Question

A glide reflection is a composition of a line reflection and a translation that is parallel to the line of reflection. Unlike some composition of transformations, a glide reflection IS ____________________. Therefore, the line reflection and translation can be done in any ____________.

Answer 1

" a glide reflection IS commutative. Therefore, the line reflection and translation can be done in any order."

Explanation:

Transformations can be written as operators that act on functions or shapes.

For example, we can define the horizontal translation operator as:

T(k)

And if we apply this to a function, we get:

T(k)[ f(x) ] = f(x - k)

Now, the operators are not necessarily commutative.

This means that if we have two operators:

K and L

is not the same:

K[ L[ f(x) ]]

than:

L[ K[ f(x) ]]

An example of this can be a reflection across the y-axis and a horizontal translation, that composition is non-commutative.

Now, let's go to our case.

A glide reflection is commutative, so we can perform the transformations in any order, then the statement is:

" a glide reflection IS commutative. Therefore, the line reflection and translation can be done in any order."