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Poincaré disk model
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models of hyperbolic geometry
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Poincaré disk model

geometry
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models of hyperbolic geometry
models of hyperbolic geometry In the Klein-Beltrami model for the hyperbolic plane, the shortest paths, or geodesics, are chords (several examples, labeled k, l, m, n, are shown). In the Poincaré disk model, geodesics are portions of circles that intersect the boundary of the disk at right angles; and in the Poincaré upper half-plane model, geodesics are semicircles with their centres on the boundary.

The Poincaré disk model is a way to visualize non-Euclidean geometry, specifically hyperbolic geometry. In this model, the "space" is represented by the interior of a circle.

Key Features of the Poincaré Disk Model:

  • Lines: Straight lines in hyperbolic geometry are represented as circular arcs within the disk that meet the boundary of the disk at right angles. Diameters of the disk are also considered straight lines.
  • Angles: Angles are measured as they appear within the disk, preserving the angle measure of hyperbolic geometry.
  • Distances: Distances are distorted compared to Euclidean geometry, but the model accurately represents the relationships and properties of hyperbolic space.

This model, developed by Henri Poincaré, is one of several ways to represent hyperbolic geometry, which differs from Euclidean geometry primarily in its treatment of parallel lines and the sum of angles in a triangle. In hyperbolic geometry, through a point not on a given line, there are infinitely many lines parallel to the given line, and the sum of interior angles of a triangle is less than 180 degrees.1

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2 Britannica Sources
  1. non-Euclidean geometry
  2. projective geometry

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