Non-parallel Glide Reflections

Date: 10/21/98 at 17:54:00
From: Stacy Shubert
Subject: reflections of a line

I am in training to be a future teacher. We received some questions 
that were asked by high school students, and we are supposed to answer 
them. A lot of them I can answer but this one really stumped me.  
Can you help me?

"If a glide reflection is defined to be the composition of a line 
reflection and a translation (or glide) in a direction parallel to the 
axis of reflection, what is the composition when the translation is 
not parallel to the axis of reflection?"

I'll thank you in advance for any help you may be able to give me on 
this.  

Thanks,
Stacy Shubert

Date: 10/22/98 at 16:55:55
From: Doctor Peterson
Subject: Re: reflections of a line

Hi, Stacy. I was a little confused by this myself, because glide 
reflections are always defined this way, so it's a good question what 
happens if you relax the definition. But then I experimented a little 
(using the Geometer's Sketchpad) and found that in fact if you 
translate by any vector the result is still a glide reflection! I 
probably should have known this, but discovering it was fun and an 
experience that I would recommend sharing with a student.

If I reflect object 1 in line L (2) and translate by vector V (3):

                      | 3
                      +--


         _*
         /|
        /
      V/       | 2
      /        +--
     /                              L
    +-----------------------------------------

               +--
               | 1

the result is the same as if I reflected it in line M parallel to L, 
which is the perpendicular bisector of segment QR, and translated it 
by vector PR parallel to L:

               | 2   | 3
               +--   +--

         _+Q
         /|
        / |                           M
   - -V/ -|- - - - - - - - - - - - - - - - - -
      /   |
     /    |                           L
    +---->+-----------------------------------
    P     R
               +--
               | 1

So we define glide reflection as we do only for convenience. It allows 
us to have a unique description of any glide reflection defined by a 
single directed line segment PR. If I were talking to the student who 
asked the question, I would ask the class to experiment with this 
without revealing the outcome, so they could discover it themselves; 
then perhaps they could try to prove it.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/