Reflection (mathematics)

<h2 id="construction">Construction</h2>

<p>In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a <a href="/facts/Perpendicular/u0k8xqx4">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 P through the line AB using <a href="/facts/Compass_and_straightedge/lbaHf2m7">compass and straightedge</a>, proceed as follows (see figure):
</p>
<ul><li>Step 1 (red): construct a <a href="/facts/Circle/9fy5ggKn">circle</a> with center at P and some fixed radius <i>r</i> to create points A′ and B′ on the line AB, which will be <a href="/facts/Equidistant/muTk20sY">equidistant</a> from P.</li>
<li>Step 2 (green): construct circles centered at A′ and B′ having radius <i>r</i>. P and Q will be the points of intersection of these two circles.</li></ul>
<p>Point Q is then the reflection of point P through line AB.
</p>
<h2 id="properties">Properties</h2>
<p>The <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> for a reflection is <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal</a> with <a href="/facts/Determinant/eGYqJ2TF">determinant</a> −1 and <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every <a href="/facts/Improper_rotation/kcLdCD9r">improper rotation</a> is the result of reflecting in an odd number. Thus reflections generate the <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a>, and this result is known as the <a href="/facts/Cartan%25E2%2580%2593Dieudonn%25C3%25A9_theorem/RRvPLzGX">Cartan–Dieudonné theorem</a>.
</p><p>Similarly the <a href="/facts/Euclidean_group/VdTSdwzl">Euclidean group</a>, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes.  In general, a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> generated by reflections in affine hyperplanes is known as a <a href="/facts/Reflection_group/7sFyhkwH">reflection group</a>. The <a href="/facts/Finite_group/vv2tvogB">finite groups</a> generated in this way are examples of <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a>.
</p>
<h2 id="reflection-across-a-line-in-the-plane">Reflection across a line in the plane</h2>
<p class="note">Further information on reflection of light rays: <a href="/facts/Specular_reflection/dwVIqLVb">Specular reflection § Direction of reflection</a></p>
<p class="note">See also: <a href="/facts/180-degree_rotation/yChkz49x">180-degree rotation</a></p>
<p>Reflection across an arbitrary line through the origin in <a href="/facts/Two_dimensions/f4b397W8">two dimensions</a> can be described by the following formula
</p>

Ref
          
            l
          
        
        ⁡
        (
        v
        )
        =
        2
        
          
            
              v
              ⋅
              l
            
            
              l
              ⋅
              l
            
          
        
        l
        −
        v
        ,
      
    
    {\displaystyle \operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,}

<p>where 
  
    
      
        v
      
    
    {\displaystyle v}
  
 denotes the vector being reflected, 
  
    
      
        l
      
    
    {\displaystyle l}
  
 denotes any vector in the line across which the reflection is performed, and 
  
    
      
        v
        ⋅
        l
      
    
    {\displaystyle v\cdot l}
  
 denotes the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 with 
  
    
      
        l
      
    
    {\displaystyle l}
  
. Note the formula above can also be written as
</p>

Ref
          
            l
          
        
        ⁡
        (
        v
        )
        =
        2
        
          Proj
          
            l
          
        
        ⁡
        (
        v
        )
        −
        v
        ,
      
    
    {\displaystyle \operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,}

<p>saying that a reflection of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 across 
  
    
      
        l
      
    
    {\displaystyle l}
  
 is equal to 2 times the <a href="/facts/Vector_projection/ENlzoaCx">projection</a> of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 on 
  
    
      
        l
      
    
    {\displaystyle l}
  
, minus the vector 
  
    
      
        v
      
    
    {\displaystyle v}
  
.  Reflections in a line have the eigenvalues of 1, and −1.
</p>
<h2 id="reflection-through-a-hyperplane-in-ndimensions">Reflection through a hyperplane in <i>n</i> dimensions</h2>
<p>Given a vector 
  
    
      
        v
      
    
    {\displaystyle v}
  
 in <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
, the formula for the reflection in the <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> through the origin, <a href="/facts/Orthogonal/JVIjYPog">orthogonal</a> to 
  
    
      
        a
      
    
    {\displaystyle a}
  
, is given by
</p>

Ref
          
            a
          
        
        ⁡
        (
        v
        )
        =
        v
        −
        2
        
          
            
              v
              ⋅
              a
            
            
              a
              ⋅
              a
            
          
        
        a
        ,
      
    
    {\displaystyle \operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,}

<p>where 
  
    
      
        v
        ⋅
        a
      
    
    {\displaystyle v\cdot a}
  
 denotes the <a href="/facts/Dot_product/tNz8MLjT">dot product</a> of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 with 
  
    
      
        a
      
    
    {\displaystyle a}
  
. Note that the second term in the above equation is just twice the <a href="/facts/Vector_projection/ENlzoaCx">vector projection</a> of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 onto 
  
    
      
        a
      
    
    {\displaystyle a}
  
. One can easily check that
</p>
<ul><li>Ref<i>a</i>(<i>v</i>) = −<i>v</i>, if 
  
    
      
        v
      
    
    {\displaystyle v}
  
 is parallel to 
  
    
      
        a
      
    
    {\displaystyle a}
  
, and</li>
<li>Ref<i>a</i>(<i>v</i>) =  <i>v</i>, if 
  
    
      
        v
      
    
    {\displaystyle v}
  
 is perpendicular to <i>a</i>.</li></ul>
<p>Using the <a href="/facts/Geometric_product/yykKMNYx">geometric product</a>, the formula is
</p>

Ref
          
            a
          
        
        ⁡
        (
        v
        )
        =
        −
        
          
            
              a
              v
              a
            
            
              a
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.}

<p>Since these reflections are isometries of Euclidean space fixing the origin they may be represented by <a href="/facts/Orthogonal_matrices/WDXBXYB5">orthogonal matrices</a>. The orthogonal matrix corresponding to the above reflection is the <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>
</p>

R
        =
        I
        −
        2
        
          
            
              a
              
                a
                
                  T
                
              
            
            
              
                a
                
                  T
                
              
              a
            
          
        
        ,
      
    
    {\displaystyle R=I-2{\frac {aa^{T}}{a^{T}a}},}

<p>where 
  
    
      
        I
      
    
    {\displaystyle I}
  
 denotes the 
  
    
      
        n
        ×
        n
      
    
    {\displaystyle n\times n}
  
 <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> and 
  
    
      
        
          a
          
            T
          
        
      
    
    {\displaystyle a^{T}}
  
 is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of a. Its entries are
</p>

R
          
            i
            j
          
        
        =
        
          δ
          
            i
            j
          
        
        −
        2
        
          
            
              
                a
                
                  i
                
              
              
                a
                
                  j
                
              
            
            
              
                ‖
                a
                ‖
              
              
                2
              
            
          
        
        ,
      
    
    {\displaystyle R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},}

<p>where <i>δ</i><i>ij</i> is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>.
</p><p>The formula for the reflection in the affine hyperplane 
  
    
      
        v
        ⋅
        a
        =
        c
      
    
    {\displaystyle v\cdot a=c}
  
 not through the origin is
</p>

Ref
          
            a
            ,
            c
          
        
        ⁡
        (
        v
        )
        =
        v
        −
        2
        
          
            
              v
              ⋅
              a
              −
              c
            
            
              a
              ⋅
              a
            
          
        
        a
        .
      
    
    {\displaystyle \operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Additive_inverse/n8gKUmCf">Additive inverse</a></li>
<li><a href="/facts/Coordinate_rotations_and_reflections/a9HmKQAr">Coordinate rotations and reflections</a></li>
<li><a href="/facts/Householder_transformation/1L87qsqE">Householder transformation</a></li>
<li><a href="/facts/Inversive_geometry/VqFmbfDI">Inversive geometry</a></li>
<li><a href="/facts/Plane_of_rotation/rClmWuga">Plane of rotation</a></li>
<li><a href="/facts/Reflection_mapping/oPL8X8Br">Reflection mapping</a></li>
<li><a href="/facts/Reflection_group/7sFyhkwH">Reflection group</a></li>
<li><a href="/facts/Reflection_symmetry/eSeR8nbD">Reflection symmetry</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, Harold Scott MacDonald</a> (1969), <i>Introduction to Geometry</i> (2nd ed.), New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-50458-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0123930">0123930</a></li>
<li><a href="/facts/Vladimir_L._Popov/U9VGpfRW">Popov, V.L.</a> (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Reflection">"Reflection"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Reflection.html">"Reflection"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.cut-the-knot.org/Curriculum/Geometry/Reflection.shtml">Reflection in Line</a> at <a href="/facts/Cut-the-knot/WLgV43jd">cut-the-knot</a></li>
<li><a href="http://demonstrations.wolfram.com/Understanding2DReflection/">Understanding 2D Reflection</a> and <a href="http://demonstrations.wolfram.com/Understanding3DReflection/">Understanding 3D Reflection</a> by Roger Germundsson, <a href="/facts/The_Wolfram_Demonstrations_Project/l0sh4gln">The Wolfram Demonstrations Project</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Reflexion" is an archaic spelling <a href="https://web.archive.org/web/20120829214317/http://oxforddictionaries.com/definition/english/reflexion" target="_blank">https://web.archive.org/web/20120829214317/http://oxforddictionaries.com/definition/english/reflexion</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p. 251, ISBN 9780387745275 <a href="9780387745275" target="_blank">9780387745275</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32, ISBN 978-1285402734 <a href="978-1285402734" target="_blank">978-1285402734</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 6, ISBN 9780821847992 <a href="9780821847992" target="_blank">9780821847992</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Reflection (mathematics) open-in-new

Reflection (mathematics)