﻿

# 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Reflection — or reflexion may refer to:Computers* in computer graphics, the techniques for simulating optical Reflection. * Reflection, a programming language feature for metaprogramming * Reflection , a piece of installation art by Shane Cooper also called… …   Wikipedia

• Reflection (mathematics) — This article is about reflection in geometry. For reflexivity of binary relations, see reflexive relation. A reflection through an axis followed by a reflection across a second axis parallel to the first one results in a total motion which is a… …   Wikipedia

• Reflection (linear algebra) — In linear algebra, a reflection is a linear transformation that squares to the identity ( R 2 = I , where R is in K dimensional space), also known as an involution in the general linear group. In addition to reflections across hyperplanes, the… …   Wikipedia

• Reflection (physics) — Reflection is the change in direction of a wave front at an between two different media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. Law of… …   Wikipedia

• Reflection high energy electron diffraction — (RHEED) is a technique used to characterize the surface of crystalline materials. RHEED systems gather information only from the surface layer of the sample, which distinguishes RHEED from other materials characterization methods that rely also… …   Wikipedia

• Reflection seismology — (or seismic reflection) is a method of exploration geophysics that uses the principles of seismology to estimate the properties of the Earth s subsurface from reflected seismic waves. The method requires a controlled seismic source of energy,… …   Wikipedia

• Reflection symmetry — Reflection symmetry, line symmetry, mirror symmetry, mirror image symmetry, or bilateral symmetry is symmetry with respect to reflection.In 2D there is an axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable …   Wikipedia

• Point (coat color) — Point coloration refers to animal coat coloration with a pale body and relatively darker extremities, i.e. the face, ears, feet, tail, and (in males) scrotum. It is most recognized as the coloration of Siamese and other breeds of cat, but can be… …   Wikipedia

• Point groups in three dimensions — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… …   Wikipedia

• Reflection mapping — An example of reflection mapping. In computer graphics, environment mapping, or reflection mapping, is an efficient Image based lighting technique for approximating the appearance of a reflective surface by means of a precomputed texture image.… …   Wikipedia