1. The Euclidean group of all rigid motions ( conjugate of the original translation ). 2. See also subgroups of the Euclidean group . 3. The set of Euclidean plane isometries forms a composition : the Euclidean group in two dimensions. 4. The Euclidean group SE ( d ) of direct isometries is generated by translations and rotations. 5. The set consisting of all point reflections and translations is Lie subgroup of the Euclidean group . 6. It is precisely the subgroup of the Euclidean group that fixes the line at infinity pointwise. 7. These visualizations show the fundamental domains for 2D and 3D Euclidean groups , and 2D spherical groups. 8. A " rigid body motion " is in effect the same as a curve in the Euclidean group . 9. Similarly the Euclidean group , which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. 10. The Euclidean group for SE ( 3 ) is used for the kinematics of a rigid body, in classical mechanics.
