View source for Point reflection

{{Short description|Geometric symmetry operation}}
{{redirect-distinguish|Central inversion|Circle inversion}}
{{distinguish|Reflection point}}
{{refimprove|date=May 2024}}
[[File:Cyclic symmetry 2.svg|thumb|Example of a 2-dimensional figure with central symmetry, invariant under point reflection]]
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In [[geometry]], a '''point reflection''' (also called a '''point inversion''' or '''central inversion''') is a [[geometric transformation]] of [[affine space]] in which every [[point (geometry)|point]] is reflected across a designated '''inversion center''', which remains [[Fixed point (mathematics)|fixed]]. In [[Euclidean space|Euclidean]] or [[pseudo-Euclidean space]]s, a point reflection is an [[isometry]] (preserves [[Euclidean distance|distance]]).<ref>{{Cite web |title=Reflections in Lines |url=https://new.math.uiuc.edu/public403/isometries/reflections.html#_geometrical_definition_of_a_reflection |access-date=2024-04-27 |website=new.math.uiuc.edu}}</ref> In the [[Euclidean plane]], a point reflection is the same as a [[half turn|half-turn]] [[rotation]] (180° or {{mvar|π}} [[radian]]s), while in three-dimensional Euclidean space a point reflection is an [[improper rotation]] which preserves distances but [[orientation-reversing|reverses orientation]]. A point reflection is an [[involution (mathematics)|involution]]: applying it twice is the [[identity transformation]].

An object that is invariant under a point reflection is said to possess '''point symmetry''' (also called '''inversion symmetry''' or '''central symmetry'''). A [[point group]] including a point reflection among its symmetries is called ''[[centrosymmetry|centrosymmetric]]''. Inversion symmetry is found in many [[Crystal structure|crystal structures]] and [[molecule]]s, and has a major effect upon their physical properties.<ref name=":0" />

== 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 [[Involution (mathematics)|involution]]s, meaning that they have order 2 – they are their own inverse: applying them twice yields the [[Identity function|identity map]] – which is also true of other maps called ''reflections''. More narrowly, a ''[[reflection (linear algebra)|reflection]]'' refers to a reflection in a [[hyperplane]] (<math>n-1</math> dimensional [[affine subspace]] – a point on the [[line (geometry)|line]], a line in the [[plane (geometry)|plane]], 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 <math>1 \leq k \leq n-1</math>) 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]] maps with all [[eigenvalue]]s either 1 or −1. Reflection in a hyperplane has a single &minus;1 eigenvalue (and multiplicity <math>n-1</math> 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 ==
{| class=wikitable align=right
|+ 2D examples
|- align=center
|[[File:Hexagonal parallelogon.png|180px]]<BR />Hexagonal [[parallelogon]]
|[[File:Octagon g2 symmetry.png|120px]]<BR />[[Octagon]]
|}
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 [[composition of functions|composed]] with reflection across the plane of rotation, perpendicular to the axis of rotation.  In dimension ''n'', point reflections are [[orientation (mathematics)|orientation]]-preserving if ''n'' is even, and orientation-reversing if ''n'' is odd.

== Formula ==
Given a vector '''a''' in the Euclidean space '''R'''<sup>''n''</sup>, the formula for the reflection of '''a''' across the point '''p''' is

:<math>\mathrm{Ref}_\mathbf{p}(\mathbf{a}) = 2\mathbf{p} - \mathbf{a}.</math>

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 (geometry)|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 (geometric)|vector]] from ''X'' to ''P'' is the same as the vector from ''P'' to ''X''*.

The formula for the inversion in ''P'' is

:'''x'''* = 2'''p''' − '''x'''

where '''p''', '''x''' and '''x'''* are the position vectors of ''P'', ''X'' and ''X''* respectively.

This [[function (mathematics)|mapping]] is an [[isometry|isometric]] [[Involution (mathematics)|involutive]] [[affine transformation]] which has exactly one [[Fixed point (mathematics)|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]].)

==Point reflection group {{anchor|Group}}==
[[Image:Composing Point Reflections.png|thumbnail|The composition of two offset point reflections in 2-dimensions is a translation.]]

The [[composition of functions|composition]] of two point reflections is a [[translation (geometry)|translation]].<ref>{{Cite web |title=Lab 9 Point Reflection |url=https://sites.math.washington.edu/~king/coursedir/m444a02/lab/lab09.html |access-date=2024-04-27 |website=sites.math.washington.edu}}</ref>  Specifically, point reflection at '''p''' followed by point reflection at '''q''' is translation by the vector 2('''q'''&nbsp;− '''p''').

The set consisting of all point reflections and translations is [[Lie subgroup]] of the [[Euclidean group]].  It is a [[semidirect product]] of '''R'''<sup>''n''</sup> with a [[cyclic group]] of order 2, the latter acting on '''R'''<sup>''n''</sup> 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 [[Euclidean group|isometry group]] of the line.

==Point reflections in mathematics==
* Point reflection across the center of a sphere yields the [[antipodal map]].
* A [[Riemannian symmetric space|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]]s and [[Riemannian geometry]].

==Point reflection in analytic geometry==
Given the point <math>P(x,y)</math> and its reflection <math>P'(x',y')</math> with respect to the point <math>C(x_c,y_c)</math>, the latter is the [[midpoint]] of the segment <math>\overline{PP'}</math>;

: <math>\begin{cases}x_c=\frac{x+x'}{2} \\ y_c=\frac{y+y'}{2}\end{cases}</math>

Hence, the equations to find the coordinates of the reflected point are

: <math>\begin{cases}x'=2x_c-x \\ y'=2y_c-y\end{cases}</math>

Particular is the case in which the point C has coordinates <math>(0,0)</math> (see the [[Point reflection#Inversion with respect to the origin|paragraph below]])

: <math>\begin{cases}x'=-x \\ y'=-y\end{cases}</math>

== Properties ==
In even-dimensional [[Euclidean space]], say 2''N''-dimensional space, the inversion in a point ''P'' is equivalent to ''N'' rotations over angles {{pi}} 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 [[Euclidean group|direct isometry]].

In odd-dimensional [[Euclidean space]], say (2''N''&nbsp;+&nbsp;1)-dimensional space, it is equivalent to ''N'' rotations over {{pi}} 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 (mathematics)|orientation]], it is an [[Euclidean group|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 (rigid body)|orientation]] (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are <math>\overline{1}</math>, ''C''<sub>''i''</sub>, ''S''<sub>2</sub>, and 1&times;. The group type is one of the three [[symmetry group]] types in 3D without any pure [[rotational symmetry]], see [[cyclic symmetries]] with ''n''&nbsp;=&nbsp;1.

The following [[point groups in three dimensions]] contain inversion:
*''C''<sub>''n''h</sub> and ''D''<sub>''n''h</sub> for even ''n''
*''S''<sub>2''n''</sub> and ''D''<sub>''n''d</sub> for odd ''n''
*''T''<sub>h</sub>, ''O''<sub>h</sub>, and ''I''<sub>h</sub>

Closely related to inverse in a point is [[reflection (mathematics)|reflection]] in respect to a [[plane (geometry)|plane]], which can be thought of as a "inversion in a plane".

==Inversion centers in crystals and molecules==
Inversion symmetry plays a major role in the properties of materials, as also do other symmetry operations.<ref name=":0" />

Some molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In many cases they can be considered as polyhedra, categorized by their coordination number and bond angles. For example, four-coordinate polyhedra are classified as [[Tetrahedral molecular geometry|tetrahedra]], while five-coordinate environments can be [[Square pyramidal molecular geometry|square pyramidal]] or [[Trigonal bipyramidal molecular geometry|trigonal bipyramidal]] depending on the bonding angles. 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 non-centrosymmetric as an inversion through the central atom would result in a reversal of the polyhedron. Polyhedra with an odd (versus even) coordination number are not centrosymmtric. Polyhedra containing inversion centers are known as centrosymmetric, while those without are non-centrosymmetric. The presence or absence of an inversion center has a strong influence on the optical properties;<ref>{{cite book |author=Harris and Bertolucci |title=Symmetry and Spectroscopy |publisher=Dover Publications |year=1989 |isbn=978-0-486-66144-5}}</ref> for instance molecules without inversion symmetry have a [[Dipole moments of molecules|dipole moment]] and can directly interact with photons, while those with inversion have no dipole moment and only interact via [[Raman scattering]].<ref name="raman1928">{{cite journal |last=Raman |first=C. V. |year=1928 |title=A new radiation |journal=Indian Journal of Physics |volume=2 |pages=387–398 |quote=Inaugural Address delivered to the South Indian Science Association on Friday, the 16th March, 1928 |hdl=10821/377}}</ref> The later is named after [[C. V. Raman]] who was awarded the 1930 [[Nobel Prize in Physics]] for his discovery.<ref>{{Cite journal |last1=Singh |first1=R. |year=2002 |title=C. V. Raman and the Discovery of the Raman Effect |journal=Physics in Perspective |volume=4 |issue=4 |pages=399–420 |bibcode=2002PhP.....4..399S |doi=10.1007/s000160200002 |s2cid=121785335}}</ref> 

In addition, in [[crystallography]], the presence of inversion centers for periodic structures distinguishes between [[centrosymmetric]] and non-centrosymmetric compounds. 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. In many materials such as oxides these polyhedra can link together via corner-, edge- or face sharing, depending on which atoms share common bonds and also the valence. In other cases such as for [[metals]] and [[alloys]] the structures are better considered as arrangements of close-packed atoms. Crystals which do not have inversion symmetry also display the [[piezoelectric effect]]. The presence or absence of inversion symmetry also has numerous consequences for the properties of solids,<ref name=":0">{{Cite book |last=Nye |first=J. F. |title=Physical properties of crystals: their representation by tensors and matrices |date=1984 |publisher=Clarendon Press ; Oxford University Press |isbn=978-0-19-851165-6 |edition=1st published in pbk. with corrections, 1984 |location=Oxford [Oxfordshire] : New York}}</ref> as does the mathematical relationships between the different crystal symmetries.<ref>{{Cite book |last=Müller |first=Ulrich |title=Symmetry relationships between crystal structures: applications of crystallographic group theory in crystal chemistry |last2=Wondratschek |first2=Hans |last3=Bärnighausen |first3=Hartmut |date=2017 |publisher=Oxford University Press |isbn=978-0-19-880720-9 |edition=first published in paperback |series=International Union of Crystallography texts on crystallography |location=Oxford}}</ref> 

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 interactions between heteroatoms or electronic effects such as [[Jahn–Teller effect|Jahn–Teller distortions]]. 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]] 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, not just individual polyhedra. Crystals are classified into thirty-two [[crystallographic point groups]] 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 non-centrosymmetric 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.

==Inversion with respect to the origin==

Inversion with respect to the origin corresponds to [[additive inverse|additive inversion]] of the position vector, and also to [[scalar multiplication]] by &minus;1. The operation commutes with every other [[linear transformation]], but not with [[Translation (geometry)|translation]]: it is in the [[Center (group theory)|center]] 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 (physics)|parity transformation]].

In mathematics, '''reflection through the origin''' refers to the point reflection of [[Euclidean space]] '''R'''<sup>''n''</sup> across the [[Origin (mathematics)|origin]] of the [[Cartesian coordinate system]].  Reflection through the origin is an [[orthogonal transformation]] corresponding to [[scalar multiplication]] by <math>-1</math>, and can also be written as <math>-I</math>, where <math>I</math> is the [[identity matrix]].  In three dimensions, this sends <math>(x, y, z) \mapsto (-x, -y, -z)</math>, and so forth.

===Representations===
As a [[scalar matrix]], it is represented in every basis by a matrix with <math>-1</math> on the diagonal, and, together with the identity, is the [[Center (group theory)|center]] of the [[orthogonal group]] <math>O(n)</math>.

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 <math>2n</math>, it is rotation by 180 degrees in ''n'' orthogonal planes;{{efn|"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]] 90°.}} note again that rotations in orthogonal planes commute.

===Properties===
It has determinant <math>(-1)^n</math> (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''&nbsp;+&nbsp;1) and instead providing a [[split short exact sequence|splitting]] of the map <math>O(2n+1) \to \pm 1</math>, showing that <math>O(2n + 1) = SO(2n + 1) \times \{\pm I\}</math> as an [[internal direct product]].

* Together with the identity, it forms the [[center (group theory)|center]] of the [[orthogonal group]].
* It preserves every quadratic form, meaning <math>Q(-v) = Q(v)</math>, 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 a Coxeter group|longest element]] of the [[Coxeter group]] of [[signed permutations]].

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 function|length]] at most ''n'' with respect to the generating set of reflections,{{efn|This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the [[spectral theorem]], 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 the other component.

===Clifford algebras and spin groups===
{{Further|Clifford algebra}}
{{Further|Spin group}}
It should ''not'' be confused with the element <math>-1 \in \mathrm{Spin}(n)</math> in the [[spin group]]. This is particularly confusing for even spin groups, as <math>-I \in SO(2n)</math>, and thus in <math>\operatorname{Spin}(n)</math> there is both <math>-1</math> and 2 lifts of <math>-I</math>.

Reflection through the identity extends to an automorphism of a [[Clifford algebra]], called the ''main involution'' or ''grade involution.''

Reflection through the identity lifts to a [[pseudoscalar (Clifford algebra)|pseudoscalar]].

==See also==
* [[Affine involution]]
* [[Circle inversion]]
* [[Clifford algebra]]
* [[Congruence (geometry)]]
* [[Estermann measure]]
* [[Euclidean group]]
* [[Kovner–Besicovitch measure]]
* [[Orthogonal group]]
* [[Parity (physics)]]
* [[Reflection (mathematics)]]
* [[Riemannian symmetric space]]
* [[Spin group]]

==Notes==
{{notelist}}

==References==
{{Reflist}}

{{Authority control}}

[[Category:Euclidean symmetries]]
[[Category:Functions and mappings]]
[[Category:Clifford algebras]]
[[Category:Quadratic forms]]