ISO(n)
E518472
ISO(n) is the n-dimensional Euclidean group consisting of all distance-preserving transformations—combinations of rotations, reflections, and translations—of n-dimensional Euclidean space.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical group
ⓘ
matrix group ⓘ topological group ⓘ |
| actionType | isometric action ⓘ |
| actsOn |
R^n
ⓘ
n-dimensional Euclidean space ⓘ |
| applicationArea |
classical mechanics
ⓘ
computer vision ⓘ crystallography ⓘ rigid body mechanics ⓘ robotics ⓘ |
| componentGroup | O(n)/SO(n) NERFINISHED ⓘ |
| connectedComponentOfIdentity | SO(n) ⋉ R^n ⓘ |
| containsTransformations |
reflections
ⓘ
rotations ⓘ translations ⓘ |
| dimension | n(n+1)/2 ⓘ |
| elementType |
combination of reflection and translation
ⓘ
combination of rotation and translation ⓘ distance-preserving transformation ⓘ isometry of R^n ⓘ |
| field | mathematics ⓘ |
| groupOperation | composition of maps ⓘ |
| hasSubgroup |
E(n)
ⓘ
SO(n) ⓘ orientation-preserving Euclidean group ⓘ |
| isometryGroupOf | Euclidean n-space NERFINISHED ⓘ |
| isomorphicTo | O(n) ⋉ R^n NERFINISHED ⓘ |
| normalSubgroup | R^n ⓘ |
| notationVariant |
E(n)
ⓘ
Isom(R^n) NERFINISHED ⓘ |
| preserves |
Euclidean distance
ⓘ
inner product up to translation ⓘ orientation (for orientation-preserving subgroup) ⓘ |
| quotientBy | R^n ⓘ |
| quotientIsomorphicTo | O(n) ⓘ |
| relatedConcept |
Euclidean geometry
NERFINISHED
ⓘ
Killing vector field on Euclidean space ⓘ rigid motion ⓘ |
| semidirectProductOf |
O(n)
ⓘ
R^n ⓘ |
| structure |
non-compact Lie group
ⓘ
non-semisimple Lie group ⓘ non-simple Lie group ⓘ |
| subfield |
Lie theory
ⓘ
geometry ⓘ group theory ⓘ |
| topology | Lie group topology induced from O(n) × R^n ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.