Isometry group of Euclidean space
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space formula_1; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension "n" of the space, and is commonly denoted E("n") or ISO("n"), for "inhomogeneous special orthogonal" group.
The Euclidean group E("n") comprises all translations, rotations, and reflections of formula_1; 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, often denoted SE("n") and E+("n"), 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("n") is "n"("n" + 1)/2, which gives 3 in case "n" = 2, and 6 for "n" = 3. Of these, "n" can be attributed to available translational symmetry, and the remaining "n"("n" − 1)/2 to rotational symmetry.
Direct and indirect isometries.
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 formula_1 implies a topology for the Euclidean group E("n"). Namely, a sequence "f""i" of isometries of formula_1 (formula_5) is defined to converge if and only if, for any point "p" of formula_1, the sequence of points "p""i" converges.
From this definition it follows that a function formula_7 is continuous if and only if, for any point "p" of formula_1, the function formula_9 defined by "f""p"("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 formula_1, 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 transformation "I" of formula_11, 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".
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("n") is a subgroup of the affine group for "n" dimensions. Both groups have a structure as a semidirect product of the group of Euclidean translations with a group of origin-preserving transformations, and this product structure is respected by the inclusion of the Euclidean group in the affine group. This gives, "a fortiori", two ways of writing elements in an explicit notation. These are:
Details for the first representation are given in the next section.
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("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: formula_12 where "A" is an orthogonal matrix
or the same orthogonal transformation followed by a translation: formula_13 with "c" = "Ab"
T("n") is a normal subgroup of E("n"): for every translation "t" and every isometry "u", the composition formula_14 is again a translation.
Together, these facts imply that E("n") is the semidirect product of O("n") extended by T("n"), which is written as formula_15. In other words, O("n") is (in the natural way) also the quotient group of E("n") by T("n"): formula_16
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).
This relation is commonly written as: formula_17
or, equivalently: formula_18
Subgroups.
Types of subgroups of E("n"):
*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:
Overview of isometries in up to three dimensions.
E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:
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.
Commuting isometries.
For some isometry pairs composition does not depend on order:
Conjugacy classes.
The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.
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: