try hard to understand everything I can read about it, but with some help I surely learn a lot faster.

I'm rotating in a projective three-dimensional space, that's why I use a 4x4 matrix.
To give a more specific example, I have a transformation matrix that is the following:

$$\begin{bmatrix}0.893 & 0.060 & -0.447 & 20 \\ -0.157 & 0.97 & -0.184 & 15 \\ -0.423 & -0.235 & -0.875 & 45 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

This transformation matrix should transform the object with a translation of 20 15 45 and a rotation of -15 25 -10 (xyz).

Now the eigenvalues. I don't know if I've understood the meaning of them correctly, but if yes the eigenvalues for this matrix should be in the identity matrix which is:

$$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

So if I'm still on the right track, the eigenvectors, which are the axes of rotation, are simply
$$\left(1, 0, 0, 0)$$ for x
$$\left(0, 1, 0, 0)$$ for y
$$\left(0, 0, 1, 0)$$ for z

Am I still on the right track or am I totally and fatally wrong?