Isotropic line

In the geometry of <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>, an isotropic line or null line is a <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a> for which the quadratic form applied to the <a href="/facts/Displacement_(geometry)/jkmEfS3C">displacement vector</a> between any pair of its points is zero. An isotropic line occurs only with an <a href="/facts/Isotropic_quadratic_form/vMMamxqE">isotropic quadratic form</a>, and never with a <a href="/facts/Definite_quadratic_form/utlBuVpN">definite quadratic form</a>.
Using <a href="/facts/Complex_geometry/ekCsxAXb">complex geometry</a>, <a href="/facts/Edmond_Laguerre/4MUzHfC4">Edmond Laguerre</a> first suggested the existence of two isotropic lines through the point (α, β) that depend on the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a> i:

First system: 
 
 
 
 (
 y
 −
 β
 )
 =
 (
 x
 −
 α
 )
 i
 ,
 
 
 {\displaystyle (y-\beta )=(x-\alpha )i,}

Second system: 
  
    
      
        (
        y
        −
        β
        )
        =
        −
        i
        (
        x
        −
        α
        )
        .
      
    
    {\displaystyle (y-\beta )=-i(x-\alpha ).}

Laguerre then interpreted these lines as <a href="/facts/Geodesic/fbWR6l80">geodesics</a>:

An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> ds2 = 0. On an arbitrary <a href="/facts/Surface_(differential_geometry)/cslVnmd6">surface</a> one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>: 90 
In the <a href="/facts/Complex_projective_plane/5WFpDyeV">complex projective plane</a>, points are represented by <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a> 
 
 
 
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 
 x
 
 3
 
 
 )
 
 
 {\displaystyle (x_{1},x_{2},x_{3})}
 
 and lines by homogeneous coordinates 
 
 
 
 (
 
 a
 
 1
 
 
 ,
 
 a
 
 2
 
 
 ,
 
 a
 
 3
 
 
 )
 
 
 {\displaystyle (a_{1},a_{2},a_{3})}
 
. An isotropic line in the complex projective plane satisfies the equation:

a
          
            3
          
        
        (
        
          x
          
            2
          
        
        ±
        i
        
          x
          
            1
          
        
        )
        =
        (
        
          a
          
            2
          
        
        ±
        i
        
          a
          
            1
          
        
        )
        
          x
          
            2
          
        
        .
      
    
    {\displaystyle a_{3}(x_{2}\pm ix_{1})=(a_{2}\pm ia_{1})x_{2}.}

In terms of the affine subspace x3 = 1, an isotropic line through the origin is

x
          
            2
          
        
        =
        ±
        i
        
          x
          
            1
          
        
        .
      
    
    {\displaystyle x_{2}=\pm ix_{1}.}

In projective geometry, the isotropic lines are the ones passing through the <a href="/facts/Circular_points_at_infinity/BS6eH0Hn">circular points at infinity</a>.
In the real orthogonal geometry of <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a>, isotropic lines occur in pairs:

A non-singular plane which contains an isotropic vector shall be called a <a href="/facts/Isotropic_quadratic_form/vMMamxqE">hyperbolic plane</a>. It can always be spanned by a pair n, m of vectors which satisfy 
 
 
 
 
 
 
 n
 
 
 
 2
 
 
 =
 
 
 
 m
 
 
 
 2
 
 
 =
 0
 ,
 
 
 
 n
 m
 
 
 =
 1.
 
 
 {\displaystyle {\mathbf {n}}^{2}={\mathbf {m}}^{2}=0,\quad {\mathbf {nm}}=1.}

We shall call any such ordered pair n, m a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and n ≠ 0 is an isotropic vector of V, then there exists precisely one m in V such that n, m is a hyperbolic pair. The vectors xn and ym are then the only isotropic vectors of V.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

Isotropic line open-in-new

Isotropic line