Reflection (mathematics)
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.
For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a vertical reflection) would look like q.
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.
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.
The set of fixed points (the "mirror") of such an isometry is an affine subspace, but is possibly smaller than a hyperplane.
For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as a central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space.
Other examples include reflections in a line in three-dimensional space.
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 a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side.
The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1.
The product of two such matrices is a special orthogonal matrix that represents a rotation.
Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem.
Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.
Reflection across an arbitrary line through the origin in two dimensions can be described by the following formula where
denotes any vector in the line across which the reflection is performed, and
Note the formula above can also be written as saying that a reflection of
, the formula for the reflection in the hyperplane through the origin, orthogonal to
Note that the second term in the above equation is just twice the vector projection of
One can easily check that Using the geometric product, the formula is Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices.
