13.6 Glide-Reflections

Since reflections and translations are both isometries, the composition of a reflections and a translation is also an isometry.

13.6.1 Synthetic Plane

Definition 13.19 Let \(A\) and \(B\) be distinct points in the plane. The function \[g_{\overrightarrow{AB}} = r_{\overleftrightarrow{AB}} \circ T_{\overrightarrow{AB}}\] is called the glide-reflection of \(\overrightarrow{AB}\).

Since the translation is parallel to the line of reflection, the order of the reflection and translation are interchangeable (the proof is left as an exercise). This means that \[\begin{align*} g_{\overrightarrow{AB}} \circ g_{\overrightarrow{BA}} & = \left( r_{\overleftrightarrow{AB}} \circ T_{\overrightarrow{AB}}\right) \circ \left( r_{\overleftrightarrow{AB}} \circ T_{\overrightarrow{BA}}\right) \\ & = r_{\overleftrightarrow{AB}} \circ \left( T_{\overrightarrow{AB}} \circ T_{\overrightarrow{BA}}\right) \circ r_{\overleftrightarrow{AB}} \\ & = r_{\overleftrightarrow{AB}} \circ r_{\overleftrightarrow{AB}} = \mbox{identity} \end{align*}\] and so \(g_{\overrightarrow{AB}}^{-1} = g_{\overrightarrow{BA}}\).

Theorem 13.19 Every glide-reflection is a composition of three reflections.

Proof. Let \(A\) and \(B\) be distinct points in the plane and let \(C\) be the midpoint of \(\overline{AB}\). Let \(m\) and \(n\) be lines through \(A\) and \(C\), respectively, such that \(m\) and \(n\) are both perpendicular to \(\overleftrightarrow{AB}\). The two-reflection theorem implies that \(T_{\overrightarrow{AB}}= r_n \circ r_m\) and so \[g_{\overrightarrow{AB}}= r_{\overleftrightarrow{AB}} \circ r_n \circ r_m\] and so every glide-reflection is a composition of three reflections.

13.6.2 Complex Plane

We know from Theorem 13.7 that every isometry \(f:\mathbb{C}\rightarrow \mathbb{C}\) is of the form \[f(z)=az+b \quad \mbox{or} \quad f(z)=a\overline{z}+b\] for some \(a,b\in \mathbb{C}\) with \(|a|=1\). We have shown that isometries of the form \(f(z)=az+b\) are either translations or rotations and isometries of the form \(f(z)=a\overline{z}+b\) are reflections if \(a\overline{b}+b=0\). The following theorem shows that the glide-reflections complete our types of isometries.

Theorem 13.20 Let \(f(z)=a\overline{z} + b\) be a function such that \(|a|=1\). Then \(f\) is a glide-reflection with a non-zero glide if and only if \(a\overline{b}+b\neq 0\).

Proof. In the proof of Theorem 13.16 we showed that for complex numbers \(a\) and \(b\), with \(|a|=1\), that \[f(z)=a\overline{z}+b = a \overline{ \left( z-\frac{b}{2}\right)} + \frac{b}{2} + \frac{a\overline{b}+b}{2}.\] Since \(|a|=1\), \(a = e^{i2 \theta}\) for some angle \(\theta\).

We can then let \(m\) be the line through \(\frac{b}{2}\) making an angle of \(\theta\) with the horizontal.

Furthermore, \[\frac{a\overline{b}+b}{2} = \frac{e^{i2\theta}\overline{b} + b}{2} = e^{i\theta} \frac{ e^{i \theta} \overline{b} + e^{-i\theta} b}{2} = e^{i \theta} \cdot \mbox{Re}\left( e^i \theta \overline{b}\right).\] Therefore, this complex number is in the same direction as \(m\) and we see that \[f(z)=a \overline{z} + b = r_m \circ T_{\frac{a\overline{b}+b}{2}}\] is a glide-reflection with a non-zero glide.

Let \[G = \left\{ f : \mathbb{C} \rightarrow \mathbb{C} \vert \: f(z)= a \overline{z}+b \mbox{ for some } a,b\in \mathbb{C} \mbox{ with } |a|=1 \right\}\] be the set of glide reflections. Then we can see that the set of isometries, \(I\), is the union of the set of orientation preserving isometries, \(P\), and \(G\). So all orientation reversing isometries are glide reflections, with or without the glide.

13.6.3 Exercises

  1. Prove that for \(g_{\overrightarrow{AB}}\), \[r_{\overleftrightarrow{AB}}\circ T_{\overrightarrow{AB}} = T_{\overrightarrow{AB}} \circ r_{\overleftrightarrow{AB}}.\]

  2. Let \(g\) be the glide-reflection about the vector \(\langle 1,1\rangle\).

    1. Write an algebraic representation for the glide-reflection from the Cartesian, Vector, and Complex perspectives.
    2. Find the image of the circle \(C=\left\{ (x,y)\in \mathbb{R}^2 \: \vert \: (x-2)^2+(y+3)^2 =9\right\}\) under this glide-reflection.
    3. Let \(A=(1,2)\), \(B=(-2,3)\) and \(C=(0,0)\). Find the image of the triangle \(\triangle ABC\) under this glide-reflection.

  3. Let \(A\) and \(B\) be any two distinct points. Prove that the composition \[R_{B,\pi} \circ r_{\overleftrightarrow{AB}}\circ R_{A,\pi}\] is a glide-reflection and find its axis and the vector of translation.

  4. Is \(G\) a normal subgroup of \(I\)? If so, describe the elements of the factor group and determine if the factor group is isomorphic to another known group.