Reflection (mathematics)

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

Related Topics:
Binary relation - Reflexive relation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In mathematics, a reflection (also spelt reflexion) is inversion with respect to a point (in 1D), a line (in 2D) or plane (in 3D), in 2D and 3D not to be confused with "inversion" period, which means inversion in a point, in particular the origin/center. In 1D it is the same as inversion. The result is a mirror image. So for example, a reflection of a small English letter p, respect to the vertical line, would look like q. Reflection in this sense preserves the distance, and the operation is thus said to be isometric. The most common type of symmetry is reflection symmetry.

Related Topics:
Mathematics - Inversion - Mirror image - Isometric - Symmetry - Reflection symmetry

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

More generally, a reflection can be defined as an involutive automorphism of a space which leaves invariant a subspace of codimension 1 (i.e. applying it twice gives the identity, and the fixed points form a space of dimension one less than the whole space).

Related Topics:
Involutive - Automorphism - Space - Invariant - Subspace - Codimension

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that this applies to more than just Euclidean geometry. Reflections in affine geometry with respect to a given hyperplane are not unique, for example. Also, an inversion in inversive geometry is considered a reflection by this definition.

Related Topics:
Euclidean geometry - Affine geometry - Hyperplane - Inversion - Inversive geometry

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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.

Related Topics:
LAPACK - Reflector - Block reflector - Elementary reflector - Householder transformation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~