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  2. Hyperbolic space - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_space

    Hyperbolic 2-space, H 2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry.

  3. Hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_geometry

    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 ...

  4. Hyperbolic metric space - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_metric_space

    In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov , generalizes the metric properties of classical hyperbolic geometry and of trees .

  5. Non-Euclidean geometry - Wikipedia

    en.wikipedia.org/wiki/Non-Euclidean_geometry

    In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.

  6. Hyperboloid model - Wikipedia

    en.wikipedia.org/wiki/Hyperboloid_model

    In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...

  7. Constructions in hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Constructions_in...

    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]

  8. Gromov boundary - Wikipedia

    en.wikipedia.org/wiki/Gromov_boundary

    The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov boundary. In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space.

  9. Hyperbolic triangle - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_triangle

    In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices . Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane.

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