Rigid Motion
Geometry & MeasurementA rigid motion is a transformation that preserves distances and angle measures, so the image is always congruent to the preimage.
Definition
A rigid motion is a transformation that moves a shape without changing its size or shape. The three rigid motions are translations (slides), reflections (flips), and rotations (turns). A shape and its image are always congruent after a rigid motion.
Example
Sliding, flipping, or turning a puzzle piece are all rigid motions. The piece never gets bigger or smaller, and it always keeps its exact shape. That is why puzzle pieces always fit no matter how you move them.
Key Insight
Dilation is NOT a rigid motion because it changes the size. Only transformations that preserve exact distances between all pairs of points are rigid motions. Two figures are congruent if and only if one can be obtained from the other by a rigid motion.
Definition
A rigid motion (isometry) is a transformation that preserves all distances: for any two points $A$ and $B$, the distance $AB$ equals the distance $A'B'$ between their images. The four isometries of the plane are translations, rotations, reflections, and glide reflections (a combination of a reflection and a translation along the mirror line).
Example
A glide reflection maps a left footprint to a right footprint and then to a left footprint again, forming the pattern of footprints in sand. It is the composition of a reflection and a translation parallel to the mirror line.
Key Insight
Two triangles are congruent if and only if there is a rigid motion mapping one to the other. The congruence theorems (SSS, SAS, etc.) are essentially descriptions of when such a rigid motion exists. Rigid motions formalize the intuitive idea that congruent shapes are "the same up to position."
Definition
An isometry of $\mathbb{R}^2$ is a map $f: \mathbb{R}^2 \to \mathbb{R}^2$ satisfying $|f(x) - f(y)| = |x - y|$ for all $x, y$. Every isometry is an affine map $T(x) = Ax + b$ where $A$ is orthogonal ($A^TA = I$). If $\det(A) = 1$, $T$ is orientation-preserving (translation or rotation); if $\det(A) = -1$, $T$ is orientation-reversing (reflection or glide reflection). The group of isometries is the Euclidean group $E(2)$.
Example
The Mazur-Ulam theorem extends this to infinite-dimensional spaces: every surjective isometry between normed spaces is affine. This deep result shows that preserving distance forces near-linearity, even without assuming linearity.
Key Insight
In Riemannian geometry, isometries of curved surfaces are defined as distance-preserving diffeomorphisms. On the sphere, the isometry group is $O(3)$. Understanding isometry groups of geometric spaces is central to differential geometry, crystallography, and the classification of symmetric spaces.