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searchaux" style="display:none">Mapping from a Euclidean space to itself</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about reflection in geometry. For reflexivity of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>, see <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive relation</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SimmetriainvOK.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/SimmetriainvOK.svg/220px-SimmetriainvOK.svg.png" decoding="async" width="220" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/SimmetriainvOK.svg/330px-SimmetriainvOK.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/SimmetriainvOK.svg/440px-SimmetriainvOK.svg.png 2x" data-file-width="390" data-file-height="362" /></a><figcaption>A reflection through an axis.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>reflection</b> (also spelled <b>reflexion</b>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">mapping</a> from a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> to itself that is an <a href="/wiki/Isometry" title="Isometry">isometry</a> with a <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> as the set of <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a>; this set is called the <a href="/wiki/Axis_of_symmetry" class="mw-redirect" title="Axis of symmetry">axis</a> (in dimension 2) or <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a> (in dimension 3) of reflection. The image of a figure by a reflection is its <a href="/wiki/Mirror_image" title="Mirror image">mirror image</a> in the axis or plane of reflection. For example the mirror image of the small Latin letter <b>p</b> for a reflection with respect to a <a href="/wiki/Vertical_axis" class="mw-redirect" title="Vertical axis">vertical axis</a> (a <i>vertical reflection</i>) would look like <b>q</b>. Its image by reflection in a <a href="/wiki/Horizontal_axis" class="mw-redirect" title="Horizontal axis">horizontal axis</a> (a <i>horizontal reflection</i>) would look like <b>b</b>. A reflection is an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a>: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. </p><p>The term <i>reflection</i> 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 <a href="/wiki/Affine_subspace" class="mw-redirect" title="Affine subspace">affine subspace</a>, but is possibly smaller than a hyperplane. For instance a <a href="/wiki/Point_reflection" title="Point reflection">reflection through a point</a> is an involutive isometry with just one fixed point; the image of the letter <b>p</b> under it would look like a <b>d</b>. This operation is also known as a <a href="/wiki/Point_reflection" title="Point reflection">central inversion</a> (<a href="#CITEREFCoxeter1969">Coxeter 1969</a>, §7.2), and exhibits Euclidean space as a <a href="/wiki/Symmetric_space" title="Symmetric space">symmetric space</a>. In a <a href="/wiki/Euclidean_vector_space" class="mw-redirect" title="Euclidean vector space">Euclidean vector space</a>, 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 <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a>. </p><p>Some mathematicians use "<b>flip</b>" as a synonym for "reflection".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=1" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Perpendicular-construction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Perpendicular-construction.svg/236px-Perpendicular-construction.svg.png" decoding="async" width="236" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Perpendicular-construction.svg/354px-Perpendicular-construction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Perpendicular-construction.svg/472px-Perpendicular-construction.svg.png 2x" data-file-width="678" data-file-height="566" /></a><figcaption>Point <span class="texhtml mvar" style="font-style:italic;">Q</span> is the reflection of point <span class="texhtml mvar" style="font-style:italic;">P</span> through the line <span class="texhtml mvar" style="font-style:italic;">AB</span>.</figcaption></figure> <p>In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> 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. </p><p>To reflect point <span class="texhtml">P</span> through the line <span class="texhtml">AB</span> using <a href="/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>, proceed as follows (see figure): </p> <ul><li>Step 1 (red): construct a <a href="/wiki/Circle" title="Circle">circle</a> with center at <span class="texhtml">P</span> and some fixed radius <span class="texhtml"><i>r</i></span> to create points <span class="texhtml">A′</span> and <span class="texhtml">B′</span> on the line <span class="texhtml">AB</span>, which will be <a href="/wiki/Equidistant" title="Equidistant">equidistant</a> from <span class="texhtml">P</span>.</li> <li>Step 2 (green): construct circles centered at <span class="texhtml">A′</span> and <span class="texhtml">B′</span> having radius <span class="texhtml"><i>r</i></span>. <span class="texhtml">P</span> and <span class="texhtml">Q</span> will be the points of intersection of these two circles.</li></ul> <p>Point <span class="texhtml">Q</span> is then the reflection of point <span class="texhtml">P</span> through line <span class="texhtml">AB</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> for a reflection is <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal</a> with <a href="/wiki/Determinant" title="Determinant">determinant</a> −1 and <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a> is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every <a href="/wiki/Improper_rotation" title="Improper rotation">improper rotation</a> is the result of reflecting in an odd number. Thus reflections generate the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a>, and this result is known as the <a href="/wiki/Cartan%E2%80%93Dieudonn%C3%A9_theorem" title="Cartan–Dieudonné theorem">Cartan–Dieudonné theorem</a>. </p><p>Similarly the <a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean group</a>, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> generated by reflections in affine hyperplanes is known as a <a href="/wiki/Reflection_group" title="Reflection group">reflection group</a>. The <a href="/wiki/Finite_group" title="Finite group">finite groups</a> generated in this way are examples of <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Reflection_across_a_line_in_the_plane">Reflection across a line in the plane</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=3" title="Edit section: Reflection across a line in the plane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information on reflection of light rays: <a href="/wiki/Specular_reflection#Direction_of_reflection" title="Specular reflection">Specular reflection § Direction of reflection</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/180-degree_rotation" class="mw-redirect" title="180-degree rotation">180-degree rotation</a></div> <p>Reflection across an arbitrary line through the origin in <a href="/wiki/Two_dimensions" class="mw-redirect" title="Two dimensions">two dimensions</a> can be described by the following formula </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ref</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>⋅</mo> <mi>l</mi> </mrow> <mrow> <mi>l</mi> <mo>⋅</mo> <mi>l</mi> </mrow> </mfrac> </mrow> <mi>l</mi> <mo>−</mo> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c312b54a667389fe3a6a99b8b33ac358fb4d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.019ex; height:5.509ex;" alt="{\displaystyle \operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> denotes the vector being reflected, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> denotes any vector in the line across which the reflection is performed, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\cdot l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>⋅</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\cdot l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf84b970c87e878ba334b0c5088ce33269dbfb1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.5ex; height:2.176ex;" alt="{\displaystyle v\cdot l}"></span> denotes the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>. Note the formula above can also be written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ref</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>Proj</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416fdb1891fbc688b287d1a563fd66a52e576753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.405ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,}"></span></dd></dl> <p>saying that a reflection of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> across <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> is equal to 2 times the <a href="/wiki/Vector_projection" title="Vector projection">projection</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>, minus the vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>. Reflections in a line have the eigenvalues of 1, and −1. </p> <div class="mw-heading mw-heading2"><h2 id="Reflection_through_a_hyperplane_in_n_dimensions">Reflection through a hyperplane in <i>n</i> dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=4" title="Edit section: Reflection through a hyperplane in n dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, the formula for the reflection in the <a href="/wiki/Hyperplane" title="Hyperplane">hyperplane</a> through the origin, <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ref</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>⋅</mo> <mi>a</mi> </mrow> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23921e8eb7259dcad8e78eecc76cbb77501cff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.574ex; height:4.676ex;" alt="{\displaystyle \operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\cdot a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2ef8f805cef6b84e3874e0b4c44bd0a01eaa12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.037ex; height:1.676ex;" alt="{\displaystyle v\cdot a}"></span> denotes the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. Note that the second term in the above equation is just twice the <a href="/wiki/Vector_projection" title="Vector projection">vector projection</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> onto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. One can easily check that </p> <ul><li><span class="texhtml">Ref<sub><i>a</i></sub>(<i>v</i>) = −<i>v</i></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is parallel to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, and</li> <li><span class="texhtml">Ref<sub><i>a</i></sub>(<i>v</i>) = <i>v</i></span>, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is perpendicular to <span class="texhtml mvar" style="font-style:italic;"><i>a</i></span>.</li></ul> <p>Using the <a href="/wiki/Geometric_product" class="mw-redirect" title="Geometric product">geometric product</a>, the formula is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ref</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>v</mi> <mi>a</mi> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7ec26e62f218c312da6382c2b864da7c39aa77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.47ex; height:5.009ex;" alt="{\displaystyle \operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.}"></span></dd></dl> <p>Since these reflections are isometries of Euclidean space fixing the origin they may be represented by <a href="/wiki/Orthogonal_matrices" class="mw-redirect" title="Orthogonal matrices">orthogonal matrices</a>. The orthogonal matrix corresponding to the above reflection is the <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=I-2{\frac {aa^{T}}{a^{T}a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>I</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=I-2{\frac {aa^{T}}{a^{T}a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15afa1f51a887c05c7a38054314051f4cbc05437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.369ex; height:6.009ex;" alt="{\displaystyle R=I-2{\frac {aa^{T}}{a^{T}a}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> denotes the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60016a20ce32487efc7b8cda83fe316ea75b9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.619ex; height:2.676ex;" alt="{\displaystyle a^{T}}"></span> is the <a href="/wiki/Transpose" title="Transpose">transpose</a> of a. Its entries are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mi>a</mi> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eab23a0f5e4e60281193ab50edf634530e05c25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.944ex; height:6.343ex;" alt="{\displaystyle R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},}"></span></dd></dl> <p>where <span class="texhtml"><i>δ</i><sub><i>ij</i></sub></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. </p><p>The formula for the reflection in the affine hyperplane <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\cdot a=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\cdot a=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/153ae5b4c99aa71dfb09e8562194f691d6c0bbcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.142ex; height:1.676ex;" alt="{\displaystyle v\cdot a=c}"></span> not through the origin is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ref</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo>⋅</mo> <mi>a</mi> <mo>−</mo> <mi>c</mi> </mrow> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff75485900d89e8333c70704affec106875dbfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.488ex; height:5.009ex;" alt="{\displaystyle \operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Additive_inverse" title="Additive inverse">Additive inverse</a></li> <li><a href="/wiki/Coordinate_rotations_and_reflections" class="mw-redirect" title="Coordinate rotations and reflections">Coordinate rotations and reflections</a></li> <li><a href="/wiki/Householder_transformation" title="Householder transformation">Householder transformation</a></li> <li><a href="/wiki/Inversive_geometry" title="Inversive geometry">Inversive geometry</a></li> <li><a href="/wiki/Plane_of_rotation" title="Plane of rotation">Plane of rotation</a></li> <li><a href="/wiki/Reflection_mapping" title="Reflection mapping">Reflection mapping</a></li> <li><a href="/wiki/Reflection_group" title="Reflection group">Reflection group</a></li> <li><a href="/wiki/Reflection_symmetry" title="Reflection symmetry">Reflection symmetry</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output 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.citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFChilds2009" class="citation cs2">Childs, Lindsay N. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qyDAKBr_I2YC&q=flip&pg=PA251"><i>A Concrete Introduction to Higher Algebra</i></a> (3rd ed.), Springer Science & Business Media, p. 251, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387745275" title="Special:BookSources/9780387745275"><bdi>9780387745275</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Concrete+Introduction+to+Higher+Algebra&rft.pages=251&rft.edition=3rd&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=9780387745275&rft.aulast=Childs&rft.aufirst=Lindsay+N.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqyDAKBr_I2YC%26q%3Dflip%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2012" class="citation cs2"><a href="/wiki/Joseph_Gallian" title="Joseph Gallian">Gallian, Joseph</a> (2012), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ef4KAAAAQBAJ&q=flip&pg=PA32"><i>Contemporary Abstract Algebra</i></a> (8th ed.), Cengage Learning, p. 32, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1285402734" title="Special:BookSources/978-1285402734"><bdi>978-1285402734</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Abstract+Algebra&rft.pages=32&rft.edition=8th&rft.pub=Cengage+Learning&rft.date=2012&rft.isbn=978-1285402734&rft.aulast=Gallian&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEf4KAAAAQBAJ%26q%3Dflip%26pg%3DPA32&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsaacs1994" class="citation cs2"><a href="/wiki/Martin_Isaacs" title="Martin Isaacs">Isaacs, I. Martin</a> (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5tKq0kbHuc4C&q=flip&pg=PA6"><i>Algebra: A Graduate Course</i></a>, American Mathematical Society, p. 6, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821847992" title="Special:BookSources/9780821847992"><bdi>9780821847992</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra%3A+A+Graduate+Course&rft.pages=6&rft.pub=American+Mathematical+Society&rft.date=1994&rft.isbn=9780821847992&rft.aulast=Isaacs&rft.aufirst=I.+Martin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5tKq0kbHuc4C%26q%3Dflip%26pg%3DPA6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxeter1969" class="citation cs2"><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter, Harold Scott MacDonald</a> (1969), <i>Introduction to Geometry</i> (2nd ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50458-0" title="Special:BookSources/978-0-471-50458-0"><bdi>978-0-471-50458-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0123930">0123930</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Geometry&rft.place=New+York&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=1969&rft.isbn=978-0-471-50458-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D123930%23id-name%3DMR&rft.aulast=Coxeter&rft.aufirst=Harold+Scott+MacDonald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPopov2001" class="citation cs2"><a href="/wiki/Vladimir_L._Popov" class="mw-redirect" title="Vladimir L. Popov">Popov, V.L.</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Reflection">"Reflection"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reflection&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Popov&rft.aufirst=V.L.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DReflection&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Reflection"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Reflection.html">"Reflection"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Reflection&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FReflection.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReflection+%28mathematics%29" class="Z3988"></span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reflection_(mathematics)&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" 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