Product of two glide reflections about two perpendicular lines

Question

If we consider two glide reflections:

$G_{\vec{v_1},l_1}$ and $G_{\vec{v_2},l_2}$, assuming that $l_1$ $\perp$ $l_2$.

Now, if I want to describe the relation between these two glide reflections' product ($G_2 . G_1$).

Here I am not sure if it is simply another glide reflection?

I apologize if the answer is too trivial, I just always have difficulty visualizing analytic geometry. If I am incorrect any advice is greatly appreciated.

Answer 1

No, it's not another glide reflection. It's a half-turn, that is, a point reflection.

For instance, suppose that $G_1(x,y)=(x,-y)+(1,0)=(x+1,-y)$ and that $G_2(x,y)=(-x,y)+(0,2)=(-x,y+2)$. Then$$G_2\bigl(G_1(x,y)\bigr)=G_2(x+1,-y)=(-x-1,-y+2)=-(x,y)+(-1,2),$$which is a half-turn around $\left(-\frac12,1\right)$.