Euclidean group

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Euclidean group

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
of n-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
. Its elements, the isometries

Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
associated with the Euclidean metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
, are called Euclidean moves.

These group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
s 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.

Dimensionality
The number of degrees of freedom

Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
for E(n) is

n(n + 1)/2,

which gives 3 in case n = 2, and 6 for n = 3.

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Encyclopedia

In mathematics

Mathematics

Symmetry group

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
. Its elements, the isometries

Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
associated with the Euclidean metric

Metric (mathematics)

Group (mathematics)

Overview

Dimensionality

The number of degrees of freedom

Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
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

Translational symmetry

In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation....
, and the remaining n(n − 1)/2 to rotational symmetry

Rotational symmetry

File:The armoured triskelion on the flag of the Isle of Man.svgGenerally 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

Orientation (mathematics)

In mathematics, an orientation on a real number vector space is a choice of which ordered basis are "positively" oriented and which are "negatively" oriented....
, also called rigid motions; they are the rigid body

Rigid body

In physics, a rigid body is an idealization of a solid Physical body of finite size in which deformation is neglected. In other words, the distance between any two given Point s of a rigid body remains constant in time regardless of external forces exerted on it....
moves. These include the translations, and the rotation

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
s, which together generate E⁺(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

Coset

In mathematics, if G is a group , H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G....
of E⁺(n). Given any

Universal quantification

In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing.The resulting statement is a universally quantified statement, and we have universally quantified over the predicate....
indirect isometry, for example a given reflection

Reflection

Reflection or reflexion may refer to:...
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

Rigid body

Classical mechanics

Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies....
. A rigid body motion is in effect the same as a curve

Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
in E⁺(3). 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

Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
of the transformation cannot jump from +1 to −1.

The Euclidean groups are not only topological group

Topological group

In mathematics, a topological group is a group G together with a topological space on G such that the group's binary operation and the group's inverse function are continuous function ....
s, they are Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
s, so that calculus

Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
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

Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself....
for n dimensions, and in such a way as to respect the semidirect product

Semidirect product

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 be put together from two subgroups, one of which is a normal subgroup....
structure of both groups. 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
Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
, and b a real column vector
Column vector

In linear algebra, a column vector or column matrix is an m × 1 matrix , i.e. a matrix consisting of a single column of elements....
of size n; or
by a single square matrix of size n + 1, as explained for the affine group
Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself....
.

Details for the first representation are given in the next section.

In the terms of Felix Klein

Felix Klein

Felix Christian Klein was a Germany mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory....
's Erlangen programme, we read off from this that Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, the geometry of the Euclidean group of symmetries, is therefore a specialisation of affine geometry

Affine geometry

In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and Translation s....
. All affine theorems apply. The extra factor in Euclidean geometry is the notion of distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
, from which angle

Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
can then be deduced.

Detailed discussion

Subgroup structure, matrix and vector representation

The Euclidean group is a subgroup of the group of affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
s.

It has as subgroups the translational

Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
group T, and the orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
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

Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...

or an orthogonal transformation followed by a translation:

.

T is a normal subgroup

Normal subgroup

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they can be used to construct quotient groups from a given group ....
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).

Together, these facts imply that E(n) is the semidirect product

Semidirect product

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 be put together from two subgroups, one of which is a normal subgroup....
of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group

Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
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

Index (mathematics)

The word index is used in variety of senses in mathematics.* In perhaps the most frequent sense, an index is a superscript or subscript to a symbol....
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

Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
, rather than a translation followed by some kind of reflection

Reflection (mathematics)

In mathematics, a reflection is a function that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q....
(in dimensions 2 and 3, these are the familiar reflections in a mirror

Mirror

A mirror is an object with one surface polished, which leads to reflection and another opaque. The most familiar type of mirror is the plane mirror, which has a flat surface....
line or plane, which may be taken to include the origin

Origin (mathematics)

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 for the geometry of the surrounding space....
, or in 3D, a rotoreflection

Improper rotation

In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis....
).

We have:

SO(n) E⁺(n) / T.

Subgroups

Types of subgroups of E(n):

Finite group
Finite group

In mathematics, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
s. 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
Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "Isolated point" from each other in a certain sense....
. E.g. for 1 ≤ m ≤ n a group generated by m translations in independent directions, and possibly a finite point group. This includes lattice
Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which linear span the real number vector space Rn....
s. Examples more general than those are the discrete space group
Space group

The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the group , which is used to build the set of space groups....
s.
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
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
)
all isometries that keep the origin fixed, or more generally, some point (the orthogonal group
Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....
)
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
Helix

A helix is a special kind of space curve, i.e. a Differentiable manifold curve in three-space. As a mental image of a helix one may take the spring ....
; 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
Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
of R³, Dih(R³).

Overview of isometries in up to three dimensions

E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom

Degrees of freedom (physics and chemistry)

Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters....
:

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
Glide reflection

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 along that line....
) - 3

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

Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
; 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:

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.

Euclidean group

Overview

Dimensionality

Direct and indirect isometries

Relation to the affine group

Detailed discussion

Subgroup structure, matrix and vector representation

Subgroups

Overview of isometries in up to three dimensions

Commuting isometries

Conjugacy classes

See also