The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1. Construction Ī reflection across an axis followed by a reflection in a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes, by an angle twice the angle between the axes. Some mathematicians use ' flip' as a synonym for 'reflection'. Typically, however, unqualified use of the term 'reflection' means reflection in a hyperplane. Other examples include reflections in a line in three-dimensional space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. This operation is also known as a central inversion ( Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. For instance a reflection through a point is an involutive isometry with just one fixed point the image of the letter p under it Such isometries have a set of fixed points (the 'mirror') that is an affine subspace, but is possibly smaller than a hyperplane. 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. 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.

Its image by reflection in a horizontal axis would look like b. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. In mathematics, a reflection (also spelled reflexion) 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. A reflection through an axis (from the red object to the green one) followed by a reflection (green to blue) across a second axis parallel to the first one results in a total motion that is a translation - by an amount equal to twice the distance between the two axes.

Reflection geometry definition