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.^[1] 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.^[2] 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.

References

^ Hubbard, John H. "2" (PDF). Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Ithaca, NY: Matrix Editions. p. 25. ISBN 9780971576629. OCLC 57965863.
^ Bowman, Joshua. "612 CLASS LECTURE: HYPERBOLIC GEOMETRY" (PDF). Retrieved August 12, 2018.

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[1] Hubbard, John H. "2" (PDF). Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Ithaca, NY: Matrix Editions. p. 25. ISBN 9780971576629. OCLC 57965863.

[2] Bowman, Joshua. "612 CLASS LECTURE: HYPERBOLIC GEOMETRY" (PDF). Retrieved August 12, 2018.

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See also

References

External links