What is special about glide reflection?

Question

In wikipedia article https://en.wikipedia.org/wiki/Translation_(geometry) it is written:

A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections.

Knowing that a glide reflection is just a composition of a reflection with a translation, I ask why citing this composition and not the other compositions like the composition of a translation and rotation or a composition of rotation and reflection ? what is special about the composition of a reflection and a translation that makes it one of the four rigid motions but not the other compositions ? Thank you for your help!

The composition of a reflection and a general translation will not be a glide reflection; it'll just be a reflection about a different line. But glide reflections are special because you're translating parallel to the line of reflection, and this is not again a reflection. — Ted Shifrin, Jun 25 '16 at 23:10
In 2D Euclidean space, the composition of rotation about a point and reflection in a line may be a reflection in a different line; the composition of a translation and rotation may be a rotation about a different point — Henry, Jun 25 '16 at 23:18
So basically you are saying that glide reflection is the only composition from translations, rotations and reflections that will not give one of these three rigid motions, and any other composition other then glide reflection will give one of the three rigid motions.. is this correct ? — palio, Jun 25 '16 at 23:21
This is pretty much the same question as the one here: https://math.stackexchange.com/questions/3055087/why-are-glide-reflections-one-of-four-isometries/3056129#3056129 — Moishe Kohan, Feb 23 '20 at 00:06
"A translation can be described as a rigid motion: the other rigid motions are rotations, reflections and glide reflections." I can't find this in that wikipedia article. — john, Jan 25 '21 at 08:43
@TedShifrin The composition of a reflection and a general translation will not be a glide reflection; it'll just be a reflection about a different line." or it will be a glide reflection about a different line of reflection ? or a composition of reflections (i am thinking of the general translation) — john, Jan 25 '21 at 09:10

score 3 · Answer 1 · answered Jun 25 '16 at 23:28

3

Because a composition between a translation and a rotation is already listed there, since such a composition is a rotation (typically with a different center). And a composition of a rotation and a reflection is another reflection. In other words, the list given is complete: every isometry is one of the things in that list, not just a composition of things in that list.

answered Jun 25 '16 at 23:28

Michael Hardy

1

I don't have a job working in collaboration with other mathematicians so I don't know whether it goes without saying in the mathematical community that when you talk about a composition of two functions in an English sentence, it means in the order you wrote them. Also it's not the case that when you apply a reflection then a rotation in the second dimension, you always get a reflection nor is it the case that when you apply a rotation then a reflection in the second dimension, you always get a reflection. What is true is that when ever you apply a rotation about the origin then a reflection – Timothy Feb 22 '20 at 23:36
about a line through the origin or you apply a reflection about a line through the origin then a rotation about the origin in the second dimension, you always get a reflection about a line through the origin. What ever you would have written if you had figured out how it really works, I'm not sure it would have answered the question anyway because it wouldn't explain the reason people consider a glide reflection special. I think a downvote is intended for when the answer can't be fixed up into an answer that actually answers the question without rewriting it in a fundamentally different way. – Timothy Feb 22 '20 at 23:41
That's how I believe your answer is so I downvoted it. The Stack Exchange community has a lot of things on their plate and by downvoting it, I can divert their attention to answers that are more worthy of their time. – Timothy Feb 22 '20 at 23:44
@Timothy Since we're working in a plane (implied by the fact that screw translations and rotatory reflections are not listed), how is "rotation in the second dimension" different from just "rotation"? – David K Oct 16 '21 at 15:59

Timothy · Answer 2 · 2020-02-26T18:06:02.860

$\mathbb{R}^2$ is the set of all ordered pairs of real numbers. A rigid motion on $\mathbb{R}^2$ is defined to be a transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ that preserves distances. It can be shown that a transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$ is a rigid motion if and only if it can be gotten either by applying a rotation about the origin then a translation or by applying an inversion about the x-axis then a rotation about the origin then a translation. In this answer, I define a rigid motion to be noninverting when it can be gotten by applying a rotation about the origin then a translation and inverting when it can be gotten by applying an inversion about the x-axis then a rotation about the origin then a translation.

I don't know why people consider a glide reflection special so I will make a guess. I'm guessing that sometimes people make the assumption that all inverting rigid motions are reflections, and when they discover that that's not the case, they find it interesting, and that's what made the glide reflection special.

In the past, I independently thought all by myself about how it's interesting that not all inverting rigid motions are reflections. The inverting rigid motions that are not reflections are glide reflections. Maybe some people consider glide reflections special for that reason.

A question is not just the question stated in the title. While you have given an answer to the question in the title "What is special about glide reflection?", you do not appear to have made any attempt to address the real question as clarified in the body of the question, which is why glide reflections are included in the list of types of rigid motions and other compositions are not. (Or if you are attempting to answer that question, your answer is just wrong.) — Eric Wofsey, Feb 26 '20 at 18:02
"Rotation" in the classification given in the question is considered to be a rotation about any point, not just the origin. A reflection may be about any line, not just the $x$ axis or even a line through the origin. In a plane, a rotation followed by a translation is a rotation. You have correctly developed a way of classifying isometries in the plane, but that does not show that someone else's classification is incorrect. The classification in the question is designed to be independent of the coordinate system. — David K, Oct 16 '21 at 16:10

What is special about glide reflection?

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