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Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature? Lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of Sirius and treating Sirius as the ideal point of an ...
Sometimes the qualificative "real" is added to distinguish it from complex hyperbolic spaces. Hyperbolic space serves as the prototype of a Gromov hyperbolic space , which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinate systems in use for the two-dimensional hyperbolic plane.
Many hyperbolic lines through point P not intersecting line a in the Beltrami Klein model A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection. In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit ...
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2).
Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic ...
In mathematics, relatively hyperbolic groups form an important class of groups of interest for geometric group theory.The main purpose in their study is to extend the theory of Gromov-hyperbolic groups to groups that may be regarded as hyperbolic assemblies of subgroups , called peripheral subgroups, in a way that enables "hyperbolic reduction" of problems for to problems for the s.
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