Glide Reflection (Euclidean Plane)
A glide reflection is a plane isometry which is the result of a translation, and reflection about the line of translation.
Given lines \(a\), \(b\) and \(c\), the transformation \(\sigma_{c} \circ \sigma_{b} \circ \sigma_{a}\) is a glide reflection if \(a\) and \(b\) are distinct lines, both perpendicular to \(c\).
 1.png)
This transformation can be thought of as a translation in the direction of \(c\) (given that a translation can be expressed as the composition of two reflections in parallel lines) followed by a reflection on the axis of translation (as shown with \(\Omega\) above). Alternatively, it can be viewed as a reflection followed by a point reflection.
An isometry is a glide reflection if and only if it can be expressed as the product of three reflections about lines \(p\), \(q\) and \(r\) (in that order) where the lines are not concurrent and not all parallel.
First assume \(p\), \(q\) and \(r\) are not concurrent and not parallel.
We will show this by replacing these reflections with reflections about different lines, until we have the form above.
The idea of the method, is to take a pair of intersecting lines, and rotate them such that the line that corresponds with the middle reflection is perpendicular to the third line, and then rotating the perpendicular pair of lines so that the first and third lines are parallel.
Proof
Case 1
Let \(\tau = \sigma_{r} \circ \sigma_{q} \circ \sigma_{p}\).
In the first case we assume \(p\) and \(q\) are not parallel, and that \(R\) is their point of intersection.
 2.png)
First we can replace \(p\) and \(q\) with lines \(p'\) and \(q'\) such that \(q'\) is perpendicular to \(r\). This can be done because of the form of a rotation in terms of reflections, that is, we choose \(q'\) to be perpendicular to \(r\) and pass through \(Q\), and choose \(p'\) to pass through \(R\) and have the same directed angle as was between \(p\) and \(q\).
Now, \(\tau = \sigma_{r} \circ \sigma_{q'} \circ \sigma_{p'}\).
 3.png)
Next we note that \(\sigma_{r} \circ \sigma_{q'}\) is a half-turn, because \(r\) is perpendicular to \(q'\). This means that we can replace \(q'\) with \(q''\) and \(r\) with \(r'\) such that \(q''\) is perpendicular to \(p'\).
 4.png)
This gives us the known form for a glide reflection.
Case 2
In the second case, if \(p\) is parallel to \(q\), then a similar construction follows:
 5.png)