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Reflection (mathematics)
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a reflection (also spelled reflexion) is a map
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 that transforms an object into its mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
. 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
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.

Geometrically, to find the reflection of a point one drops a perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 from the point onto the line (plane) used for reflection, and continues the same distance on the other side.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a reflection (also spelled reflexion) is a map
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 that transforms an object into its mirror image
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
. 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
Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
 for a mirror. Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.

Geometrically, to find the reflection of a point one drops a perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 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 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
Involution

In mathematics, an involution, or an involutary function, is a function that is its own inverse function, so that...
 isometry
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 of an Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 whose set of fixed point
Fixed point

"Fixed point" has many meanings in science, most of them mathematical.*Fixed point *Fixed point combinator*Fixed-point arithmetic, a manner of doing arithmetic on computers...
s is an affine subspace of codimension
Codimension

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....
 1.

A figure which does not change upon undergoing a certain reflection is said to have reflection symmetry
Reflection symmetry

The triangles with this symmetry are isosceles. The quadrilaterals with this symmetry are the kite s and the isosceles trapezoids.For each line or plane of reflection, the symmetry group is isomorphic with Cs , one of the three types of order two , hence algebraically C2....
.

Closely related to reflections are oblique reflection
Oblique reflection

In Euclidean geometry, oblique reflections generalize ordinary reflection by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations....
s and circle inversions. These transformations are still involutions with the set of fixed points having codimension 1, but they are no longer isometries.

Formulas


Given a vector a in Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
 Rn, the formula for the reflection in the hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
 through the origin, orthogonal to a, is given by where v·a denotes the dot product
Dot product

In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
 of v with a. Note that the second term in the above equation is just twice the projection
Projection

Projection can be any of:* The display of an image by devices such as:**Movie projector**Video projector**Overhead projector**Slide projector...
 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
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 whose entries are where dij is the Kronecker delta
Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , 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


  • coordinate rotations and reflections
    Coordinate rotations and reflections

    In geometry, 2D coordinate rotations and reflection s are two kinds of Euclidean plane isometry which are related to one another.A rotation in the plane can be formed by composing a pair of reflections....
  • improper rotation
    Improper rotation

    In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis....
  • point reflection
    Point reflection

    In geometry, a point reflection is a type of isometry of Euclidean space. It is a reflection whose mirror is a single point. An object that is invariant under a point reflection is said to possess point symmetry....
  • reflection (linear algebra)
    Reflection (linear algebra)

    In linear algebra, a reflection is a linear transformation that squares to the identity , also known as an involution in the general linear group....


External links

  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • and by Roger Germundsson, The Wolfram Demonstrations Project.