What kind of transformation is that?

Question

We have the glide reflection \begin{equation*}\kappa \begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}x+2\\ -y+2\end{pmatrix}\end{equation*} and the rotation \begin{equation*}\delta \begin{pmatrix} x\\ y\end{pmatrix}=\begin{pmatrix} -x +2\\ -y+2\end{pmatrix}\end{equation*} The composition of these maps is $$\kappa\circ\delta\begin{pmatrix}x \\ y\end{pmatrix}= \kappa \left (\delta \begin{pmatrix}x \\ y\end{pmatrix}\right )=\kappa \begin{pmatrix} -x +2\\ -y+2\end{pmatrix}=\begin{pmatrix} -x +4\\ -y+4\end{pmatrix}$$ What kind of transformation is that?

Answer 1

As I pointed out in my comment, the right map is $$(\kappa \circ \delta)(x,y) = (-x+4,y).$$ Note that you can write $(\kappa \circ \delta)(x,y) = (-x,y)+(4,0)$, or equivalently using matrices $$(\kappa \circ \delta) \left( \begin{matrix} x \\ y \end{matrix} \right) = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) \left( \begin{matrix} x \\ y \end{matrix} \right)+ \left( \begin{matrix} 4 \\ 0 \end{matrix} \right) = A \left( \begin{matrix} x \\ y \end{matrix} \right) + b. $$ So $\kappa \circ \delta$ îs an affine transformation of $\mathbb R^2$ and is a composition of an improper ($\det A = -1$) rotation $A \in \mathrm{O}(2)$ and a translation by a vector $b = (4,0)$ (horizontal translation in positive direction). The transformation fixes the line given by points $(2,y)$, because $(\kappa \circ \delta)(2,y) = (2,y)$ for all $y \in \mathbb R$.