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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:
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. Laplace's equations are important in many areas of physics, including ...
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 ...
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.
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.
For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. [8] Many textbooks erroneously state that a flat or hyperbolic universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe ...
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
Paraboloid. In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola.