Inmathematics, areflection (also spelledreflexion)[1] is amapping from aEuclidean space to itself that is anisometry with ahyperplane as the set offixed points; this set is called theaxis (in dimension 2) orplane (in dimension 3) of reflection. The image of a figure by a reflection is itsmirror image in the axis or plane of reflection. For example the mirror image of the small Latin letterp for a reflection with respect to avertical axis (avertical reflection) would look likeq. Its image by reflection in ahorizontal axis (ahorizontal reflection) would look likeb. A reflection is aninvolution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
The termreflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isometry is anaffine subspace, but is possibly smaller than a hyperplane. For instance areflection through a point is an involutive isometry with just one fixed point; the image of the letterp under itwould look like ad. This operation is also known as acentral inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as asymmetric space. In aEuclidean 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 ahyperplane.
Some mathematicians use "flip" as a synonym for "reflection".[2][3][4]
In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop aperpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
To reflect pointP through the lineAB usingcompass and straightedge, proceed as follows (see figure):
PointQ is then the reflection of pointP through lineAB.
Thematrix for a reflection isorthogonal withdeterminant −1 andeigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Everyrotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and everyimproper rotation is the result of reflecting in an odd number. Thus reflections generate theorthogonal group, and this result is known as theCartan–Dieudonné theorem.
Similarly theEuclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, agroup generated by reflections in affine hyperplanes is known as areflection group. Thefinite groups generated in this way are examples ofCoxeter groups.
Reflection across an arbitrary line through the origin intwo dimensions can be described by the following formula
where denotes the vector being reflected, denotes any vector in the line across which the reflection is performed, and denotes thedot product of with. Note the formula above can also be written as
saying that a reflection of across is equal to 2 times theprojection of on, minus the vector. Reflections in a line have the eigenvalues of 1, and −1.
Given a vector inEuclidean space, the formula for the reflection in thehyperplane through the origin,orthogonal to, is given by
where denotes thedot product of with. Note that the second term in the above equation is just twice thevector projection of onto. One can easily check that
Using thegeometric product, the formula is
Since these reflections are isometries of Euclidean space fixing the origin they may be represented byorthogonal matrices. The orthogonal matrix corresponding to the above reflection is thematrix
where denotes theidentity matrix and is thetranspose of a. Its entries are
whereδij is theKronecker delta.
The formula for the reflection in the affine hyperplane not through the origin is
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