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.[1] Distance is preserved along one line through the middle of the band. Assuming the band is given by [math]\displaystyle{ \{z \in \mathbb C: \left|\operatorname {Im} z\right| \lt \pi / 2\} }[/math], the metric is given by [math]\displaystyle{ |dz| \sec (\operatorname{Im} z) }[/math].
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. All geodesics have ends which either are orthogonal to the boundaries of the band or which approach [math]\displaystyle{ \plusmn \infty }[/math].[2] Lines parallel to the boundaries of the band within the band are hypercycles whose centers are the line through the middle of the band.
See also
References
- ↑ Hubbard, John H.. "2". Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Ithaca, NY: Matrix Editions. p. 25. ISBN 9780971576629. OCLC 57965863. http://matrixeditions.com/TVol1.Chap2.pdf.
- ↑ Bowman, Joshua. "612 CLASS LECTURE: HYPERBOLIC GEOMETRY". http://pi.math.cornell.edu/~bowman/metrics.pdf.
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