page_title: Beltrami–Klein model
section_title: Relation to the Poincaré disk model
hierarchical_section_title: Beltrami–Klein model / The Klein disk model / Relation to the Poincaré disk model
caption_attribution_description: Français&#160;: En géométrie hyperbolique, on passe du modèle de Klein,(où les droites hyperboliques sont représentées par des segments) au modèle de Poincaré (où les droites hyperboliques sont représentées par des arcs de cercles) de la façon suivante&#160;: projeter un point du modèle de Klein sur la demi-sphère dont le modèle de Klein est le disque équatorial, puis procéder à une projection stéréographique depuis le pôle sud pour ramener le point dans le plan équatorial. Le schéma illustre cette transformation géométrique
context_page_description: 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 disk and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
  The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley.
  The Beltrami–Klein model is analogous to the gnomonic projection of spherical geometry, in that geodesics are mapped to straight lines.
  This model is not conformal, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these.
  In this model, lines and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally.
context_section_description: Both the Poincaré disk model and the Klein disk model are models of the hyperbolic plane. An advantage of the Poincaré disk model is that it is conformal (circles and angles are not distorted); a disadvantage is that lines of the geometry are circular arcs orthogonal to the boundary circle of the disk.
  The two models are related through a projection on or from the hemisphere model. The Klein model is an orthographic projection to the hemisphere model while the Poincaré disk model is a stereographic projection.
  When projecting the same lines in both models on one disk both lines go through the same two ideal points. (the ideal points remain on the same spot) also the pole of the chord is the centre of the circle that contains the arc.
  If P is a point a distance  from the centre of the unit circle in the Beltrami–Klein model, then the corresponding point on the Poincaré disk model a distance of u on the same radius:
  Conversely, If P is a point a distance  from the centre of the unit circle in the  Poincaré disk model, then the corresponding point of the Beltrami–Klein model is a distance of s on the same radius:
caption: Combined projections from the Klein disk model (yellow) to the Poincaré disk model (red) via the hemisphere model (blue)
exif: {}