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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)
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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.)
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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.
