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/
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