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Concept
Beltrami–Klein model
Summary
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 (or n-dimensional unit ball) 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 (great circles in spherical geometry) 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.
Cayley–Klein metric
This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension n = 2 and then for general n, these essays proved the equiconsistency of hyperbolic geometry with ordinary Euclidean geometry.
The papers of Beltrami remained little noticed until recently and the model was named after Klein ("The Klein disk model"). This happened as follows. In 1859 Arthur Cayley used the cross-ratio definition of angle due to Laguerre to show how Euclidean geometry could be defined using projective geometry. His definition of distance later became known as the Cayley metric.
In 1869, the young (twenty-year-old) Felix Klein became acquainted with Cayley's work. He recalled that in 1870 he gave a talk on the work of Cayley at the seminar of Weierstrass and he wrote:
"I finished with a question whether there might exist a connection between the ideas of Cayley and Lobachevsky. I was given the answer that these two systems were conceptually widely separated.
Official source
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