Band model

<p>The band model is a <a href="/facts/Conformal_geometry/tuU181cE">conformal</a> <a href="/facts/Hyperbolic_geometry/DjTMKdgy">model</a> of the <a href="/facts/Hyperbolic_plane/DjTMKdgy">hyperbolic plane</a>. The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by 
  
    
      
        {
        z
        ∈
        
          C
        
        :
        
          |
          
            Im
            ⁡
            z
          
          |
        
        <
        π
        
          /
        
        2
        }
      
    
    {\displaystyle \{z\in \mathbb {C} :\left|\operatorname {Im} z\right|<\pi /2\}}
  
, the metric is given by 
  
    
      
        
          |
        
        d
        z
        
          |
        
        sec
        ⁡
        (
        Im
        ⁡
        z
        )
      
    
    {\displaystyle |dz|\sec(\operatorname {Im} z)}
  
.
</p>

<p>Geodesics include the line along the middle of the band, and any open line segment perpendicular to boundaries of the band connecting the sides of the band. Every end of a geodesic either meets a boundary of the band at a right angle or is asymptotic to the midline; the midline itself is the only geodesic that does not meet a boundary. Lines parallel to the boundaries of the band within the band are <a href="/facts/Hypercycle_(geometry)/5w8YYW78">hypercycles</a> whose common axis is the line through the middle of the band.
</p>

Band model open-in-new

Band model