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Euclidean group - Wikipedia, the free encyclopedia Page 1 of 6 Euclidean group From Wikipedia, the free encyclopedia In mathematics, the Euclidean group E(n), sometimes called ISO( n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves . 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 known. Contents 1 Overview 1.1 Dimensionality 1.2 Direct and indirect isometries 1.3 Relation to the affine group 2 Detailed discussion 2.1 Subgroup structure, matrix and vector representation 2.2 Subgroups 2.3 Overview of isometries in up to three dimensions 2.4 Commuting isometries 2.5 Conjugacy classes 3 See also 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 There is a subgroup E+(n) of the direct isometries , i.e., isometries preserving orientation, also called rigid motions ; they are the rigid body moves. These include the translations, and the rotations, which together generate E+(n). E+(n) is also called a special Euclidean group , and denoted SE (n). The others are the indirect isometries . The subgroup E+(n) is of index 2. In other words, the indirect isometries form a single coset of E+(n). Given any indirect isometry, for example a given reflection R that reverses orientation , all indirect isometries are given as DR , where D is a direct isometry. The Euclidean group for n = 3 is used for the kinematics of a rigid body , in classical mechanics . A rigid http://en.wikipedia.org/wiki/Euclidean_group 5/23/2011 Euclidean group - Wikipedia, the free encyclopedia Page 2 of 6 body motion is in effect the same as a curve in the Euclidean group. Starting with a body B oriented in a certain way at time t = 0, its orientation at any other time is related to the starting orientation by a Euclidean motion, say f(t). Setting t = 0, we have f(0) = I, the identity transformation. This means that the curve will always lie inside E+(3), in fact: starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. This is for simple topological reasons: the determinant of the transformation cannot jump from +1 to −1. 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, and in such a way as to respect the semidirect product structure of both groups. This gives, a fortiori , two ways of writing down elements in an explicit notation. These are: 1. by a pair ( A, b), with A an n×n orthogonal matrix, and b a real column vector of size n; or 2. by a single square matrix of size n + 1, as explained for the affine group. 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 extra factor in Euclidean geometry is 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(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 an orthogonal transformation followed by a translation: . T is a normal subgroup of E(n): for any translation t and any isometry u, we have u−1tu again a translation (one can say, through a displacement that is u acting on the displacement of t; a translation does not affect a displacement, so equivalently, the displacement is the result of the linear http://en.wikipedia.org/wiki/Euclidean_group 5/23/2011 Euclidean group - Wikipedia, the free encyclopedia Page 3 of 6 part of the isometry acting on t). Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In other words O (n) is (in the natural way) also the quotient group of E(n) by T: O(n) E(n) / T . 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). We have: SO (n) E+(n) / T . Subgroups Types of subgroups of E(n) : Finite groups. 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 1 ≤ m ≤ n 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 √2, 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. E.g. 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: http://en.wikipedia.org/wiki/Euclidean_group 5/23/2011 Euclidean group - Wikipedia, the free encyclopedia Page 4 of 6 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 ( k ≥ 1); 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 R 3, Dih(R 3). Overview of isometries in up to three dimensions E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom: E(1) - 1: + E (1): identity - 0 translation - 1 those not preserving orientation: reflection in a point - 1 E(2) - 3: + E (2): identity - 0 translation - 2 rotation about a point - 3 those not preserving orientation: reflection in a line - 2 reflection in a line combined with translation along that line (glide reflection) - 3 See also Euclidean plane isometry. E(3) - 6: + E (3): identity - 0 translation - 3 rotation about an axis - 5 rotation about an axis combined with translation along that axis (screw operation) - 6 those not preserving orientation: reflection in a plane - 3 reflection in a plane combined with translation in that plane (glide plane operation) - 5 rotation about an axis by an angle not equal to 180°, combined with reflection in a plane http://en.wikipedia.org/wiki/Euclidean_group 5/23/2011 Euclidean group - Wikipedia, the free encyclopedia Page 5 of 6 perpendicular to that axis ( roto -reflection ) - 6 inversion in a point - 3 See also 3D isometries which leave the origin fixed, space group, involution.
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