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 ${\displaystyle \{z\in \mathbb {C} :\left|\operatorname {Im} z\right|<\pi /2\}}${\displaystyle \{z\in \mathbb {C} :\left|\operatorname {Im} z\right|<\pi /2\}}, the metric is given by ${\displaystyle |dz|\sec(\operatorname {Im} z)}${\displaystyle |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.

• Mercator projection

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

• Conformal geometry
• Hyperbolic geometry
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