Hyperbolic geometry

mathematical function related with trigonometric functions

type of non-Euclidean geometry

In geometry, a pseudosphere is a surface in \mathbb{R}^3. It is the most famous example of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in \mathbb{R}^3 with constant negative Gaussian curvature. A "pseudospherical surface of radius " is a surface in \mathbb{R}^3 having curvature −1/R2 at each point. Its name comes from the analogy with the sphere of radius , which is a surface of curvature 1/R2. Examples include the tractroid, Dini's surfaces, breather surfaces, and the Kuen surface.

limiting point in some geometric spaces

triangle in the hyperbolic plane, possibly having ideal vertices

maximally symmetric Lorentzian manifold with negative cosmological constant, which appears in the AdS/CFT correspondence

model of hyperbolic geometry

fractal generated from three mutually tangent circles by repeatedly placing a tangent circle into the gap between three circles

homogeneous space that has a constant negative curvature (not any hyperbolic manifold)

quadrilateral with two equal sides perpendicular to the base

failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would

complex numbers with positive imaginary part

upper-half plane model of hyperbolic non-Euclidean geometry

discrete subgroup of the real projective special linear group of dimension 2

model of hyperbolic geometry

curve in hyperbolic plane whose points have the same orthogonal distance from a given straight line

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: \mbox{SL}(2,\mathbf{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \colon a,b,c,d \in \mathbf{R}\mbox{ and }ad-bc=1\right\}.

quadrilateral with three right angles

Mathematical concept

220px|right|thumb| A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central [[ideal point.]]

in hyperbolic geometry, the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides

model of n-dimensional hyperbolic geometry

Group realized geometrically by reflections across the sides of a triangle

coordinate system

hyperbolic triangle such that all three of its vertices are ideal points

point at infinity in hyperbolic geometry

{|class="wikitable" align="right" style="text-align:center" |+Split-quaternion multiplication |- !width=15| × !width=15| 1 !width=15| i !width=15| j !width=15| k |- ! 1 | 1 | i | j | k |- !i |i |−1 |k |−j |- !j |j |−k |1 |−i |- !k |k |j |i |1 |}

metric tensor describing constant negative (hyperbolic) curvature

mathematical tree in the hyperbolic plane

220px|right|thumb|A horosphere within the Poincaré disk model tangent to the edges of a [[hexagonal tiling cell of a hexagonal tiling honeycomb]] thumb|Apollonian sphere packing can be seen as showing horospheres that are tangent to an outer sphere of a [[Poincaré disk model]] In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic n-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of tangency. For n = 2 a horosphere is called a horocycle.

Concept in mathematics