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### 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?"

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/
```
Associated Topics:
High School Equations, Graphs, Translations
High School Euclidean/Plane Geometry
High School Geometry
Middle School Geometry
Middle School Graphing Equations
Middle School Two-Dimensional Geometry

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