Reflection (mathematics)

Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.

The term "reflection" is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.

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

Read more about Reflection (mathematics):  Construction, Properties, Reflection Across A Line in The Plane, Reflection Through A Hyperplane in n Dimensions

Other articles related to "reflection, reflections":

Reflection (mathematics) - Reflection Through A Hyperplane in n Dimensions
... Given a vector a in Euclidean space Rn, the formula for the reflectionin the hyperplane through the origin, orthogonal to a, is given by where v·a denotes ... the formula is a little simpler Since these reflectionsare isometries of Euclidean space fixing the origin, they may be represented by orthogonal matrices ... The orthogonal matrix corresponding to the above reflectionis the matrix whose entries are where δij is the Kronecker delta ...

Famous quotes containing the word reflection:

    There are three principal means of acquiring knowledge available to us: observation of nature, reflection, and experimentation. Observation collects facts; reflection combines them; experimentation verifies the result of that combination. Our observation of nature must be diligent, our reflection profound, and our experiments exact. We rarely see these three means combined; and for this reason, creative geniuses are not common.
    Denis Diderot (1713–1784)