Is there a glide reflection?

Question

I am looking the following:

Let $g$ be a line that goes through the points $(-2,0)$, $(1,-2)$.

• Is there a glide reflection with fixed line $g$, that maps the point $(-1,0)$ to the point $(0,-2)$ ?
• For each point $Z$ let $\delta_Z$ be a reflection around the point $Z$. For which points $Z$ is $\sigma_g\circ\delta_Z$ a refection along a line?

I have done the following:

• We have that the line $g$ has the equation $2x+3y+4=0$. A fixed line is a line that is mapped to itself, but not every point is mapped to itself but maybe to an other point of the line.

  We have that $P=(-1,0)$ is not on $g$. $P'=(0,-2)$ is the image of that point by the glide reflection.

  We have that the segment $PP'$ gets halved by $g$. Let $Q$ be the midpoint of $PP'$. We have that $Q=\left (-\frac{1}{2},-1\right )$. We have to prove if $Q\in g$. $$2\cdot \left (-\frac{1}{2}\right )+3\cdot (-1)+4=-1-3+4=0 \Rightarrow Q\in g$$

  We have that $P$ reflects along $g$ and then we translate it by $(a,b)$.

  The reflection of $P$ along $g$ is the point $\left (-\frac{21}{13}, -\frac{12}{13}\right )$.

  We add at this point the point $(a,b)$ to get $P'$: $$\left (-\frac{21}{13}, -\frac{12}{13}\right )+(a,b)=(0,-2) \ \Rightarrow \ a=\frac{21}{13}, \ b=-\frac{14}{13}$$

  So, the glide reflection is the following:

  The glide reflection is the composirion of a reflection and a translation. The reflection is along $g$ and then we translate about $\left (\frac{21}{13}, -\frac{14}{13}\right )$.

• A reflection around a point is the composition of two reflection along a line, $\sigma_a$ and $\sigma_b$, where $a$ is perpendicular to $b$.

  So, we have that $\sigma_g\circ\delta_Z=\sigma_g\circ\sigma_a\circ\sigma_b$.

  So, we have a composition of three reflections along a line.

  We have that the composition of two parallel reflections is a translation and that the composition of a reflection and a translation is a glide reflection. So, when $g$ is not parallel to $a$ and $b$ then the composition is a reflection.

Is everything correct?

Answer 1

For the second question:

Forget about $a $ and $b $.

The composition of a reflexion and a symmetry about a point must be orientation reversing. So it is either a reflection or a glide-reflexion.

• If $Z $ is on $g $ then the point $Z $ is fixed so it cannot be a glide-reflection. Thererefore is it a reflexion.

• If not, then you can see that there is no fixed point (try it!). So it is a glide-relfexion.

Answer 2

The first question looks fine to me. For the second one, the set of points for which it is a reflection is the line $g $ itself, otherwise it is a glide-reflexion.