# Point reflection

In geometry, a **point reflection** or **inversion in a point** (or **inversion through a point**, or **central inversion**) 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. Namely, it is an isometric 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.

The term *inversion* should not be confused with inversive geometry, where *inversion* is defined with respect to a circle.

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.

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**.

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

This mapping is an isometric involutive affine transformation which has exactly one fixed point, which is *P*.

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).

The composition 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. It is a semidirect product 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 of the line.

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

In even-dimensional Euclidean space, say 2*N*-dimensional space, the inversion in a point *P* is equivalent to *N* rotations over angles π 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 π 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.

Closely related to inverse in a point is reflection in respect to a plane, which can be thought of as a "inversion in a plane".

Molecules contain an inversion center when a point exists through which all atoms can reflect while retaining symmetry. In crystallography, the presence of inversion centers distinguishes between centrosymmetric 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, while five-coordinate environments can be square pyramidal 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 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 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, KTiOPO_{4} (KTP) crystalizes in the noncentrosymmetric, orthorhombic Pna21 space group, 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. 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 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 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.

It is a product of *n* orthogonal reflections (reflection through the axes of any orthogonal basis); note that orthogonal reflections commute.

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,^{[note 2]} 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.

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.

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