Isometries are transformations the plane which don't distort
shapes. For this reason, another common name for an isometry is a
rigid motion. The most familiar isometry is probably just
translation  shifting a shape in a straight line. However,
another very familiar kind of isometry is rotation:
Unlike translation, this isometry has an obvious fixed
point. The pivot point around which everything rotates doesn't
move at all. Using the analogy of a square drawn on a large
transparent sheet of plastic, you might do a rotation by holding down
the pivot point with your finger, and whirling the sheet around it.
Translations have no fixed points at all, while rotations have
exactly one  namely, if you stay away from the pivot point,
everything else moves. This is worth pointing out, since it suggests
a way of classifying any kind of isometry, by considering their
fixed point behavior.
