A rigid motion, or isometry, is a transformation that preserves the distance between any two points on the preimage. The following diagram displays two logos. The logo with the points $A$ and $B$ is the preimage and the logo with the points $A^{\prime }$ and $B^{\prime }$ is the image. The image is the result of a rigid motion because the distances between all points are preserved. Rigid motions are also called congruence transformations because the preimage and its image under a rigid motion are congruent figures. Some examples of rigid motions are translations, reflections, and rotations.

A translation is a transformation that moves every point of a figure the same distance in the same direction. More precisely, a translation along a vector $\vec{v}$ maps every point $A$ in the plane onto its image $A^{\prime }$ such that the following statements hold true.

• The length of $\overline{A{A}^{\prime }}$ and the magnitude of $\vec{v}$ are equal.
• Segment $\overline{A{A}^{\prime }}$ and $\vec{v}$ are parallel.
• Vectors $\overrightarrow{A{A}^{\prime }}$ and $\vec{v}$ have the same direction.

These three properties imply that the quadrilateral formed by $A,$ $A^{\prime },$ the tip of $\vec{v},$ and the tail of $\vec{v}$ is a parallelogram. Since translations preserve side lengths and angle measures, they are rigid motions. Additionally, translations map lines onto parallel lines.

A reflection is a transformation in which every point of a figure is reflected across a line. The line across the points are reflected is called the line of reflection and acts like a mirror. More precisely, a reflection across a line $\ell$ maps every point $A$ in the plane onto its image $A^{\prime }$ such that one of the following statements is satisfied.

• If $A$ is on the line $\ell ,$ then $A$ and $A^{\prime }$ are the same point.
• If $A$ is not on the line $\ell ,$ then $\ell$ is the perpendicular bisector of $\overline{A{A}^{\prime }}.$

Like rotations and translations, reflections are rigid motions because they preserve the side lengths and angle measures. However, reflections can change the orientation of the preimage.

A rotation is a transformation in which a figure is turned about a fixed point $P.$ The number of degrees the figure rotates ${\alpha }^{\circ }$ is the angle of rotation. The fixed point $P$ is called the center of rotation. Rotations map every point $A$ in the plane to its image $A^{\prime }$ such that one of the following statements is satisfied.

• If $A$ is the center of rotation, then $A$ and $A^{\prime }$ are the same point.
• If $A$ is not the center of rotation, then $A$ and $A^{\prime }$ are equidistant from $P$, with $\angle AP{A}^{\prime }$ measuring ${\alpha }^{\circ }.$

Rotations are usually performed counterclockwise unless stated otherwise. Since rotations preserve side lengths and angle measures, they are rigid motions.

A figure in space is said to have axis symmetry if it can be mapped onto itself after a rotation that is strictly between $0^{\circ }$ and $360^{\circ }$ about a line. As the example above shows, a hexagonal prism can be rotated $60^{\circ },$ $120^{\circ },$ $180^{\circ },$ $240{,}^{\circ }$ or $300^{\circ }$ about the axis of symmetry $\ell$ and it would still look the same. It is worth noting that rotating any figure $360^{\circ }$ about any axis will not modify how it looks.

A glide reflection is a transformation that combines a translation and a reflection across a line parallel to the translation vector. Since a glide reflection is a composition of rigid motions, it is also a rigid motion. Thanks to the fact that the line of reflection and the translation vector are parallel, a glide reflection could instead be a reflection followed by a translation. That is, the image does not depend on the order of the transformations.