Traducción y análisis de palabras por inteligencia artificial
En esta página puede obtener un análisis detallado de una palabra o frase, producido utilizando la mejor tecnología de inteligencia artificial hasta la fecha:
- cómo se usa la palabra
- frecuencia de uso
- se utiliza con más frecuencia en el habla oral o escrita
- opciones de traducción
- ejemplos de uso (varias frases con traducción)
- etimología
Qué (quién) es hyperbolic surface - definición
![A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the [[Institute For Figuring]]](https://commons.wikimedia.org/wiki/Special:FilePath/Crochet hyperbolic kelp.jpg?width=200 "A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the [[Institute For Figuring]]")
![An [[apeirogon]] and circumscribed [[horocycle]] in the [[Poincare disk model]]](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolic apeirogon example.png?width=200 "An [[apeirogon]] and circumscribed [[horocycle]] in the [[Poincare disk model]]")
![Hypercycle and pseudogon in the [[Poincare disk model]]](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolic pseudogon example0.png?width=200 "Hypercycle and pseudogon in the [[Poincare disk model]]")
![Poincaré disk model with [[truncated triheptagonal tiling]]](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolic tiling omnitruncated 3-7.png?width=200 "Poincaré disk model with [[truncated triheptagonal tiling]]")
![A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), along with two diverging ultra-parallel lines](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolic triangle.svg?width=200 "A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), along with two diverging ultra-parallel lines")
![The "hyperbolic soccerball" is a paper model which approximates (part of) the hyperbolic plane as a [[truncated icosahedron]] approximates the sphere.](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolicsoccerball.jpg?width=200 "The \"hyperbolic soccerball\" is a paper model which approximates (part of) the hyperbolic plane as a [[truncated icosahedron]] approximates the sphere.")
![Poincaré disk, hemispherical and hyperboloid models are related by [[stereographic projection]] from −1. [[Beltrami–Klein model]] is [[orthographic projection]] from hemispherical model. [[Poincaré half-plane model]] here projected from the hemispherical model by rays from left end of Poincaré disk model.](https://commons.wikimedia.org/wiki/Special:FilePath/Relation5models.png?width=200 "Poincaré disk, hemispherical and hyperboloid models are related by [[stereographic projection]] from −1. [[Beltrami–Klein model]] is [[orthographic projection]] from hemispherical model. [[Poincaré half-plane model]] here projected from the hemispherical model by rays from left end of Poincaré disk model.")
![[[Rhombitriheptagonal tiling]] of the hyperbolic plane, seen in the [[Poincaré disk model]]](https://commons.wikimedia.org/wiki/Special:FilePath/Rhombitriheptagonal tiling.svg?width=200 "[[Rhombitriheptagonal tiling]] of the hyperbolic plane, seen in the [[Poincaré disk model]]")
![gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.](https://commons.wikimedia.org/wiki/Special:FilePath/Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola.png?width=200 "gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.")
![E<sup>3</sup>]]''](https://commons.wikimedia.org/wiki/Special:FilePath/Hyperbolic orthogonal dodecahedral honeycomb.png?width=200 "E<sup>3</sup>]]''")
Wikipedia
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:
- For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.
(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)
The hyperbolic plane is a plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.
A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.
When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry. In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.
This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions.
