In wikipedia article (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!
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$\begingroup$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.
$\endgroup$ 4 $\begingroup$$\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.
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