General Form of Reflections and Glide Reflections

Question

I am reading a set of notes about isometries of the plane. The section about reflections and glide reflections says that the general form for these isometries is: $$x'=x\cos\alpha+y\sin\alpha+h$$ $$y'=x\sin\alpha-y\cos\alpha+k$$ But, the notes don't given any explanation of what $\alpha$, $h$, and $k$ are. These equations seem similar to a rotation, with some sign changes, but I'm not seeing how they are related. If anyone can help that would be wonderful. Thanks

Answer 1

The equations that you write actually define the composition of three isometries of the plane:

1. A rotation around the origin by an angle $\alpha$
2. A reflection about the rotated $X$-axis (flip the $y'$ coordinate)
3. A translation over a vector $(h,k).$

Answer 2

A reflection across a line through the origin making an angle of $\alpha/2$ with the horizontal axis is given by $$ \begin{align*} x' = x \cos \alpha + y \sin \alpha \\y'=x \sin\alpha - y \cos\alpha\end{align*}$$

You can check this by verifying that the basis vectors $(1,0)$ and $(0,1)$ are mapped to their reflected selves: a reflection, being linear, is entirely determined by where the basis vectors are sent.

Then, the transformation

$$ \begin{align*} x' = x + h\\y'=y + k\end{align*}$$

where $h,k$ are real numbers, is simply a translation.

Since a glide reflection can be expressed as a reflection across a line through the origin followed by a translation, the claim follows.