Band model

File:Order 7-3 rhombic tiling in the Band Model.png shown in a portion of the band model.]]

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.{{cite book|url=http://matrixeditions.com/TeichmullerVol1.html|title=Teichmüller Theory and Applications to Geometry, Topology, and Dynamics|date=|publisher=Matrix Editions|last=Hubbard|first=John H.|year=2006|authorlink=John H. Hubbard|isbn=9780971576629|location=Ithaca, NY|oclc=57965863|chapter=2|chapter-url=http://matrixeditions.com/TVol1.Chap2.pdf|page=25}} Distance is preserved along one line through the middle of the band. Assuming the band is given by , the metric is given by $|dz|\; \backslash sec\; (\backslash operatorname\{Im\}\; z)$.

File:Geodesics in the Band Model.pngs shown in a portion of the band model.]]

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.{{Cite web|url=http://pi.math.cornell.edu/~bowman/metrics.pdf|title=612 CLASS LECTURE: HYPERBOLIC GEOMETRY|last=Bowman|first=Joshua|date=|website=|archive-url=|archive-date=|access-date=August 12, 2018}} Lines parallel to the boundaries of the band within the band are hypercycles whose common axis is the line through the middle of the band.