An isometry of Euclidean space that can be realized as a composition of rotations and translations, and depending on convention possibly reflections, which preserves shape and size of figures.

Principle

Rigid motions preserve all pairwise distances and (when reflections are excluded) orientation; in Euclidean n-space they form the special Euclidean group SE(n) when orientation-preserving, and the full Euclidean group E(n) when reflections are allowed.

Demonstration

Moving a solid object in R^3 by rotating it about an axis and then translating it results in a congruent placement—the mapping is a rigid motion and preserves interpoint distances and angles within the object.

Misapplication

Applying a uniform or nonuniform scaling, bending, or shear to an object is not a rigid motion; treating any distance-preserving map on a non-Euclidean metric as a Euclidean rigid motion is a category error.

Consequence

Rigid motions preserve congruence: two figures related by a rigid motion are congruent. They generate symmetry groups of rigid bodies and underlie notions of mechanical rigidity and allowable kinematic motions in rigid-body mechanics.

Reversal

Opposite operations are deformations that change internal distances (elastic or plastic changes) or affine transformations that alter angles or parallelism; the inverse of a rigid motion is another rigid motion (inverse rotation and translation).

Boundary

Term applies to Euclidean affine spaces and their standard inner-product metrics; in curved (Riemannian) spaces local isometries may act analogously but global rigid motions as group elements may not exist; excludes any map that changes distances.

Semantic Tension

Overlap with the term isometry (isometry is broader and applies to arbitrary metric spaces) and with 'Euclidean motion' which sometimes is taken to exclude reflections; also contrasted with physical notions of rigid body that include mass and inertia beyond pure geometry.

Synthesis

A rigid motion is a distance-preserving Euclidean transformation realizable by rotations and translations (and optionally reflections), the geometric operation capturing congruent repositioning without deformation.