Glide Reflection
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When a translation (a slide or glide) and a reflection are performed one after the other, a transformation called a glide reflection is produced.  In a glide reflection, the line of reflection is parallel to the direction of the translation.  It does not matter whether you glide first and then reflect, or reflect first and then glide.  This transformation is commutative.

When two or more transformations are combined to form a new transformation, the result is called a composition of transformations.  Since translations and reflections are both isometries, a glide reflection is also an isometry.  (The composition of isometries is an isometry.)
 

Properties preserved (invariant) under a glide reflection:
(Since these properties are preserved under both the reflection and the translation, they are preserved under the glide reflection.)
1.  distance is preserved (lengths of segments are the same)
2.  angle measures (remain the same)
3.  parallelism (parallel lines remain parallel)
4.  colinearity (points stay on the same lines)
5.  midpoint (midpoints remain the same in each figure)
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6.  orientation NOT preserved (lettering order does not remain the same)


A glide reflection is an opposite isometry.
(A translation is a direct isometry and a reflection is an opposite isometry.  Their composition is an opposite isometry.  The composition of a direct isometry and an opposite isometry is an opposite isometry.)

 

Definition:  A glide reflection is a transformation in the plane that is the composition of a line reflection and a translation through a line (a vector) parallel to that line of reflection.

is the image of  under a glide reflection
that is a composition of a reflection over the line l
and a translation through the vector v.