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Topic:  Glide Reflections... order count? 
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Subject:  RE: Glide Reflections... order count? 
Author:  Craig 
Date:  Mar 16 2005 
> When I teach Glide Reflections I usually demonstrate with Geometer's
> Sketchpad by Translating and then reflecting across a line parallel
> to the direction of the translation. I was surprised to see that
> some textbooks teach it that way, and others teach it reflect and
> then translate. Which do you think is the best way to approach it
> with the students?
I vote for using both ways, revisiting the idea of commutative operations
along the way. The glide reflection is really a composition of two simple
transformations (translation and reflection, as you point out). Normally,
translation and reflection are NOT commutative: if the translation is not
parallel to the mirror, you don't wind up in the same spot if you do them in a
different order.
I just did something with Sketchpad to illustrate: I drew a small triangle,
then a line segment. I marked the line segment as my mirror for the reflection,
and I also used the endpoints of the segment to mark a vector for the
translation. I reflected the vertices only of the triangle, then constructed a
triangle interior in the reflection. I chose a color for the interior that
contrasts with the line color in the original triangle. I then translated both
the original triangle and its reflection by the marked vector, then reflected
the translated copy of the triangle outline. Behold, the translated image of
the reflection now has a contrasting boundary, the reflected image of the
translation!
 
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