In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Euclidean group is the group of (Euclidean)
isometries
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
; that is, the transformations of that space that preserve the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
between any two points (also called
Euclidean transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations ...
s). The group depends only on the dimension ''n'' of the space, and is commonly denoted E(''n'') or ISO(''n'').
The Euclidean group E(''n'') comprises all
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
,
rotations, and
reflections of
; and arbitrary finite combinations of them. The Euclidean group can be seen as the
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
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''), whose elements are called
rigid motion
Rigid or rigidity may refer to:
Mathematics and physics
*Stiffness, the property of a solid body to resist deformation, which is sometimes referred to as rigidity
*Structural rigidity, a mathematical theory of the stiffness of ensembles of rig ...
s or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections.
These
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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 invented.
Overview
Dimensionality
The number of
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
for E(''n'') is , which gives 3 in case , and 6 for . Of these, ''n'' can be attributed to available
translational symmetry
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
, and the remaining to
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
.
Direct and indirect isometries
The direct isometries (i.e., isometries preserving the
handedness
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
of
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
subsets) comprise a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is 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
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of E
+(''n''), which can be denoted by E
−(''n''). It follows that the subgroup E
+(''n'') is of
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
2 in E(''n'').
Topology of the group
The natural
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
of Euclidean space
implies a topology for the Euclidean group E(''n''). Namely, a sequence ''f''
''i'' of isometries of
(
) is defined to converge if and only if, for any point ''p'' of
, the sequence of points ''p''
''i'' converges.
From this definition it follows that a function
is continuous if and only if, for any point ''p'' of
, the function
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
, 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
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, because they describe the physically possible movements of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
in three-dimensional space over time. One takes ''f''(0) to be the
identity transformation
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
''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 structure
The Euclidean groups are not only
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
s, they are
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s, so that
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
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
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Relat ...
for ''n'' dimensions, and in such a way as to respect the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
structure of both groups. This gives, ''a fortiori'', two ways of writing elements in an explicit notation. These are:
# by a pair , with ''A'' an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
, and ''b'' a real
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
of size ''n''; or
# by a single
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
of size , as explained for the
affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
Relat ...
.
Details for the first representation are given in the next section.
In the terms of
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's
Erlangen programme
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
, we read off from this that
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, the geometry of the Euclidean group of symmetries, is, therefore, a specialisation of
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of ''parallel lines'' is one of the main properties that is inde ...
. All affine theorems apply. The origin of Euclidean geometry allows definition of the notion of
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
, from which
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
can then be deduced.
Detailed discussion
Subgroup structure, matrix and vector representation
The Euclidean group is a subgroup of the group of
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s.
It has as subgroups the
translational group T(''n''), and the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
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
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
or the same orthogonal transformation followed by a translation:
with
T(''n'') is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of E(''n''): for every translation ''t'' and every isometry ''u'', the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
is again a translation.
Together, these facts imply that E(''n'') is the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in w ...
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
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
of E(''n'') by T(''n''):
Now SO(''n''), the
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
, is a subgroup of O(''n'') of
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
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, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
, rather than a translation followed by some kind of
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
(in dimensions 2 and 3, these are the familiar reflections in a
mirror
A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
line or plane, which may be taken to include the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, or in 3D, a
rotoreflection
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
).
This relation is commonly written as:
or, equivalently:
Subgroups
Types of subgroups of E(''n''):
;
Finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
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 may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
(e.g., for a group generated by ''m'' translations in independent directions, and possibly a finite point group). This includes
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
s. Examples more general than those are the discrete
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
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 , 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
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
)
:*all isometries that keep the origin fixed, or more generally, some point (the
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
)
:*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 (); the set of images of a point under the isometries is a ''k''-fold
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
; 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
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
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
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
:
Chasles' theorem asserts that any element of E
+(3) is a
screw displacement
A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw ...
.
See also
3D isometries that leave the origin fixed,
space group
In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unchan ...
,
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
.
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
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
; 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.
*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.
See also
*
Fixed points of isometry groups in Euclidean space A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more g ...
*
Euclidean plane isometry 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 properties such as length. There are four types: translations, rotations, reflections, ...
*
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
*
Coordinate rotations and reflections
In geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point ''P'' to its im ...
*
Reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
*
Plane of rotation
In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as ...
References
*
*
William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thurston ...
. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp.
{{DEFAULTSORT:Euclidean Group
Lie groups
*