Band model

In geometry, the band model is a conformal model of the hyperbolic plane. 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 ${\displaystyle \{z\in \mathbb {C}$ :\left|\operatorname {Im} z\right|<\pi /2\}} , the metric is given by ${\displaystyle |dz|\sec(\operatorname {Im} z)}$.

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 hypercycles whose centers are the line through the middle of the band.