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Isometries: Reflection
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In thinking about translation and rotation, we discovered that a useful question to ask about an isometry is whether or not it has fixed points, i.e. points that do not move. Translations have no fixed points, while rotations have exactly one, namely the pivot point around which everything else turns. Reflections are isometries that have infinitely many fixed points. No point lying in the red line moves as a result of doing a reflection:

On the other hand, reflections are notable for introducing another general property of isometries. We say that reflection is orientation reversing.

One way of describing what it means for an isometry to be orientation reversing is to think of the square as being drawn on a large clear sheet of plastic laying on a white table top. To translate or rotate the square, you just have to slide the plastic sheet around on the table top. By contrast, to do a reflection, you have to pick up the sheet, and turn it over, before sliding the square into its final position.

Another way of describing the difference between and orientation reversing and preserving transformations is that an orientation reversing transformation gives you back a mirror image of your original shape. In the case illustrated above, the color pattern on the square after the reflection is what you would see if you looked at the original square in a mirror set with one edge along the red line, sometimes called the mirror line.

Next: "Glide Reflection"

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