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Not to be confused with inversive geometry, in which inversion is through a circle instead of a point.
Contents
- Terminology
- Examples
- Formula
- Point reflection as a special case of uniform scaling or homothety
- Point reflection group
- Point reflections in mathematics
- Properties
- Inversion with respect to the origin
- Representations
- Geometry
- Clifford algebras and Spin groups
- References
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.
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 – which is also true of other maps called "reflections". More narrowly, a "reflection" refers to a reflection in a hyperplane (
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity
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 Rn, 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−xwhere 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.
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
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 Rn with a cyclic group of order 2, the latter acting on Rn 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.
Point reflections in mathematics
Properties
In even-dimensional Euclidean space, say 2N-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 (2N + 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 2N-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
The following point groups in three dimensions contain inversion:
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".
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 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 mathematics, reflection through the origin refers to the point reflection of Euclidean space Rn across the origin of the Cartesian coordinate system. Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by
Representations
As a scalar matrix, it is represented in every basis by a matrix with
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
Properties
It has determinant
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, 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(2r), reflection through the origin is the farthest point from the identity element with respect to the usual metric. In O(2r+1), reflection through the origin is not in SO(2r+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
It should not be confused with the element
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.