Composition of isometries

Question

I'm having trouble wrapping my head around the composition of isometries.

For example, we've learnt that that the composition of three reflections is a glide reflection if they are not all parallel or all concurrent. How does one find the axis of the glide reflection geometrically in the following case (composition of $\sigma_a\sigma_e\sigma_d$). Clearly, $e \parallel d$, but the resulting $\tau_u \not\parallel l$

Answer 1

$\sigma_d\circ\sigma_e\circ\sigma_a$ can be geometrically buit the following way with the help of remarquable points & lines:

1. Build $f = \sigma_d\circ\sigma_e\circ\sigma_a(a)$.

   $f \parallel a$

2. Build $l$ as the middle of $a$ and $f$.

   This is the axis of the glide reflection.

3. Let $F = a \cap e$

   $\sigma_e \circ \sigma_a (F) = F$

   Build $G=\sigma_d\circ\sigma_e\circ\sigma_a(F) = \sigma_d(F)$ which is simply the symetrical of $F$ through $d$.

4. Build $H = \sigma_l(F)$ which is the symetrical of $F$ through $l$. $\overrightarrow{HG}$ is the vector of the translation of this glide reflection.

Then: $\sigma_d\circ\sigma_e\circ\sigma_a = \tau_\overrightarrow{HG}\circ\sigma_l$