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In erik, 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. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle. Partial plot of a function f. ...
This article is about the Twilight Zone episode. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ...
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. Fig. ...
A reflection done twice brings us back where we started. A reflection preserves the distance between points. 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. These observations allow one to formalize the definition of reflection: a reflection is an involutive isometry of an Euclidean space whose set of fixed points is an affine subspace of codimension 1. In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...
An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ...
A figure which does not change upon undergoing a certain reflection is said to have reflection symmetry. Figures with the axes of symmetry drawn in. ...
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. In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. ...
In mathematics, inversive geometry is the geometry of circles and the set of transformations that map all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ...
On a somewhat unrelated note, 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. LAPACK, the Linear Algebra PACKage, is a software library for numerical computing written in Fortran 77. ...
In mathematics, a reflection (also spelt reflexion) is to invert a geometric figure, respect to a line or plane (but not a point). ...
A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one. ...
An elementrary reflector is a vector that implements reflection (mathematics). ...
In mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane. ...
Formulas
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 Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
A hyperplane is a concept in geometry. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
 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 In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...
The word projection can mean more than one thing. ...
- 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 In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, The definition can be given for matrices with entries from any field, but the most common case is the one of matrices with real entries, and only that case will be considered in the...
 where δij is the Kronecker delta. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...
The formula for the reflection in the affine hyperplane  is given by
See also In geometry, coordinate rotations and reflections are two kinds of isometry which are related to each to other. ...
In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ...
External links - Reflection in Line at cut-the-knot
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