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Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...
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 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.
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
Hyperbolic 3-manifold; Hyperbolic coordinates; Hyperbolic Dehn surgery; Hyperbolic functions; Hyperbolic group; Hyperbolic law of cosines; Hyperbolic manifold; Hyperbolic metric space; Hyperbolic motion; Hyperbolic space; Hyperbolic tree; Hyperbolic volume; Hyperbolization theorem; Hyperboloid model; Hypercycle (geometry) HyperRogue
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.
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary ), cubic equations , and Laplace's equation in Cartesian coordinates .
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).