The set of proper rigid transformation is called special Euclidean group, denoted SE ( " n " ).
12.
It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections.
13.
The smallest subgroup of the Euclidean group containing all translations by a given distance is the set of " all " translations.
14.
The geometry associated with this pseudo-metric was investigated by translations and plays the same role as Euclidean groups of ordinary Euclidean spaces.
15.
Is a subgroup of the Euclidean group, the group of-sphere and all objects with spherical symmetry, if the origin is chosen at the center.
16.
Another classical case occurs when M is the cotangent bundle of \ mathbb { R } ^ 3 and G is the Euclidean group generated by rotations and translations.
17.
For "'su "'( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.
18.
As the group of all isometries,, the Euclidean group is important because it makes Euclidean geometry a case of Klein geometry, a theoretical framework including many alternative geometries.
19.
In the terms of Felix Klein's Erlangen programme, we read off from this that Euclidean geometry, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry.
20.
The even isometries identity, rotation, and translation never do; they correspond to " rigid motions ", and form a normal subgroup of the full Euclidean group of isometries.
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