Why are glide reflections one of four isometries?

Question

It says here that there are four types of Euclidean plane isometries which I am going to list below:

1. Reflections

2. Translations

3. Rotations

4. Glide Reflections

$\textbf{Question:}$ Why are "glide reflections" considered to be different than the rest?

My reasoning is (as far as I am aware) that you can combine a reflection and translation and get a glide reflection anyway. So, why is it on the list as being different?

Answer 1

This is to convert my comment to an answer. First of all, the set of planar isometries forms a group $Isom(E^2)$, being a group means that the composition of isometries (product in a group) is again an isometry and the inverse of an isometry is again an isometry. This group sits inside of a larger group, the group of Euclidean similarities, $Sim(E^2)$ consisting of compositions of Euclidean isometries and dilations. Two elements $g_1, g_2$ of a group $G$ are called conjugate if there is a third element $h\in G$ such that $g_2= h g_1 h^{-1}$. This conjugation operation should be similar for linear algebra when you diagonalize square matrices $A$ by replacing them with conjugate matrices $D=PAP^{-1}$. From the viewpoint of coordinates, conjugating one isometry of $E^2$ by an element of $Sim(E^2)$ amounts to changing one Cartesian coordinate system with another Cartesian coordinate system (the center of coordinates is likely to change as well).

Now, let's see what conjugation does to various types of isometries of $E^2$: The conjugate of a rotation $R_{\phi,o}$ by the angle $\phi$ and center $o$ after conjugation becomes a rotation $R_{\pm\phi,o'}$ with the new center. (The angle of the rotation changes to its opposite if you conjugate by an orientation-reversing transformation.) The conjugate of a translation $T_{L,D}$ along an oriented line $L$ by the distance $D\ge 0$ is still a translation $T_{L',D'}$, but, in general, $D\ne D'$ (since we are allowing conjugation by similarities). It is a nice exercise to check that any (nonzero) translation can be conjugate to any other (nonzero) translation. A reflection $S_L$ in a line $L$ is conjugate to a reflection $S_{L'}$ is another line $L'$; again, all reflections are conjugate. Lastly, for a glide-reflection $G_{L,D}$ ($D$ is the distance of translation along a line $L$ and $L$ is the line of translation and of reflection), the conjugate is another
glide-reflection $G_{L',D'}$. More specifically, if you conjugate by a Euclidean similarity $h$ with the dilation factor $\lambda$ then $$ h G_{L,D} h^{-1} = G_{h(L),\lambda D}. $$

As before, any two glide-reflections are conjugate. Lastly, conjugating the identity map does not change it at all: $$ h \cdot 1 \cdot h^{-1}= 1. $$

Another nice exercise to work out is that two linear transformations of different types are never conjugate to each other. For instance, if $g_1$ is and isometry fixing a subset $F\subset E^2$ (say, a line), then $hg_1h^{-1}$ fixes a subset $h(F)\subset E^2$. Since a reflection fixes a line and a glide reflection has no fixed points, the two are never conjugate to each other. A glide-reflection is not conjugate to a translation since one is orientation-reversing and the other preserves orientation of $E^2$.

Thus, glide-reflections constitute a class of planar isometries which is manifestly different from the other classes.

Answer 2

Every isometry can be described uniquely as belonging to one of the following five classes:

1. A reflection,
2. A translation,
3. A rotation,
4. A glide reflection, or
5. The identity map.

In particular: an (ordinary) reflection is not a glide reflection, and a glide reflection is not a reflection, despite the overlap of terminology.

Now, there are a lot of interesting descriptions of what happens when you combine two of them. Here's some examples:

• The composition of two reflections in the same line is the identity.
• The composition of two reflections in different but parallel lines is a translation.
• The composition of two reflections in non-parallel lines is a rotation.
• The composition of a reflection across a line and a translation along a non-perpendicular line is a glide reflection (and not a reflection).
• The composition of a reflection across a line and a translation along a perpendicular line is a reflection (and not a glide reflection).

I could go on, but I think you might see the point now: there are interesting things to say about what happens when you combine two isometries, based on the types of the isometries that you combine, but that does not affect the fact that the five types are all different from each other.

Answer 3

The fact that a reflection and a translation can be combined (the technical term is composed) to give a glide reflection does not mean that glide reflections do not require a separate place in in classification. Similarly two reflections can be composed the give either a rotation (if the axes were intersecting lines) or a translation (if the axes were parallel) does not mean we need no translations or rotations in the classification once we have admitted reflections. A glide reflection is quite different from a rotation or a translation; for instance a glide reflection does not conserve orientation (so in this respect is more like a reflection, but a glide reflection does not have any fixed points). Also glide reflections can be characterised as being those isometries that cannot be obtained as composition of less that three reflections.

As noted by others, the identity logically belongs in a class of its own; although it can be seen as a limiting case of a translation or of a rotation, this does not take away its unique characteristics. There is nothing sacred about the five classes; for instance it could be argued that a half-turn is in a class by itself (unlike other reflections it can be realised by a scalar multiplication, namely by$~{-}1$) and that the remaining rotations can be partitioned into clockwise and counter-clockwise rotations. However the given classification results from looking at two parameters of the isometry: the minimal number of reflections from which it can be composed, and whether the set of fixed points is empty or not. This classification works in spaces of any dimension$~n$, and is as follows: isometries with fixed points can be a composition of $i~$reflections with $0\leq i\leq n$ and then have a set of fixed points of dimension$~n-i$; isometries without fixed points can be a composition of $i+2~$reflections with $0\leq i\leq n-1$, in which case they can be uniquely written as a composition of an isometry with a set of fixed points of dimension $n-i$ followed by a translation by a vector parallel to this set of fixed points (here the former accounts for $i$ reflections and the translation for $2$ more reflections; note that $i=n$ is excluded here, as it would not allow any nonzero translation to be chosen). For $n=2$ the isometries with fixed points for $i=0,1,2$ respectively give the identity, a reflection or a rotation, and the isometries without fixed points for $i=0,1$ respectively give a translation or a glide reflection.