# Point reflection

In geometry, a point reflection is a type of isometry of Euclidean space. It is a reflection whose mirror is a single point. An object that is invariant under a point reflection is said to possess point symmetry.

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

:$mathrm\left\{Ref\right\}_mathbf\left\{p\right\}\left(mathbf\left\{a\right\}\right) = 2mathbf\left\{p\right\} - mathbf\left\{a\right\}.$

In the case where p is the origin, point reflection is simply the negation of the vector a.

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 &ndash; 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.

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.

ee also

* congruence (geometry)
* Euclidean group
* reflection (linear algebra)
* point symmetry
* Riemannian symmetric space

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