Euclidean group

In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves. For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set.

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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

1 Overview
2 Detailed discussion
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. For information on degrees of freedom in other sciences see Degrees of freedom. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation.

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. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected These include the translations, and the rotations, which together generate E⁺(n). In Euclidean geometry, a translation is moving every point a constant distance in a specified direction A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

The others are the indirect isometries. The subgroup E⁺(n) is of index 2. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In other words, the indirect isometries form a single coset of E⁺(n). In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH 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. In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every

The Euclidean group for n = 3 is used for the kinematics of a rigid body, in classical mechanics. In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects A rigid body motion is in effect the same as a curve in E⁺(3). In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object Starting at the identity transformation I, such a continuous curve can certainly never reach anything other than a direct isometry. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that This is for simple topological reasons: the determinant of the transformation cannot jump from +1 to −1. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

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. In Mathematics, the affine group or general affine group of any Affine space over a field K is the group of all invertible In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can This gives, a fortiori, two ways of writing down elements in an explicit notation. These are:

by a pair (A, b), with A an n×n orthogonal matrix, and b a real column vector of size n; or
by a single square matrix of size n + 1, as explained for the affine group. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

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. Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance, from which angle can then be deduced. Distance is a numerical description of how far apart objects are In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

Detailed discussion

Subgroup structure, matrix and vector representation

The Euclidean group is a subgroup of the group of affine transformations. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector

It has as subgroups the translational group T, and the orthogonal group O(n). In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way:

$x \mapsto A (x+ b)$

where A is an orthogonal matrix

or an orthogonal transformation followed by a translation:

$x \mapsto A x+ b$ . In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

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 part of the isometry acting on t). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup.

Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In other words O(n) is (in the natural way) also the quotient group of E(n) by T:

O(n) $\cong$ E(n) / T

Now SO(n), the special orthogonal group, is a subgroup of O(n), of index two. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript 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). A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. A mirror is an object with a surface that has good Specular reflection; that is it is smooth enough to form an Image. In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or

We have:

SO(n) $\cong$ E⁺(n) / T

Subgroups

Types of subgroups of E(n):

Finite groups. In Mathematics, a finite group is a group which has finitely many elements 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: O_h and I_h. The groups I_h 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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " E. g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite point group. This includes lattices. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of Examples more general than those are the discrete space groups. The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure
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:

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. This article is about rotations in three-dimensional Euclidean space In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n A helix (pl helixes or helices) from the Greek word έλιξ, is a special kind of Space curve, i
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). In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections

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. For information on degrees of freedom in other sciences see Degrees of freedom. In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation In Geometry, a Euclidean plane isometry is an Isometry of the Euclidean plane, or more informally a way of transforming the plane that preserves geometrical

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 perpendicular to that axis (roto-reflection) - 6
- inversion in a point - 3

See also 3D isometries which leave the origin fixed, space group, involution. The screw axis ( helical axis or twist axis) of an Object is a Parameter for describing Simultaneous Rotation and Translation In Crystallography, a glide plane is symmetry operation describing how a reflection in a plane followed by a translation parallel with that plane may In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or In Euclidean geometry, the inversion of a point X in respect to a point P is a point X * such that P is the midpoint of In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n The space group of a Crystal or crystallographic group is a mathematical description of the Symmetry inherent in the structure

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 (results in rotation by 180° about the axis perpendicular to both)
two rotoreflections about the same axis, with respect to the same plane
two glide reflections with respect to the same plane

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 Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class

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, and in corresponding direction (right-hand or left-hand screw).
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.

Dictionary

-noun

(mathematics) the set of rigid motions that are also affine transformations

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