Hyperbolic space (original) (raw)
En matemàtiques, l'espai hiperbòlic és un espai, introduït al segle xix pels matemàtics János Bolyai i Nikolai Ivànovitx Lobatxevski de manera independent, que es defineix en una geometria no euclidiana anomenada geometria hiperbòlica. Es tracta, juntament amb la geometria el·líptica, de l'exemple més important de la geometria no euclidiana.
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| dbo:abstract | En matemàtiques, l'espai hiperbòlic és un espai, introduït al segle xix pels matemàtics János Bolyai i Nikolai Ivànovitx Lobatxevski de manera independent, que es defineix en una geometria no euclidiana anomenada geometria hiperbòlica. Es tracta, juntament amb la geometria el·líptica, de l'exemple més important de la geometria no euclidiana. (ca) In der Geometrie ist der hyperbolische Raum ein Raum mit konstanter negativer Krümmung. Er erfüllt die Axiome der euklidischen Geometrie mit Ausnahme des Parallelenaxioms. Der zweidimensionale hyperbolische Raum mit konstanter Krümmung heißt hyperbolische Ebene. (de) En matematiko, hiperbola n-spaco, s Hn, estas la maksimume simetria, simple koneksa, n-dimensia rimana sternaĵo kun konstanta -1. Ĝi estas la negative kurbeca analogo de la n-sfero. Kvankam hiperbola spaco Hn estas difeomorfa al eŭklida spaco Rn ĝia negativa kurbeca metriko donas ĝi tre malsamajn geometriajn propraĵojn. Hiperbola 2-spaco, H2 estas . Hiperbola spaco estas la ĉefa speco de spaco en hiperbola geometrio. (eo) In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai--Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, and the which are the other symmetric spaces of negative curvature. Hyperbolic space serves as the prototype of a Gromov hyperbolic space which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a CAT(-1) space. (en) En matemáticas, un espacio hiperbólico es un con negativa, donde la curvatura se refiere a la . Es geometría hiperbólica en más de 2 dimensiones, y se distingue de los espacios euclídeos con curvatura cero, que definen la geometría euclídea, y de la geometría elíptica, que tiene curvatura constante positiva. Al embeberse en un espacio euclídeo (de mayor dimensión), todo punto de un espacio hiperbólico es un punto de silla. Otra propiedad importante es la cubierta por la n-bola en el n-espacio hiperbólico, que aumenta exponencialmente con respecto al radio de la bola para radios grandes, en lugar de polinómicamente. (es) In matematica, lo spazio iperbolico è uno spazio introdotto indipendentemente dai matematici Bolyai e Lobachevsky nel XIX secolo, su cui è definita una particolare geometria non euclidea, detta geometria iperbolica. Si tratta dell'esempio più importante di geometria non euclidea, assieme alla geometria ellittica. Lo spazio iperbolico ha dimensione arbitraria ed è indicato con . Può essere realizzato tramite vari modelli equivalenti, quali ad esempio il disco, il semispazio di Poincaré o il . Come nella geometria euclidea, gli spazi più studiati sono il piano iperbolico e lo spazio iperbolico tridimensionale . (it) 쌍곡 기하학에서 쌍곡공간(雙曲空間, 영어: hyperbolic space)은 균일한 음의 곡률을 갖는 동차공간이다. (ko) In de meetkunde, een deelgebied van de wiskunde, is een hyperbolische ruimte een soort van niet-euclidische ruimte. Overwegende dat de bolmeetkunde een constante positieve kromming heeft, kent de hyperbolische meetkunde een negatieve kromming: elk punt in de hyperbolische ruimte is een zadelpunt. Evenwijdige lijnen zijn in de hyperbolische ruimte niet op unieke wijze gekoppeld: gegeven een lijn en een punt dat niet op die lijn ligt, kan er een oneindige aantal lijnen worden getekend die door dit punt gaan, die met de eerste in dit vlak liggen en het niet snijden. Dit contrasteert met zowel de Euclidische meetkunde, waar evenwijdige lijnen een uniek paar vormen, als de bolmeetkunde, waar evenwijdige lijnen niet bestaan, omdat alle lijnen, die in de bolmeetkunde grootcirkels zijn, elkaar kruisen. Een andere kenmerkende eigenschap is de hoeveelheid ruimte die door een in de hyperbolische n-ruimte wordt afgedekt - deze neemt in relatie tot de straal van de bal exponentieel in plaats van polynomiaal toe. (nl) Пространство Лобачевского, или гиперболическое пространство — пространство с постоянной отрицательной кривизной. Двумерным пространством Лобачевского является плоскость Лобачевского. Отрицательная кривизна отличает пространство Лобачевского от евклидова пространства с нулевой кривизной, описываемого евклидовой геометрией, и от сферы — пространства с постоянной положительной кривизной, описываемого геометрией Римана. n-мерное пространство Лобачевского обычно обозначается или . (ru) Em matemática, um espaço hiperbólico é um que possui uma curvatura negativa constante, onde neste caso a curvatura é a curvatura seccional. É uma geometria hiperbólica em mais de 2 dimensões e distingue-se dos espaços euclidianos com curvatura zero que definem a geometria euclidiana e a geometria elíptica que possui uma curvatura positiva constante. Quando incorporado a um espaço euclidiano (de maior dimensão), todo ponto de um espaço hiperbólico é um ponto de sela. Outra propriedade distintiva é a coberto pela bola n no espaço n hiperbólico: aumenta exponencialmente em relação ao raio da bola para raios grandes, em vez de polinomialmente. (pt) Простір Лобачевського або гіперболічний простір, — це простір із постійною негативною кривиною. Двовимірним простором Лобачевського є площина Лобачевського. Від'ємна кривина відрізняє простір Лобачевського від евклідового простору з нульовою кривиною, описуваного евклідовою геометрією, і від сфери — простору з постійною додатною кривиною, описуваного геометрією Рімана. n-вимірний простір Лобачевського зазвичай позначають або . (uk) |
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| rdfs:comment | En matemàtiques, l'espai hiperbòlic és un espai, introduït al segle xix pels matemàtics János Bolyai i Nikolai Ivànovitx Lobatxevski de manera independent, que es defineix en una geometria no euclidiana anomenada geometria hiperbòlica. Es tracta, juntament amb la geometria el·líptica, de l'exemple més important de la geometria no euclidiana. (ca) In der Geometrie ist der hyperbolische Raum ein Raum mit konstanter negativer Krümmung. Er erfüllt die Axiome der euklidischen Geometrie mit Ausnahme des Parallelenaxioms. Der zweidimensionale hyperbolische Raum mit konstanter Krümmung heißt hyperbolische Ebene. (de) En matematiko, hiperbola n-spaco, s Hn, estas la maksimume simetria, simple koneksa, n-dimensia rimana sternaĵo kun konstanta -1. Ĝi estas la negative kurbeca analogo de la n-sfero. Kvankam hiperbola spaco Hn estas difeomorfa al eŭklida spaco Rn ĝia negativa kurbeca metriko donas ĝi tre malsamajn geometriajn propraĵojn. Hiperbola 2-spaco, H2 estas . Hiperbola spaco estas la ĉefa speco de spaco en hiperbola geometrio. (eo) 쌍곡 기하학에서 쌍곡공간(雙曲空間, 영어: hyperbolic space)은 균일한 음의 곡률을 갖는 동차공간이다. (ko) Пространство Лобачевского, или гиперболическое пространство — пространство с постоянной отрицательной кривизной. Двумерным пространством Лобачевского является плоскость Лобачевского. Отрицательная кривизна отличает пространство Лобачевского от евклидова пространства с нулевой кривизной, описываемого евклидовой геометрией, и от сферы — пространства с постоянной положительной кривизной, описываемого геометрией Римана. n-мерное пространство Лобачевского обычно обозначается или . (ru) Простір Лобачевського або гіперболічний простір, — це простір із постійною негативною кривиною. Двовимірним простором Лобачевського є площина Лобачевського. Від'ємна кривина відрізняє простір Лобачевського від евклідового простору з нульовою кривиною, описуваного евклідовою геометрією, і від сфери — простору з постійною додатною кривиною, описуваного геометрією Рімана. n-вимірний простір Лобачевського зазвичай позначають або . (uk) In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. (en) En matemáticas, un espacio hiperbólico es un con negativa, donde la curvatura se refiere a la . Es geometría hiperbólica en más de 2 dimensiones, y se distingue de los espacios euclídeos con curvatura cero, que definen la geometría euclídea, y de la geometría elíptica, que tiene curvatura constante positiva. (es) In matematica, lo spazio iperbolico è uno spazio introdotto indipendentemente dai matematici Bolyai e Lobachevsky nel XIX secolo, su cui è definita una particolare geometria non euclidea, detta geometria iperbolica. Si tratta dell'esempio più importante di geometria non euclidea, assieme alla geometria ellittica. (it) In de meetkunde, een deelgebied van de wiskunde, is een hyperbolische ruimte een soort van niet-euclidische ruimte. Overwegende dat de bolmeetkunde een constante positieve kromming heeft, kent de hyperbolische meetkunde een negatieve kromming: elk punt in de hyperbolische ruimte is een zadelpunt. Evenwijdige lijnen zijn in de hyperbolische ruimte niet op unieke wijze gekoppeld: gegeven een lijn en een punt dat niet op die lijn ligt, kan er een oneindige aantal lijnen worden getekend die door dit punt gaan, die met de eerste in dit vlak liggen en het niet snijden. Dit contrasteert met zowel de Euclidische meetkunde, waar evenwijdige lijnen een uniek paar vormen, als de bolmeetkunde, waar evenwijdige lijnen niet bestaan, omdat alle lijnen, die in de bolmeetkunde grootcirkels zijn, elkaar kru (nl) Em matemática, um espaço hiperbólico é um que possui uma curvatura negativa constante, onde neste caso a curvatura é a curvatura seccional. É uma geometria hiperbólica em mais de 2 dimensões e distingue-se dos espaços euclidianos com curvatura zero que definem a geometria euclidiana e a geometria elíptica que possui uma curvatura positiva constante. (pt) |
| rdfs:label | Espai hiperbòlic (ca) Hyperbolischer Raum (de) Hiperbola spaco (eo) Espacio hiperbólico (es) Spazio iperbolico (it) Hyperbolic space (en) 쌍곡공간 (ko) Hyperbolische ruimte (nl) Espaço hiperbólico (pt) Пространство Лобачевского (ru) Простір Лобачевського (uk) |
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