**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

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*n*Dimensions

... a in Euclidean space Rn, the formula for the

**reflection**in the hyperplane through the origin, orthogonal to a, is given by where v·a denotes the dot ... Using the geometric product the formula is a little simpler Since these

**reflections**are isometries of Euclidean space fixing the origin, they may be represented by orthogonal ... The orthogonal matrix corresponding to the above

**reflection**is the matrix whose entries are where δij is the Kronecker delta ...

### Famous quotes containing the word reflection:

“The Americans ... have invented so wide a range of pithy and hackneyed phrases that they can carry on an amusing and animated conversation without giving a moment’s *reflection* to what they are saying and so leave their minds free to consider the more important matters of big business and fornication.”

—W. Somerset Maugham (1874–1965)