# Reflection (mathematics)

## Mapping from a Euclidean space to itself / From Wikipedia, the free encyclopedia

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In mathematics, a **reflection** (also spelled **reflexion**)^{[1]} is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a 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 (a *vertical reflection*) would look like **q**. Its image by reflection in a horizontal axis (a *horizontal reflection*) 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.

Some mathematicians use "**flip**" as a synonym for "reflection".^{[2]}^{[3]}^{[4]}