Glide reflection is a composite transformation which is a translation followed by a reflection in line parallel to the direction of translation. The order of the two constituent transforms (translation and reflection) is not important. One can easily verify that the same result is obtained by first reflecting and then translating the image.
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The following observations are noteworthy:
Glide reflection changes the orientation: if a polygon is traversed clockwise, its image is traversed counterclockwise, and vice versa.
Reflection is isometry: a glide reflection preserves distances.
Reflection preserves angles.
Reflection maps parallel lines onto parallel lines.
Unless the translation part of a glide reflection it trivial (defined by a 0 vector), the glide reflection has neither fixed points, nor fixed lines, save the axis of reflection itself. If the translation part is trivial, the glide reflection becomes a common reflection and inherits all its properties.
All these properties are implied by the definition of the glide reflection being a product of reflection and translation. The importance of the glide reflection lies in the fact that it is one of the four isometries of the plane.
Isometry, also called rigid motion, is a transformation (of the plane in our case) that preserves distances. It can be shown that there are only four plane isometries: translation, reflection, rotation and glide reflection. Together, the four are known as the basic rigid motions of the plane, which, in view of the fact that there are no others, is really a stupid nomenclature.
Copyright © 1996-2018 Alexander Bogomolny