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Poincaré disk model

Model of hyperbolic geometry / From Wikipedia, the free encyclopedia

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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 diameters of the unit circle.

Poincaré disk with hyperbolic parallel lines
Poincaré disk model of the truncated triheptagonal tiling.

The group of orientation preserving isometries of the disk model is given by the projective special unitary group PSU(1,1), the quotient of the special unitary group SU(1,1) by its center {I, −I}.

Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami.[1]

The Poincaré ball model is the similar model for 3 or n-dimensional hyperbolic geometry in which the points of the geometry are in the n-dimensional unit ball.

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