User:Tomruen/aa
Euclidean_group#Overview_of_isometries_in_up_to_three_dimensions
The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, 6 for n = 3, and 10 for n = 4. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.
Overview of isometries in up to three dimensions
[edit]E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:
| Type of isometry | Degrees of freedom | Preserves orientation? |
|---|---|---|
| Identity | 0 | yes |
| Translation | 1 | yes |
| Reflection in a point | 1 | no |
| Type of isometry | Degrees of freedom | Preserves orientation? |
|---|---|---|
| Identity | 0 | yes |
| Translation | 2 | yes |
| Rotation about a point | 3 | yes |
| Reflection in a line | 2 | no |
| Glide reflection Transflection |
2 | no |
| Type of isometry | Degrees of freedom | Preserves orientation? |
|---|---|---|
| Identity | 0 | yes |
| Translation | 3 | yes |
| Rotation about an axis | 5 | yes |
| Screw displacement Rotary translation |
6 | yes |
| Reflection in a plane | 3 | no |
| Glide plane operation Transflection |
5 | no |
| Improper rotation Rotary reflection |
6 | no |
| Inversion in a point | 3 | no |
Chasles' theorem asserts that any element of E+(3) is a screw displacement.
See also 3D isometries that leave the origin fixed, space group, involution.
| Type of isometry | Degrees of freedom | Preserves orientation? |
|---|---|---|
| Identity | 0 | yes |
| Translation | 4 | yes |
| Rotation | 7 | yes |
| Rotary translation | 10? | yes |
| Double rotation | 10? | yes |
| Inversion in a point | 4 | yes |
| Reflection | 4 | no |
| Transflection | 8? | no |
| Rotary reflection | 8? | no |
| Rotary transflection | 8? | no |