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Concept
Poincaré disk model
Summary
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.
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.
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.
The disk model was first described by Bernhard Riemann in an 1854 lecture (published 1868), which inspired an 1868 paper by Eugenio Beltrami. Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions, but it became widely known following Poincaré's presentation in his 1905 philosophical treatise, Science and Hypothesis. There he describes a world, now known as the Poincaré disk, in which space was Euclidean, but which appeared to its inhabitants to satisfy the axioms of hyperbolic geometry:"Suppose, for example, a world enclosed in a large sphere and subject to the following laws: The temperature is not uniform; it is greatest at their centre, and gradually decreases as we move towards the circumference of the sphere, where it is absolute zero. The law of this temperature is as follows: If be the radius of the sphere, and the distance of the point considered from the centre, the absolute temperature will be proportional to .
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