Euclidean group
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Basic notions
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Infinite dimensional Lie group
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In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; 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 ; 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. OverviewDimensionalityThe 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 isometriesThe 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 groupThe natural topology of Euclidean space implies a topology for the Euclidean group E(n). Namely, a sequence fi of isometries of () 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 transformation I 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". Lie structureThe 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 groupThe 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 discussionSubgroup structure, matrix and vector representationThe 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: where A is an orthogonal matrix or the same orthogonal transformation followed by a translation: with c = Ab T(n) is a normal subgroup of E(n): for every translation t and every isometry u, the composition 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 . 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). This relation is commonly written as: or, equivalently: SubgroupsTypes of subgroups of E(n):
Examples in 3D of combinations:
Overview of isometries in up to three dimensionsE(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 isometriesFor some isometry pairs composition does not depend on order:
Conjugacy classesThe 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:
See also
References
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