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v. t. e. A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), along with two diverging ultra-parallel lines. 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:
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.
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 ...
A hyperbolic angle has magnitude equal to the area of the corresponding hyperbolic sector, which is in standard position if a = 1. In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola ...
A hyperbola is a set of points, such that for any point of the set, the absolute difference of the distances to two fixed points (the foci) is constant, usually denoted by : 5. The midpoint of the line segment joining the foci is called the center of the hyperbola. 6 The line through the foci is called the major axis.
In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane. . Hyperbolic coordinates take values in the hyperbolic plane defined as: . These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For in take. and.
Truncated trioctagonal tiling. 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 ...
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with coordinates whose coordinate is greater than zero, the upper half-plane, and a metric ...
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