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In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.

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  • Poincaré half-plane model
  • Demi-plan de Poincaré
  • Semispazio di Poincaré
  • ポアンカレの上半平面モデル
  • Model Poincarégo
  • 庞加莱半平面模型
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  • Le demi-plan de Poincaré est un sous-ensemble des nombres complexes. Il a permis au mathématicien français Henri Poincaré d'éclairer les travaux du Russe Nikolaï Lobatchevski.
  • Il semispazio di Poincaré è un modello di geometria iperbolica, descritto dal matematico francese Jules Henri Poincaré. Un altro modello con caratteristiche simili è il disco di Poincaré.
  • 非ユークリッド幾何学におけるポアンカレ半平面模型(はんへいめんもけい、英: Poincaré half-plane model)は、上半平面(以下 H と記す)にポアンカレ計量と呼ばれる計量をあわせて考えたもので、二次元双曲幾何学のモデルを形成する。 名称はアンリ・ポアンカレに因むものだが、そもそもはベルトラミが、クライン模型・(リーマンによる)ポアンカレ円板模型とともに、双曲幾何学がユークリッド幾何学に無矛盾等価であることを示すために用いたものである。円板模型と半平面模型とは共形写像のもとで同型である。
  • 在非欧几里得几何中,庞加莱半平面模型(Poincaré half-plane model)是赋有庞加莱度量的上半平面,这是二维双曲几何的一个模型。 它以昂利·庞加莱命名,但最初是贝尔特拉米(Eugenio Beltrami)发现的,他用这个模型与克莱因模型以及庞加莱圆盘模型(属于黎曼)证明了双曲几何与欧几里得几何的相容性等价(equiconsistent)。圆盘模型与半平面模型在共形映射下是等价的。
  • In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane.
  • Model Poincarégo – jeden z modeli planimetrii hiperbolicznej odkryty przez uczonego francuskiego Henriego Poincarégo w 1882 roku. Na płaszczyźnie euklidesowej ustalona jest prosta nazywana absolutem. Punktami płaszczyzny hiperbolicznej są punkty leżące po jednej stronie absolutu, czyli płaszczyzną hiperboliczną jest półpłaszczyzna otwarta (tj. bez punktów ograniczającej ją prostej) wyznaczona przez absolut. Punkty tej półpłaszczyzny nazywamy punktami skończonymi płaszczyzny hiperbolicznej. Prostymi w tym modelu są:
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  • In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincaré half-plane model is sometimes described as a complex plane where the imaginary part (the y coordinate mentioned above) is positive. The Poincaré half-plane model is named after Henri Poincaré, but it originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann), to show that hyperbolic geometry was equiconsistent with Euclidean geometry. This model is conformal which means that the angles measured at a point are the same in the model as they are in the actual hyperbolic plane. The Cayley transform provides an isometry between the half-plane model and the Poincaré disk model. This model can be generalized to model an n+1 dimensional hyperbolic space by replacing the real number x by a vector in an n dimensional Euclidean vector space.
  • Le demi-plan de Poincaré est un sous-ensemble des nombres complexes. Il a permis au mathématicien français Henri Poincaré d'éclairer les travaux du Russe Nikolaï Lobatchevski.
  • Il semispazio di Poincaré è un modello di geometria iperbolica, descritto dal matematico francese Jules Henri Poincaré. Un altro modello con caratteristiche simili è il disco di Poincaré.
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