Reflection (mathematics)

From Wikipedia, the free encyclopedia

This article is about reflection in geometry. For reflexivity of binary relations, see reflexive relation.

In mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a plane for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.

Geometrically, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure.

A reflection done twice brings us back where we started. A reflection preserves the distance between points. A reflection does not move the points which are on the mirror, and the dimension of the mirror is by one smaller than the dimension of the space in which the reflection takes places. These observations allow one to formalize the definition of reflection: a reflection is an involutive isometry of an Euclidean space whose set of fixed points is an affine subspace of codimension 1.

A figure which does not change upon undergoing a certain reflection is said to have reflection symmetry.

Closely related to reflections are oblique reflections and circle inversions. These transformations are still involutions with the set of fixed points having codimension 1, but they are no longer isometries.

On a somewhat unrelated note, in LAPACK the term reflector with the types block reflector and elementary reflector is used to describe the functionality of the routines that implement the Householder transformation.

[edit] Formulas

Given a vector a in Euclidean space Rⁿ, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

$\mathrm{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a$

where v·a denotes the dot product of v with a. Note that the second term in the above equation is just twice the projection of v onto a. One can easily check that

Ref_a(v) = − v, if v is parallel to a, and
Ref_a(v) = v, if v is perpendicular to a.

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose entries are

$R_{ij} = \delta_{ij} - 2\frac{a_i a_j}{\|a\|^2}$