Rigid Motion
A 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 basic rigid motions are translations (slides), reflections (flips), and rotations (turns), and a shape and its image are always congruent after a rigid motion. Formally, a rigid motion (isometry) 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 reflection composed with a translation along the mirror line). An isometry of $\mathbb{R}^2$ 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), and if $\det(A) = -1$, it is orientation-reversing (reflection or glide reflection). The group of all such isometries is the Euclidean group $E(2)$.
Example
Sliding, flipping, or turning a puzzle piece are all rigid motions: the piece never gets bigger or smaller and always keeps its exact shape, which is why puzzle pieces always fit no matter how you move them. A glide reflection maps a left footprint to a right footprint and back again, forming the pattern of footprints in sand. The Mazur-Ulam theorem extends this idea to infinite-dimensional spaces: every surjective isometry between normed spaces is affine, showing that preserving distance forces near-linearity even without assuming linearity.
Key Insight
Dilation is NOT a rigid motion because it changes size; only transformations preserving exact distances between all pairs of points qualify, and two figures are congruent if and only if one can be obtained from the other by a rigid motion. The congruence theorems (SSS, SAS, and so on) are essentially descriptions of when such a rigid motion exists. In Riemannian geometry, isometries of curved surfaces are distance-preserving diffeomorphisms; on the sphere the isometry group is $O(3)$, and understanding isometry groups of geometric spaces is central to differential geometry, crystallography, and the classification of symmetric spaces.