Exercise 1.8.5

Question

A transformation \emph{T} in $\mathbb{R}^3$ is a rotation about the line $y = x + 3$ in the $x$-$y$ plane through an angle $\gamma$. Write a $4 	imes 4$ matrix corresponding to this transformation.
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You can leave the result as a product of matrices.

Jing Huang · Accepted Answer

extbf{Solution} For a general spatial rotation around a given direction (suppose the direction is given by a vector) through an angle $\gamma$, the $3 	imes 3$ rotation matrix can be given by:
%%%%%%%%%%%%%%%%%
\begin{equation*}
    R = R_x^{-1} R_y^{-1} R_z(\gamma) R_y R_x,
\end{equation*}
%%%%%%%%%%%%%%%%
where the rotation by $\gamma$ is assumed to be performed around $z$-axis. $R_x$ and $R_y$ are rotations used to align the direction with $z$-axis and can be determined by simple trigonometry.   \
For the problem given, the line $y = x + 3$ doesn't go through the origin, so extra step $T_{0}$ is needed to translate the line to make it pass the origin and homogeneous coordinates are applied:
%%%%%%%%%%%%%%%%%
\begin{equation*}
    R = T_{0}^{-1} R_x^{-1} R_y^{-1} R_z(\gamma) R_y R_x T_{0}.
\end{equation*}
%%%%%%%%%%%%%%%%
According to the description, 
%%%%%%%%%%%%%%%%%
\begin{equation*}
    \begin{bmatrix}
    T & 0 \
    0 & 1 \
    \end{bmatrix}
    = R_x^{-1} R_y^{-1} R_z(\gamma) R_y R_x.
\end{equation*}
%%%%%%%%%%%%%%%
The corresponding matrix is then
%%%%%%%%%%%%%%%%%
\begin{equation*}
    R = T_0^{-1}
    \begin{bmatrix}
    T & 0 \
    0 & 1 \
    \end{bmatrix}
    T_0.
\end{equation*}
%%%%%%%%%%%%%%%
$T_0$ is not unique for the translation to make two parallel lines align.

Exercise 1.8.5

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Exercise 1.8.5

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