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

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

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 – 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