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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 with either are orthogonal to the boundaries of the band or which approach . Lines parallel to the boundaries of the band within the band are hypercycles whose centers are the line through the middle of the band.
Models of hyperbolic geometry
Conformal Models of the Hyperbolic Geometry
Conformal geometry Hyperbolic geometry {{geometry-stub
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the band model is a conformal model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
of the hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
. The band model employs a portion of the Euclidean plane between two parallel lines. Distance is preserved along one line through the middle of the band. Assuming the band is given by , the metric is given by .
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 with either are orthogonal to the boundaries of the band or which approach . 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
The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and ...
References
External links
Models of hyperbolic geometry
Conformal Models of the Hyperbolic Geometry
Conformal geometry Hyperbolic geometry {{geometry-stub