Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 23

Recall that a glide reflection is defined as the composition of a reflection with a translation parallel to the line of reflection.

1. Let R be a reflection in a line L, and let T be a translation parallel to L. Prove that R composed with T equals T composed with R. Hint: first do this for the special case where L is the x-axis. (The point of this problem is that in the definition of "glide reflection" the order of composition doesn't matter.)
2. Prove that a glide reflection has no fixed points. Hint: same as above.
3. Let R be a reflection in a line L, and let T be a translation perpendicular to L. Prove that R composed with T is a reflection. Hint: you know the routine...!
4. Let R be a reflection in a line L, and let T be a translation not perpendicular to L. Prove that R composed with T is a glide reflection. Hint: Decompose the translation into two other translations, one perpendicular, and one parallel to L; then use the above problems.