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
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"