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Klein-Beltrami model
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models of hyperbolic geometry
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Klein-Beltrami 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 Klein-Beltrami model is a way to visualize hyperbolic geometry, a type of non-Euclidean geometry that differs from Euclidean geometry. In hyperbolic geometry, unlike Euclidean geometry, there are infinitely many lines that can be drawn through a point parallel to a given line. Also, the sum of the interior angles of a triangle is less than 180 degrees.

The Klein-Beltrami model represents hyperbolic space as the interior of a circle. In this model, straight lines in hyperbolic geometry are represented as chords within the circle. This model preserves the concept of "straightness" but distorts angles. It was developed by Eugenio Beltrami and Felix Klein in the late 19th century.1

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3 Britannica Sources
  1. non-Euclidean geometry
  2. Klein bottle
  3. hyperbolic geometry

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