Angle of parallelism

In <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a>, angle of parallelism 
 
 
 
 Π
 (
 a
 )
 
 
 {\displaystyle \Pi (a)}
 
 is the <a href="/facts/Angle/gFUTEQYq">angle</a> at the non-right angle vertex of a right <a href="/facts/Hyperbolic_triangle/hTQldn7s">hyperbolic triangle</a> having two <a href="/facts/Limiting_parallel/0zRSgp7h">asymptotic parallel</a> sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. 
Given a point not on a line, drop a perpendicular to the line from the point. Let a be the length of this perpendicular segment, and 
 
 
 
 Π
 (
 a
 )
 
 
 {\displaystyle \Pi (a)}
 
 be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel,

lim
          
            a
            →
            0
          
        
        Π
        (
        a
        )
        =
        
          
            
              1
              2
            
          
        
        π
        
        
           and 
        
        
        
          lim
          
            a
            →
            ∞
          
        
        Π
        (
        a
        )
        =
        0.
      
    
    {\displaystyle \lim _{a\to 0}\Pi (a)={\tfrac {1}{2}}\pi \quad {\text{ and }}\quad \lim _{a\to \infty }\Pi (a)=0.}

There are five equivalent expressions that relate 
 
 
 
 Π
 (
 a
 )
 
 
 {\displaystyle \Pi (a)}
 
 and a:

sin
        ⁡
        Π
        (
        a
        )
        =
        sech
        ⁡
        a
        =
        
          
            1
            
              cosh
              ⁡
              a
            
          
        
        =
        
          
            2
            
              
                e
                
                  a
                
              
              +
              
                e
                
                  −
                  a
                
              
            
          
        
         
        ,
      
    
    {\displaystyle \sin \Pi (a)=\operatorname {sech} a={\frac {1}{\cosh a}}={\frac {2}{e^{a}+e^{-a}}}\ ,}

cos
        ⁡
        Π
        (
        a
        )
        =
        tanh
        ⁡
        a
        =
        
          
            
              
                e
                
                  a
                
              
              −
              
                e
                
                  −
                  a
                
              
            
            
              
                e
                
                  a
                
              
              +
              
                e
                
                  −
                  a
                
              
            
          
        
         
        ,
      
    
    {\displaystyle \cos \Pi (a)=\tanh a={\frac {e^{a}-e^{-a}}{e^{a}+e^{-a}}}\ ,}

tan
        ⁡
        Π
        (
        a
        )
        =
        csch
        ⁡
        a
        =
        
          
            1
            
              sinh
              ⁡
              a
            
          
        
        =
        
          
            2
            
              
                e
                
                  a
                
              
              −
              
                e
                
                  −
                  a
                
              
            
          
        
         
        ,
      
    
    {\displaystyle \tan \Pi (a)=\operatorname {csch} a={\frac {1}{\sinh a}}={\frac {2}{e^{a}-e^{-a}}}\ ,}

tan
        ⁡
        
          (
          
            
              
                
                  1
                  2
                
              
            
            Π
            (
            a
            )
          
          )
        
        =
        
          e
          
            −
            a
          
        
        ,
      
    
    {\displaystyle \tan \left({\tfrac {1}{2}}\Pi (a)\right)=e^{-a},}

Π
        (
        a
        )
        =
        
          
            
              1
              2
            
          
        
        π
        −
        gd
        ⁡
        (
        a
        )
        ,
      
    
    {\displaystyle \Pi (a)={\tfrac {1}{2}}\pi -\operatorname {gd} (a),}

where sinh, cosh, tanh, sech and csch are <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic functions</a> and gd is the <a href="/facts/Gudermannian_function/bA0B2KcC">Gudermannian function</a>.

Angle of parallelism open-in-new

Angle of parallelism