Band model explained

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

\{z \in 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 centers are the line through the middle of the band.

See also

• Mercator projection

External links

• Models of hyperbolic geometry
• Conformal Models of the Hyperbolic Geometry

Notes and References

1. Book: Hubbard, John H.. John H. Hubbard

   . Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Matrix Editions. John H. Hubbard. 9780971576629. Ithaca, NY. 57965863. 2. http://matrixeditions.com/TVol1.Chap2.pdf. 25.

2. Web site: 612 CLASS LECTURE: HYPERBOLIC GEOMETRY. Bowman. Joshua. August 12, 2018.