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Rigid Motions

There are four kinds of distance preserving symmetries of the plane:

Move all points the same distance in the same direction.
by an angle about a fixed center of rotation.
in a fixed line (the mirror).
Glide Reflections:
reflection in a mirror followed by a translation parallel to that mirror line.

Applying first one symmetry s and then another t is called composition $t \circ s$ of the symmetries. We'll be interested in collections of symmetries which form a group. This means the composition of two symmetries in the collection is again in the collection. And the inverse of a symmetry in the collection is again in it. (The inverse of a symmetry involves reversing the correspondence of points in the symmetry. For example, the inverse of rotation counterclockwise is rotation clockwise by the same angle. Or the inverse of translation left is translation right by the same distance.)