# Band model

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[[File:Order 7-3 rhombic tiling in the Band Model.png|thumb|The [order 7-3 rhombic tiling](/source/order-7-3_rhombille_tiling) shown in a portion of the '''band model'''.]]
The '''band model''' is a [conformal](/source/Conformal_geometry) [model](/source/Hyperbolic_geometry) of the [hyperbolic plane](/source/hyperbolic_plane). The band model employs a portion of the Euclidean plane between two parallel lines.<ref>{{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}}</ref> Distance is preserved along one line through the middle of the band. Assuming the band is given by <math>\{z \in \mathbb C: \left|\operatorname {Im} z\right| < \pi / 2\}</math>, the metric is given by <math>|dz| \sec (\operatorname{Im} z)</math>.
[[File:Geodesics in the Band Model.png|thumb|[Geodesic](/source/Geodesic)s 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.<ref>{{Cite web|url=https://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}}</ref> Lines parallel to the boundaries of the band within the band are [hypercycles](/source/hypercycle_(geometry)) whose common axis is the line through the middle of the band.

==See also==
* [Mercator projection](/source/Mercator_projection)

==References==
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==External links==
* [http://www.roguetemple.com/z/hyper/models.php Models of hyperbolic geometry]
* [http://bulatov.org/math/1001/ Conformal Models of the Hyperbolic Geometry]

Category:Conformal geometry
Category:Hyperbolic geometry

{{hyperbolic-geometry-stub}}

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