Rigid Motions Properties and Examples

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 in what is called the line of reflection. This acts like a mirror. In more precise terms, 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. Notice that a $90^{\circ }$ counterclockwise rotation is the same as a $270^{\circ }$ clockwise rotation. 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 — strictly between $0^{\circ }$ and $360^{\circ }$ — about a line. As the given example demonstrates, 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 change how it looks.

A glide reflection is a transformation that combines a translation and a reflection across a line parallel to the translation vector. It is a composition of rigid motions — meaning it, too, is considered a rigid motion. The line of reflection and the translation vector are parallel. That means a glide reflection can be a reflection followed by a translation, rather than only a translation followed by a reflection. The image does not depend on the order of the transformations.