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Isometries: Glide Reflection
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The least well-known kind of isometry is usually called a glide reflection. It is a kind of cross between a reflection and a translation:

Unlike translation, rotation and reflection, we can't distinguish a glide reflection from the others just by considering how many fixed points there are. Just like a translation, a glide reflection doesn't have any fixed points. On the surface, you might think that points lying on the red line are left fixed, just as with a reflection. But if you think in terms of a square drawn on a large sheet of transparent plastic, you will realize that the entire red line shifts along itself in the direction of motion.

Similarly, we can't distinguish a glide reflection from the others just by asking whether is preserves or reverses orientation. Like a reflection, it reverses orientations, transforming the original square into its mirror image. Again, thinking of the plastic sheet analogy, getting to the final position requires flipping the sheet over as you shift it.

However, if we consider both orientation and fixed point behavior, each type of isometry has a unique character:

  fixed points orientation
Translation none preserving
Rotation one preserving
Glide Reflection none reversing
Reflection infinite reversing

Next: "Reflections on Reflections"

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