In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an 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 centrosymmetric. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their physical properties.
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 ( n − 1 {\displaystyle n-1} dimensional affine subspace – a point on the line, a line in the 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 1 ≤ k ≤ n − 1 {\displaystyle 1\leq k\leq n-1} ) 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 eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity n − 1 {\displaystyle n-1} 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 the plane of rotation, 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
R e f p ( a ) = 2 p − a . {\displaystyle \mathrm {Ref} _{\mathbf {p} }(\mathbf {a} )=2\mathbf {p} -\mathbf {a} .}
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* = 2p − x where p, 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 Point reflection across the center of a sphere yields the antipodal map. 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 groups and Riemannian geometry.
Point reflection in analytic geometry Given the point P ( x , y ) {\displaystyle P(x,y)} and its reflection P ′ ( x ′ , y ′ ) {\displaystyle P'(x',y')} with respect to the point C ( x c , y c ) {\displaystyle C(x_{c},y_{c})} , the latter is the midpoint of the segment P P ′ ¯ {\displaystyle {\overline {PP'}}} ;
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