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 v maps every point A in the plane onto its image A' such that the following statements hold true.

• The length of AA' and the magnitude of v are equal.
• Segment AA' and v are parallel.
• Vectors AA' and v have the same direction.

These three properties imply that the quadrilateral formed by A, A', the tip of v, and the tail of 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 l maps every point A in the plane onto its image A' such that one of the following statements is satisfied.

• If A is on the line l, then A and A' are the same point.
• If A is not on the line l, then l is the perpendicular bisector of AA'.

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 α ^(∘) 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' such that one of the following statements is satisfied.

• If A is the center of rotation, then A and A' are the same point.
• If A is not the center of rotation, then A and A' are equidistant from P, with ∠ APA' measuring α ^(∘).

Rotations are usually performed counterclockwise unless stated otherwise. Notice that a 90^(∘) counterclockwise rotation is the same as a 270^(∘) 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^(∘) and 360^(∘) — about a line. As the given example demonstrates, a hexagonal prism can be rotated 60^(∘), 120^(∘), 180^(∘), 240,^(∘) or 300^(∘) about the axis of symmetry l and it would still look the same. It is worth noting that rotating any figure 360^(∘) 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.