Composition of isometries 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 $\tau_u \not\parallel l$


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


*

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

*Build $l$ as the middle of $a$ and $f$.
This is the axis of the glide reflection.

*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 $d$.

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