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 invariant under point reflection through its center, it is said to possess central symmetry or to be
centrally symmetric.
Point reflection can be classified as an
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, ...
. Namely, it is an
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
involutive affine transformation, which has exactly one
fixed point, which is the point of inversion. It is equivalent to a
homothetic transformation with scale factor equal to −1. The point of inversion is also called
homothetic center.
Terminology
The term ''reflection'' is loose, and considered by some an abuse of language, with ''inversion'' preferred; however, ''point reflection'' is widely used. Such maps are
involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the
identity map
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, unc ...
– which is also true of other maps called ''reflections''. More narrowly, a ''
reflection'' refers to a reflection in a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
(
dimensional
affine subspace
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
– a point on the
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
, a line in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, a plane in 3-space), with the hyperplane being fixed, but more broadly ''reflection'' is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension ''k'', where
) is called the ''mirror''. In dimension 1 these coincide, as a point is a hyperplane in the line.
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
maps with all
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity
on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity ''n'').
The term ''inversion'' should not be confused with
inversive geometry, where ''inversion'' is defined with respect to a circle.
Examples
In two dimensions, a point reflection is the same as a
rotation of 180 degrees. In three dimensions, a point reflection can be described as a 180-degree rotation
composed with reflection across a plane perpendicular to the axis of rotation. In dimension ''n'', point reflections are
orientation-preserving if ''n'' is even, and orientation-reversing if ''n'' is odd.
Formula
Given a vector a in the Euclidean space R
''n'', the formula for the reflection of a across the point p is
:
In the case where p is the origin, point reflection is simply the negation of the vector a.
In
Euclidean geometry, the inversion of a
point ''X'' with respect to a point ''P'' is a point ''X''* such that ''P'' is the midpoint of the
line segment with endpoints ''X'' and ''X''*. In other words, the
vector from ''X'' to ''P'' is the same as the vector from ''P'' to ''X''*.
The formula for the inversion in ''P'' is
:x* = 2a − x
where a, x and x* are the position vectors of ''P'', ''X'' and ''X''* respectively.
This
mapping is an
isometric
The term ''isometric'' comes from the Greek for "having equal measurement".
isometric may mean:
* Cubic crystal system, also called isometric crystal system
* Isometre, a rhythmic technique in music.
* "Isometric (Intro)", a song by Madeon from ...
involutive 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, ...
which has exactly one
fixed point, which is ''P''.
Point reflection as a special case of uniform scaling or homothety
When the inversion point ''P'' coincides with the origin, point reflection is equivalent to a special case of
uniform scaling: uniform scaling with scale factor equal to −1. This is an example of
linear transformation.
When ''P'' does not coincide with the origin, point reflection is equivalent to a special case of
homothetic transformation: homothety with
homothetic center coinciding with P, and scale factor −1. (This is an example of non-linear
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, ...
.)
Point reflection group
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 ...
of two point reflections is a
translation. Specifically, point reflection at p followed by point reflection at q is translation by the vector 2(q − p).
The set consisting of all point reflections and translations is
Lie subgroup of the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations) ...
. It is a
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 ...
of R
''n'' with a
cyclic group of order 2, the latter acting on R
''n'' by negation. It is precisely the subgroup of the Euclidean group that fixes the
line at infinity pointwise.
In the case ''n'' = 1, the point reflection group is the full
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of the line.
Point reflections in mathematics
* Point reflection across the center of a sphere yields the
antipodal map
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...
.
* A
symmetric space is a
Riemannian manifold with an isometric reflection across each point. Symmetric spaces play an important role in the study of
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 ad ...
s and
Riemannian geometry.
Point reflection in analytic geometry
Given the point
and its reflection
with respect to the point
, the latter is the
midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
of the segment
;
:
Hence, the equations to find the coordinates of the reflected point are
:
Particular is the case in which the point C has coordinates
(see the
paragraph below)
:
Properties
In even-dimensional
Euclidean space, say 2''N''-dimensional space, the inversion in a point ''P'' is equivalent to ''N'' rotations over angles in each plane of an arbitrary set of ''N'' mutually orthogonal planes intersecting at ''P''. These rotations are mutually commutative. Therefore, inversion in a point in even-dimensional space is an orientation-preserving isometry or
direct isometry.
In odd-dimensional
Euclidean space, say (2''N'' + 1)-dimensional space, it is equivalent to ''N'' rotations over in each plane of an arbitrary set of ''N'' mutually orthogonal planes intersecting at ''P'', combined with the reflection in the 2''N''-dimensional subspace spanned by these rotation planes. Therefore, it ''reverses'' rather than preserves
orientation, it is an
indirect isometry.
Geometrically in 3D it amounts to
rotation about an axis through ''P'' by an angle of 180°, combined with reflection in the plane through ''P'' which is perpendicular to the axis; the result does not depend on the
orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are
, ''C''
''i'', ''S''
2, and 1×. The group type is one of the three
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 ambie ...
types in 3D without any pure
rotational symmetry, see
cyclic symmetries
In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object.
They are ...
with ''n'' = 1.
The following
point groups in three dimensions
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries th ...
contain inversion:
*''C''
''n''h and ''D''
''n''h for even ''n''
*''S''
2''n'' and ''D''
''n''d for odd ''n''
*''T''
h, ''O''
h, and ''I''
h
Closely related to inverse in a point is
reflection in respect to a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, which can be thought of as a "inversion in a plane".
Inversion centers in crystallography
Molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, the presence of inversion centers distinguishes between
centrosymmetric
In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point gr ...
and noncentrosymmetric compounds. Crystal structures are composed of various polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
, while five-coordinate environments can be
square pyramidal
In molecular geometry, square pyramidal geometry describes the shape of certain compounds with the formula where L is a ligand. If the ligand atoms were connected, the resulting shape would be that of a pyramid with a square base. The point ...
or
trigonal bipyramidal depending on the bonding angles. All crystalline compounds come from a repetition of an atomic building block known as a unit cell, and these unit cells define which polyhedra form and in what order. These polyhedra link together via corner-, edge- or face sharing, depending on which atoms share common bonds. Polyhedra containing inversion centers are known as centrosymmetric, while those without are noncentrosymmetric. Six-coordinate octahedra are an example of centrosymmetric polyhedra, as the central atom acts as an inversion center through which the six bonded atoms retain symmetry. Tetrahedra, on the other hand, are noncentrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. It is important to note that bonding geometries with odd coordination numbers must be noncentrosymmetric, because these polyhedra will not contain inversion centers.
Real polyhedra in crystals often lack the uniformity anticipated in their bonding geometry. Common irregularities found in crystallography include distortions and disorder. Distortion involves the warping of polyhedra due to nonuniform bonding lengths, often due to differing electrostatic attraction between heteroatoms. For instance, a titanium center will likely bond evenly to six oxygens in an octahedra, but distortion would occur if one of the oxygens were replaced with a more
electronegative
Electronegativity, symbolized as , is the tendency for an atom of a given chemical element to attract shared electrons (or electron density) when forming a chemical bond. An atom's electronegativity is affected by both its atomic number and the ...
fluorine. Distortions will not change the inherent geometry of the polyhedra—a distorted octahedron is still classified as an octahedron, but strong enough distortions can have an effect on the centrosymmetry of a compound. Disorder involves a split occupancy over two or more sites, in which an atom will occupy one crystallographic position in a certain percentage of polyhedra and the other in the remaining positions. Disorder can influence the centrosymmetry of certain polyhedra as well, depending on whether or not the occupancy is split over an already-present inversion center.
Centrosymmetry applies to the crystal structure as a whole, as well. Crystals are classified into thirty-two
crystallographic point groups
In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...
which describe how the different polyhedra arrange themselves in space in the bulk structure. Of these thirty-two point groups, eleven are centrosymmetric. The presence of noncentrosymmetric polyhedra does not guarantee that the point group will be the same—two noncentrosymmetric shapes can be oriented in space in a manner which contains an inversion center between the two. Two tetrahedra facing each other can have an inversion center in the middle, because the orientation allows for each atom to have a reflected pair. The inverse is also true, as multiple centrosymmetric polyhedra can be arranged to form a noncentrosymmetric point group.
Noncentrosymmetric compounds can be useful for application in
nonlinear optics. The lack of symmetry via inversion centers can allow for areas of the crystal to interact differently with incoming light. The wavelength, frequency and intensity of light is subject to change as the electromagnetic radiation interacts with different energy states throughout the structure.
Potassium titanyl phosphate
Potassium is the chemical element with the symbol K (from Neo-Latin ''kalium'') and atomic number19. Potassium is a silvery-white metal that is soft enough to be cut with a knife with little force. Potassium metal reacts rapidly with atmosp ...
, KTiOPO
4 (KTP). crystalizes in the noncentrosymmetric,
orthorhombic Pna21
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 unch ...
, and is a useful non-linear crystal. KTP is used for frequency-doubling
neodymium-doped lasers, utilizing a nonlinear optical property known as
second-harmonic generation
Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...
. The applications for nonlinear materials are still being researched, but these properties stem from the presence of (or lack thereof) an inversion center.
Inversion with respect to the origin
Inversion with respect to the origin corresponds to
additive inversion of the position vector, and also to
scalar multiplication by −1. The operation commutes with every other
linear transformation, but not with
translation: it is in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of the
general linear group. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion; in physics 3-dimensional reflection through the origin is also called a
parity transformation
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
.
In mathematics, reflection through the origin refers to the point reflection of
Euclidean space R
''n'' across 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 ...
of the
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
. Reflection through the origin is an
orthogonal transformation corresponding to
scalar multiplication by
, and can also be written as
, where
is the
identity matrix. In three dimensions, this sends
, and so forth.
Representations
As a
scalar matrix, it is represented in every basis by a matrix with
on the diagonal, and, together with the identity, is the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of the
orthogonal group .
It is a product of ''n'' orthogonal reflections (reflection through the axes of any
orthogonal basis); note that orthogonal reflections commute.
In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension
, it is rotation by 180 degrees in ''n'' orthogonal planes;
["Orthogonal planes" meaning all elements are orthogonal and the planes intersect at 0 only, not that they intersect in a line and have ]dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
90°. note again that rotations in orthogonal planes commute.
Properties
It has determinant
(from the representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the
special orthogonal group SO(2''n''), and it is orientation-reversing in odd dimension, thus not an element of SO(2''n'' + 1) and instead providing a
splitting
Splitting may refer to:
* Splitting (psychology)
* Lumpers and splitters, in classification or taxonomy
* Wood splitting
* Tongue splitting
* Splitting, railway operation
Mathematics
* Heegaard splitting
* Splitting field
* Splitting principle ...
of the map
, showing that
as an
internal direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
.
* Together with the identity, it forms the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of the
orthogonal group.
* It preserves every quadratic form, meaning
, and thus is an element of every
indefinite orthogonal group as well.
* It equals the identity if and only if the characteristic is 2.
* It is the
longest element of the
Coxeter group of
signed permutations
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the no ...
.
Analogously, it is a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have
length at most ''n'' with respect to the generating set of reflections,
[This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the ]spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
, for instance. and reflection through the origin has length ''n,'' though it is not unique in this: other maximal combinations of rotations (and possibly reflections) also have maximal length.
Geometry
In SO(2''r''), reflection through the origin is the farthest point from the identity element with respect to the usual metric. In O(2''r'' + 1), reflection through the origin is not in SO(2''r''+1) (it is in the non-identity component), and there is no natural sense in which it is a "farther point" than any other point in the non-identity component, but it does provide a
base point
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains ...
in the other component.
Clifford algebras and spin groups
It should ''not'' be confused with the element
in the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
. This is particularly confusing for even spin groups, as
, and thus in
there is both
and 2 lifts of
.
Reflection through the identity extends to an automorphism of a
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
, called the ''main involution'' or ''grade involution.''
Reflection through the identity lifts to a
pseudoscalar.
See also
*
Affine involution
In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space R''n''. Such involutions are easy to characterize and they can be described geometrically.
Linear involutions
To give a ...
*
Circle inversion
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
*
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
*
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
More formally, two sets of points are called congruent if, and only if, one can be ...
*
Estermann measure
*
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations) ...
*
Kovner–Besicovitch measure
*
Orthogonal group
*
Parity (physics)
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
*
Reflection (mathematics)
*
Riemannian symmetric space
*
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
Notes
References
{{refend
Euclidean symmetries
Functions and mappings
Clifford algebras
Quadratic forms