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 $$\{z \in \mathbb C: \left|\operatorname {Im} z\right| < \pi / 2\}$$, the metric is given by $$|dz| \sec (\operatorname{Im} z)$$. 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.