Chapter 14.3, Problem 19WE

To determine

To explain:A glide reflection is an isometry.

Answer to Problem 19WE

A glide reflection is an isometry.

Explanation of Solution

Given information :

There is a glide reflection transformation.

Concept used:

In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation.

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.

An isometry of the plane is a linear transformation which preserves length.If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections (at most two if there is a fixed point).

Since translations and reflections are both isometries, a glide reflection is also an isometry. (The composition of isometries is an isometry.)

i.e.

Glide reflection is composition of glide & reflection where translation & reflection are isometric since after translation the length or the distance remains the same.

Hence,

A glide reflection is an isometry.