How Glide Reflections Work - dummies

By Mark Ryan

A glide reflection is just what it sounds like: You glide a figure (that’s just another way of saying slide or translate) and then reflect it over a reflecting line. Or you can reflect the figure first and then slide it; the result is the same either way.

The footprints are glide reflections of each other.

A glide reflection is also called a walk because it looks like the motion of two feet, as shown here. In a sense, it’s the most complicated of the four types of isometries because it’s the composition of two other isometries: a reflection and a translation.

If you have a pre-image and an image like the two feet in the figure, it’s impossible to move the pre-image to the image with one simple reflection, one translation, or one rotation (try it—you’ll see!). The only way to get from the pre-image to the image is with a combination of one reflection and one translation. And because you can produce the translation part with two reflections, you can achieve a glide reflection with three reflections.

Some images are just one reflection away from their pre-images; other images (in translation and rotation problems) are two reflections away. And now you see that glide reflection images are three reflections away from their pre-images. It’s interesting to note that this covers all possibilities. In other words, every image—no matter where it is in the coordinate system and no matter how it’s spun around or flipped over—is either one, two, or three reflections away from its pre-image. Pretty cool, eh?