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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. ==History== This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension and then for general ''n'', these essays proved the equiconsistency of hyperbolic geometry with ordinary Euclidean geometry.〔 〕〔 〕〔 〕 Formally speaking however, this model was first constructed in 1859 by the English mathematician Arthur Cayley. But he considered it only as a construction in projective geometry and apparently did not notice the connection with non-Euclidean or hyperbolic geometry. Arthur Cayley applied the cross-ratio from projective geometry to measurement of distances and angles in spherical geometry. In 1869, the young (twenty-year-old) Felix Klein became acquainted with Cayley's work. He recalled that in 1870, and gave a talk on the work of Cayley at the seminar of the mathematician Karl Weierstrass, and, as he writes: :"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." Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane. As Klein puts it, "I allowed myself to be convinced by these objections and put aside this already mature idea." However, in 1871, he returned to this idea, formulated it mathematically, and published it.〔 〕 The distance is given by the Cayley–Klein metric and was first written down by Arthur Cayley in the context of projective and spherical geometry. Felix Klein recognized its importance for non-Euclidean geometry and popularized the subject. Arthur Cayley applied the cross-ratio from projective geometry to measurement of distances and angles in spherical geometry. Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Beltrami–Klein model」の詳細全文を読む スポンサード リンク
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