I have this problem where I have to represent analytically an image of a movement. The image is defined by: $\phi: \rho_g . \tau_p$.
$\rho_g$ is a rotation over the line g with angle $\pi/2$.
g: $\begin{cases} x=s\\ y=s\\ z=0 \end{cases}$
where s is some real number.
$\tau_p$ is a translation with a step 1.
So, I know that first I have to change the coordinate system so that g is $O_z$, represent the transformation in that coordinate system, and then turn back to the original coordinate system.
The overall the transformation would be defined by the matrix $A=TA^*T^{-1}$, where T is the change of coordinate system matrix, $A^*$ is the image matrix in the new coordinate system where the rotation is around $O_z$ and $T^{-1}$ is the inverse of the change of coordinate system matrix.
I have found the change of coordinate system matrix and it's inverse:
$$T = \frac{\sqrt{2}}{2}\begin{bmatrix}-1 & 0 & 1\\ 1 & 0 & 1 \\0 & \frac{2}{\sqrt{2}} & 0\end{bmatrix}$$
$$T^{-1} = \frac{\sqrt{2}}{2}\begin{bmatrix}-1 & 0 & 0\\ 0 & 0 & \frac{2}{\sqrt{2}} \\1 & 1 & 0\end{bmatrix}$$
The transformation matrix I got in the new coordinate system is:
$$A^*=\begin{bmatrix}0 & -1 & 0\\ 1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}+\vec{p},$$
where $\vec p$ is the translation vector.
So my questions are:
- What does it mean for the translation to have a step 1, i.e. what would $\vec p$ like in that case?
- Since the overall transformation is defined by the composition of 3 matrices, what should I do with the translation $\vec p$ which is in the middle of the matrix multiplication with the transformation $A^*$?