##  [Rigid Motion](/rigid-motion-0) 

 Definition

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

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

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

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

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

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

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

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

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.