In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to −1. [1] It is homogeneous, and satisfies the stronger property of being a symmetric space.

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:

2025年1月31日 · The best-known example of a hyperbolic space are spheres in Lorentzian four-space. The Poincaré hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood in two dimensions, but not in three dimensions.

Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one

The hyperbolic space $\mathbb H^3$ is the three-dimensional analog of the hyperbolic plane. It is an isotropic space (all the directions play the same role). Among the eight geometries, this is probably the one which has the richest class of lattices.

structures such as frameworks and cabled tensegrities in the hyperbolic plane and hyperbolic space on the one hand and the Euclidean plane, the sphere and Euclidean space on the other hand. MSC: 52C25, 52C35, 52Cxx, 52.50 Keywords: Hyperbolic plane, configurations, framework, tensegrity 1. Introduction

Non-Euclidean, or hyperbolic, geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. Einstein and Minkowski found in non-Euclidean geometry a geometric basis …

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points.

2021年10月13日 · The hyperbolic group, denoted \(\mathbf{H}\), is the subgroup of the Möbius group \(\mathbf{M}\) of transformations that map \(\mathbb{D}\) onto itself. The pair \((\mathbb{D},\mathbf{H})\) is the (Poincaré) disk model of hyperbolic geometry.

There are a variety of ways to visualize the hyperbolic plane. One example is the Poincar é disk model in which hyperbolic-space straight lines are rendered as arcs of circles orthogonal to the boundary: