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The Euclidean group E(n) comprises all translations, rotations, and reflections of ; 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 (subset) 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.
The direct isometries (i.e., isometries preserving the handedness of chiral subsets) comprise a subgroup of E(n), called the special Euclidean group and usually denoted by E+(n) or SE(n). 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, or opposite. 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+(n), which can be denoted by E-(n). It follows that the subgroup E+(n) is of index 2 in E(n).
Topology of the group
The natural topology of Euclidean space implies a topology for the Euclidean group E(n). Namely, a sequence fi of isometries of (i?) is defined to converge if and only if, for any point p of , the sequence of points pi converges.
From this definition it follows that a function is continuous if and only if, for any point p of , the function defined by fp(t) = (f(t))(p) is continuous. Such a function is called a "continuous trajectory" in E(n).
It turns out that the special Euclidean group SE(n) = E+(n) is connected in this topology. That is, given any two direct isometries A and B of , there is a continuous trajectory f in E+(n) such that f(0) = A and f(1) = B. The same is true for the indirect isometries E-(n). On the other hand, the group E(n) as a whole is not connected: there is no continuous trajectory that starts in E+(n) and ends in E-(n).
The continuous trajectories in E(3) 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(0) to be the identity transformationI of , 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(t). Since f(0)=I is in E+(3), the same must be true of f(t) for any later time. For that reason, the direct Euclidean isometries are also called "rigid motions".
The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both[clarification needed] groups. This gives, a fortiori, two ways of writing elements in an explicit notation. These are:
It has as subgroups the translational group T(n), and the orthogonal group O(n). Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:
Together, these facts imply that E(n) is the semidirect product of O(n) extended by T(n), which is written as . In other words, O(n) is (in the natural way) also the quotient group of E(n) by T(n):
Now SO(n), the special orthogonal group, is a subgroup of O(n) of index two. Therefore, E(n) has a subgroup E+(n), 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 (in dimensions 2 and 3, these are the familiar reflections in a mirror line or plane, which may be taken to include the origin, or in 3D, a rotoreflection).
They always have a fixed point. In 3D, for every point there are for every orientation two which are maximal (with respect to inclusion) 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 (e.g., for a group generated by m translations in independent directions, and possibly a finite point group). 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
(e.g., in 2D all translations in one direction, and all translations by rational distances in another direction).
Non-countable groups, where for all points the set of images under the isometries is closed
all direct isometries that keep the origin fixed, or more generally, some point (in 3D called the rotation group)
all isometries that keep the origin fixed, or more generally, some point (the orthogonal group)
all direct isometries E+(n)
the whole Euclidean group E(n)
one of these groups in an m-dimensional subspace combined with a discrete group of isometries in the orthogonal (n-m)-dimensional space
one of these groups in an m-dimensional subspace combined with another one in the orthogonal (n-m)-dimensional space
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(R3).
William Thurston. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN0-691-08304-5