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Hyperbolic geometry - Wikipedia
There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. See more
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 … See moreThough hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, it is usual to assume a scale in which the curvature K is −1.
This results in some … See moreThere exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.
By See moreEvery isometry (transformation or motion) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up … See more
Relation to Euclidean geometry
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. … See moreSince the publication of Euclid's Elements circa 300BC, many geometers tried to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a … See more
Various pseudospheres – surfaces with constant negative Gaussian curvature – can be embedded in 3-D space under the standard Euclidean … See more
Wikipedia text under CC-BY-SA license Poincaré half-plane model - Wikipedia
Hyperboloid model - Wikipedia
The Hyperbolic Plane - SpringerLink
7. The Poincaré disk model for the hyperbolic plane
The Poincaré disk model for the hyperbolic plane. The second model that we use to represent the hyperbolic plane is called the Poincaré disk model, named after the great French mathematician, Henri Poincaré (1854 - 1912). This …
5.5: The Upper Half-Plane Model - Mathematics LibreTexts
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