Euclidean group
In mathematics, a Euclidean group is the group of isometries of an Euclidean space ?n; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension n of the space, and is commonly denoted E or ISO.
The Euclidean group E comprises all translations, rotations, and reflections of ?n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure of that space.
A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections.
These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.
Overview
Dimensionality
The number of degrees of freedom for E is, which gives 3 in case, and 6 for. Of these, n can be attributed to available translational symmetry, and the remaining to rotational symmetry.Direct and indirect isometries
The direct isometries comprise a subgroup of E, called the special Euclidean group and usually denoted by E+ or SE. They include the translations and rotations, and combinations thereof; including the identity transformation, but excluding any reflections.The isometries that reverse handedness are called "indirect". For any fixed indirect isometry R, such as a reflection about some hyperplane, every other indirect isometry can be obtained by the composition of R with some direct isometry. Therefore, the indirect isometries are a coset of E+, which can be denoted by E−. It follows that the subgroup E+ is of index 2 in E.
Topology of the group
The natural topology of Euclidean space ?n implies a topology for the Euclidean group E. Namely, a sequence fi of isometries of ?n is defined to converge if and only if, for any point p of ?n, the sequence of points pi converges.From this definition it follows that a function f:→E is continuous if and only if, for any point p of ?n, the function fp:→?n defined by fp = is continuous. Such a function is called a "continuous trajectory" in E.
It turns out that the special Euclidean group SE = E+ is connected in this topology. That is, given any two direct isometries A and B of ?n, there is a continuous trajectory f in E+ such that f = A and f = B. The same is true for the indirect isometries E−. On the other hand, the group E as a whole is not connected: there is no continuous trajectory that starts in E+ and ends in E−.
The continuous trajectories in E play an important role in classical mechanics, because they describe the physically possible movements of a rigid body in three-dimensional space over time. One takes f to be the identity transformation I of ?3, which describes the initial position of the body. The position and orientation of the body at any later time t will be described by the transformation f. Since f=I is in E+, the same must be true of f for any later time. For that reason, the direct Euclidean isometries are also called "rigid motions".
Lie structure
The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting.Relation to the affine group
The Euclidean group E is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups. This gives, a fortiori, two ways of writing elements in an explicit notation. These are:- by a pair, with A an orthogonal matrix, and b a real column vector of size n; or
- by a single square matrix of size, as explained for the affine group.
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. All affine theorems apply. The origin of Euclidean geometry allows definition of the notion of distance, from which angle can then be deduced.
Detailed discussion
Subgroup structure, matrix and vector representation
The Euclidean group is a subgroup of the group of affine transformations.It has as subgroups the translational group T, and the orthogonal group O. Any element of E is a translation followed by an orthogonal transformation, in a unique way:
where A is an orthogonal matrix
or the same orthogonal transformation followed by a translation:
with
T is a normal subgroup of E: for any translation t and any isometry u, we have
again a translation.
Together, these facts imply that E is the semidirect product of O extended by T, which is written as. In other words, O is also the quotient group of E by T:
Now SO, the special orthogonal group, is a subgroup of O, of index two. Therefore, E has a subgroup E+, also of index two, consisting of direct isometries. In these cases the determinant of A is 1.
They are represented as a translation followed by a rotation, rather than a translation followed by some kind of reflection.
This relation is commonly written as:
or, equivalently:
Subgroups
Types of subgroups of E:;Finite groups.:They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal among the finite groups: Oh and Ih. The groups Ih are even maximal among the groups including the next category.
;Countably infinite groups without arbitrarily small translations, rotations, or combinations: i.e., for every point the set of images under the isometries is topologically discrete. This includes lattices. Examples more general than those are the discrete space groups.
;Countably infinite groups with arbitrarily small translations, rotations, or combinations: In this case there are points for which the set of images under the isometries is not closed. Examples of such groups are, in 1D, the group generated by a translation of 1 and one of, and, in 2D, the group generated by a rotation about the origin by 1 radian.
;Non-countable groups, where there are points for which the set of images under the isometries is not closed: .
;Non-countable groups, where for all points the set of images under the isometries is closed: e.g.:
Examples in 3D of combinations:
- all rotations about one fixed axis
- ditto combined with reflection in planes through the axis and/or a plane perpendicular to the axis
- ditto combined with discrete translation along the axis or with all isometries along the axis
- a discrete point group, frieze group, or wallpaper group in a plane, combined with any symmetry group in the perpendicular direction
- all isometries which are a combination of a rotation about some axis and a proportional translation along the axis; in general this is combined with k-fold rotational isometries about the same axis ; the set of images of a point under the isometries is a k-fold helix; in addition there may be a 2-fold rotation about a perpendicularly intersecting axis, and hence a k-fold helix of such axes.
- for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R3, Dih.
Overview of isometries in up to three dimensions
| Type of isometry | Degrees of freedom | Preserves orientation? |
| Identity | 0 | |
| Translation | 1 | |
| Reflection in a point | 1 |
| Type of isometry | Degrees of freedom | Preserves orientation? |
| Identity | 0 | |
| Translation | 2 | |
| Rotation about a point | 3 | |
| Reflection in a line | 2 | |
| Glide reflection | 3 |
| Type of isometry | Degrees of freedom | Preserves orientation? |
| Identity | 0 | |
| Translation | 3 | |
| Rotation about an axis | 5 | |
| Screw displacement | 6 | |
| Reflection in a plane | 3 | |
| Glide plane operation | 5 | |
| Improper rotation | 6 | |
| Inversion in a point | 3 |
Chasles' theorem asserts that any element of E+ is a screw displacement.
See also 3D isometries that leave the origin fixed, space group, involution.
Commuting isometries
For some isometry pairs composition does not depend on order:- two translations
- two rotations or screws about the same axis
- reflection with respect to a plane, and a translation in that plane, a rotation about an axis perpendicular to the plane, or a reflection with respect to a perpendicular plane
- glide reflection with respect to a plane, and a translation in that plane
- inversion in a point and any isometry keeping the point fixed
- rotation by 180° about an axis and reflection in a plane through that axis
- rotation by 180° about an axis and rotation by 180° about a perpendicular axis
- two rotoreflections about the same axis, with respect to the same plane
- two glide reflections with respect to the same plane
Conjugacy classes
In 1D, all reflections are in the same class.
In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.
In 3D:
- Inversions with respect to all points are in the same class.
- Rotations by the same angle are in the same class.
- Rotations about an axis combined with translation along that axis are in the same class if the angle is the same and the translation distance is the same.
- Reflections in a plane are in the same class
- Reflections in a plane combined with translation in that plane by the same distance are in the same class.
- Rotations about an axis by the same angle not equal to 180°, combined with reflection in a plane perpendicular to that axis, are in the same class.