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A reflection through an axis. 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 the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space. Technically, a reflection group is a subgroup of a linear group (or various generalizations) generated by orthogonal matrices of determinant -1.
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation R {\displaystyle R} on a set X {\displaystyle X} is reflexive if it relates every element of X {\displaystyle X} to itself.
These equations can be proved through straightforward matrix multiplication and application of trigonometric identities, specifically the sum and difference identities.. The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group.
A reflection in a line is an opposite isometry, like R 1 or R 2 on the image. Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
In terms of linear algebra, assuming the origin is fixed, involutions are exactly the diagonalizable maps with all eigenvalues either 1 or −1. Reflection in a hyperplane has a single −1 eigenvalue (and multiplicity on the 1 eigenvalue), while point reflection has only the −1 eigenvalue (with multiplicity n).
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group.
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation . It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.