So far each move kept one point fixed (the mirror line or the centre). Now we add two moves that travel.

• A translation slides every point the same distance in the same direction. It is described by a vector (a length and a direction). A translation by a non-zero vector has no fixed point: nothing stays where it was. Sliding a shape never turns it or flips it.
• A glide reflection is a reflection followed by a slide along the mirror line. The footprints of a person walking along a beach make the standard picture: the left footprint and the right footprint are mirror images of each other, and each is also shifted forward. Reflect the left print in the centre line and slide it forward by one step and it lands on the next right print.

A glide reflection is not the same thing as a translation (the shape ends up flipped), and not the same as a plain reflection (the shape has moved along the line). Like a translation, it has no fixed point.

Together, the four moves are called the isometries of the plane: reflection, rotation, translation and glide reflection. “Isometry” means “same measure”: distances and angles are kept. There are no other isometries of the plane. Every move that keeps all distances is one of these four.