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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!

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    $\begingroup$ 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. $\endgroup$ – Ted Shifrin Jun 25 '16 at 23:10
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    $\begingroup$ 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 $\endgroup$ – Henry Jun 25 '16 at 23:18
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    $\begingroup$ 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 ? $\endgroup$ – palio Jun 25 '16 at 23:21
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    $\begingroup$ Precisely, @palio. :) $\endgroup$ – Ted Shifrin Jun 26 '16 at 4:05
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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.

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