- Point reflection
In

geometry , a**point reflection**is a type ofisometry ofEuclidean space . It is a reflection whose mirror is a single point. An object that is invariant under a point reflection is said to possesspoint 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 theEuclidean group . It is asemidirect product of**R**^{"n"}with acyclic group of order 2, the latter acting on**R**^{"n"}by negation. It is precisely the subgroup of the Euclidean group that fixes theline 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 aRiemannian manifold with an isometric reflection across each point. Symmetric spaces play an important role in the study ofLie group s andRiemannian geometry .**ee also***

congruence (geometry)

*Euclidean group

*reflection (linear algebra)

*point symmetry

*Riemannian symmetric space

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