In mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q. In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a plane for a mirror.
Geometrically, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues the same distance on the other side. To find the reflection of a figure, one reflects each point in the figure.
A reflection applied twice to a geometrical object restores the object to its original state: it is an involution. A reflection preserves the distance between points: it is an isometry. A reflection does not move the points which are on the mirror, and the dimension of the mirror is by one smaller than the dimension of the space in which the reflection takes places. Thus a reflection is an involutive isometry of an Euclidean space whose set of fixed points is an affine subspace of codimension 1. This can be taken as the formal definition of a reflection.
More generally, it is possible to consider reflections in subspaces of higher codimension. For instance, reflection in the point situated at the origin is the same as vector negation. Reflection in a point is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. One may also consider, for instance, reflection in a line, or other affine subspace. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.
Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem. Similarly the Euclidean group consisting of all isometries of Euclidean space is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are the Coxeter groups.
A figure which does not change upon undergoing a certain reflection is said to have reflection symmetry.
Closely related to reflections are oblique reflections and circle inversions. These transformations are still involutions with the set of fixed points having codimension 1, but they are no longer isometries. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.
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Reflections in linear algebra
Given a vector a in Euclidean space Rn, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by
where v·a denotes the dot product of v with a. Note that the second term in the above equation is just twice the projection of v onto a. One can easily check that
- Refa(v) = − v, if v is parallel to a, and
- Refa(v) = v, if v is perpendicular to a.
Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix whose entries are
where δij is the Kronecker delta.
The formula for the reflection in the affine hyperplane 
is given by
See also
References
- Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, MR123930, ISBN 978-0-471-50458-0
- Popov, V.L. (2001), "Reflection", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Weisstein, Eric W., "Reflection" from MathWorld.
External links
- Reflection in Line at cut-the-knot
- Understanding 2D Reflection and Understanding 3D Reflection by Roger Germundsson, The Wolfram Demonstrations Project.
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