Editing Reflection (mathematics)

{{Short description|Mapping from a Euclidean space to itself}}
{{About|reflection in geometry|reflexivity of [[binary relation]]s|reflexive relation}}
[[File:SimmetriainvOK.svg|right|thumb|A reflection through an axis.]] 

In [[mathematics]], a '''reflection''' (also spelled '''reflexion''')<ref>[https://web.archive.org/web/20120829214317/http://oxforddictionaries.com/definition/english/reflexion "Reflexion" is an archaic spelling]</ref> is a [[function (mathematics)|mapping]] from a [[Euclidean space]] to itself that is an [[isometry]] with a [[hyperplane]] as the set of [[Fixed point (mathematics)|fixed point]]s; this set is called the [[Axis of symmetry|axis]] (in dimension 2) or [[plane (mathematics)|plane]] (in dimension 3) of reflection. The image of a figure by a reflection is its [[mirror image]] in the axis or plane 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'''. Its image by reflection in a [[horizontal axis]] (a ''horizontal reflection'') would look like '''b'''. A reflection is an [[involution (mathematics)|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 [[Point reflection|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 [[point reflection|central inversion]] {{harv|Coxeter|1969|loc=§7.2}}, and exhibits Euclidean space as a [[symmetric space]]. In a [[Euclidean 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 a [[hyperplane]].

Some mathematicians use "'''flip'''" as a synonym for "reflection".<ref>{{Citation |last=Childs |first=Lindsay N. |year=2009 |title=A Concrete Introduction to Higher Algebra |edition=3rd |publisher=Springer Science & Business Media |page=251 |isbn=9780387745275 |url=https://books.google.com/books?id=qyDAKBr_I2YC&q=flip&pg=PA251 }}</ref><ref>
{{Citation |last=Gallian |first=Joseph |author-link=Joseph Gallian |year=2012 |title=Contemporary Abstract Algebra |edition=8th |publisher=Cengage Learning |page=32 |isbn=978-1285402734 |url=https://books.google.com/books?id=Ef4KAAAAQBAJ&q=flip&pg=PA32 }}</ref><ref>
{{Citation |last=Isaacs |first=I. Martin |author-link=Martin Isaacs |year=1994 |title=Algebra: A Graduate Course |publisher=American Mathematical Society |page=6 |isbn=9780821847992 |url=https://books.google.com/books?id=5tKq0kbHuc4C&q=flip&pg=PA6 }}
</ref>

==Construction==
[[File:Perpendicular-construction.svg|thumb|236px|Point {{mvar|Q}} is the reflection of point {{mvar|P}} through the line {{mvar|AB}}.]]

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. To find the reflection of a figure, reflect each point in the figure.

To reflect point {{math|P}} through the line {{math|AB}} using [[compass and straightedge]], proceed as follows (see figure):

* Step 1 (red): construct a [[circle]] with center at {{math|P}} and some fixed radius {{math|''r''}} to create points {{math|A′}} and {{math|B′}} on the line {{math|AB}}, which will be [[equidistant]] from {{math|P}}.
* Step 2 (green): construct circles centered at {{math|A′}} and {{math|B′}} having radius {{math|''r''}}. {{math|P}} and {{math|Q}} will be the points of intersection of these two circles.

Point {{math|Q}} is then the reflection of point {{math|P}} through line {{math|AB}}.

==Properties==

The [[matrix (mathematics)|matrix]] for a reflection is [[orthogonal matrix|orthogonal]] with [[determinant]] −1 and [[eigenvalue]]s −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every [[Rotation (mathematics)|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]], which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.  In general, a [[group (mathematics)|group]] generated by reflections in affine hyperplanes is known as a [[reflection group]]. The [[finite group]]s generated in this way are examples of [[Coxeter group]]s.

==Reflection across a line in the plane==
{{Further|topic=reflection of light rays|Specular reflection#Direction of reflection}}
{{see also|180-degree rotation}}

Reflection across an arbitrary line through the origin in [[two dimensions]] can be described by the following formula
:<math>\operatorname{Ref}_l(v) = 2\frac{v \cdot l}{l \cdot l}l - v,</math>

where <math>v</math> denotes the vector being reflected, <math>l</math> denotes any vector in the line across which the reflection is performed, and <math>v\cdot l</math> denotes the [[dot product]] of <math>v</math> with <math>l</math>. Note the formula above can also be written as
:<math>\operatorname{Ref}_l(v) = 2\operatorname{Proj}_l(v) - v,</math>
saying that a reflection of <math>v</math> across <math>l</math> is equal to 2 times the [[vector projection|projection]] of <math>v</math> on <math>l</math>, minus the vector <math>v</math>.  Reflections in a line have the eigenvalues of 1, and −1.

==Reflection through a hyperplane in ''n'' dimensions==
Given a vector <math>v</math> in [[Euclidean space]] <math>\mathbb R^n</math>, the formula for the reflection in the [[hyperplane]] through the origin, [[orthogonal]] to <math>a</math>, is given by

:<math>\operatorname{Ref}_a(v) = v - 2\frac{v\cdot a}{a\cdot a}a,</math>

where <math>v\cdot a</math> denotes the [[dot product]] of <math>v</math> with <math>a</math>. Note that the second term in the above equation is just twice the [[vector projection]] of <math>v</math> onto <math>a</math>. One can easily check that
*{{math|1=Ref<sub>''a''</sub>(''v'') = −''v''}}, if <math>v</math> is parallel to <math>a</math>, and
*{{math|1=Ref<sub>''a''</sub>(''v'') =  ''v''}}, if <math>v</math> is perpendicular to {{mvar|''a''}}.

Using the [[geometric product]], the formula is

:<math>\operatorname{Ref}_a(v) = -\frac{a v a}{a^2} .</math>

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 (mathematics)|matrix]]

:<math>R = I-2\frac{aa^T}{a^Ta},</math>

where <math>I</math> denotes the <math>n \times n</math> [[identity matrix]] and <math>a^T</math> is the [[transpose]] of a. Its entries are

:<math>R_{ij} = \delta_{ij} - 2\frac{a_i a_j}{ \left\| a \right\| ^2 },</math>

where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]].

The formula for the reflection in the affine hyperplane <math>v\cdot a=c</math> not through the origin is
:<math>\operatorname{Ref}_{a,c}(v) = v - 2\frac{v \cdot a - c}{a\cdot a}a.</math>

==See also==
* [[Additive inverse]]
* [[Coordinate rotations and reflections]]
* [[Householder transformation]]
* [[Inversive geometry]]
* [[Plane of rotation]]
* [[Reflection mapping]]
* [[Reflection group]]
* [[Reflection symmetry]]

==Notes==
{{Reflist}}

==References==
*{{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Introduction to Geometry | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50458-0 | mr=123930 | year=1969}}
*{{springer|title=Reflection|first=V.L.|last=Popov|authorlink=Vladimir L. Popov|id=R/r080510}}
*{{MathWorld |title=Reflection |urlname=Reflection}}

==External links==
* [http://www.cut-the-knot.org/Curriculum/Geometry/Reflection.shtml Reflection in Line] at [[cut-the-knot]]
* [http://demonstrations.wolfram.com/Understanding2DReflection/ Understanding 2D Reflection] and [http://demonstrations.wolfram.com/Understanding3DReflection/ Understanding 3D Reflection] by Roger Germundsson, [[The Wolfram Demonstrations Project]].

{{Authority control}}

[[Category:Euclidean symmetries]]
[[Category:Functions and mappings]]
[[Category:Linear operators]]
[[Category:Transformation (function)]]