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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hyperboloid model ： ウィキペディア英語版 Hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model (after Hermann Minkowski and Hendrik Lorentz), is a model of ''n''-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space and ''m''-planes are represented by the intersections of the (''m''+1)-planes in Minkowski space with ''S''+. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the ''n''-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group. == Minkowski quadratic form == (詳細はquadratic form is defined to be :$Q(x\_0,\; x\_1,\; \backslash ldots,\; x\_n)\; =\; x\_0^2\; -\; x\_1^2\; -\; \backslash ldots\; -\; x\_n^2.$ The vectors such that form an ''n''-dimensional hyperboloid ''S'' consisting of two connected components, or ''sheets'': the forward, or future, sheet ''S''+, where ''x''0>0 and the backward, or past, sheet ''S''−, where ''x''0<0. The points of the ''n''-dimensional hyperboloid model are the points on the forward sheet ''S''+. The Minkowski bilinear form ''B'' is the polarization of the Minkowski quadratic form ''Q'', :$B(\backslash mathbf,\; \backslash mathbf)\; =\; (Q(\backslash mathbf+\backslash mathbf)\; -\; Q(\backslash mathbf)\; -\; Q(\backslash mathbf))\; /\; 2\; .$ Explicitly, :$B((x\_0,\; x\_1,\; \backslash ldots,\; x\_n),\; (y\_0,\; y\_1,\; \backslash ldots,\; y\_n))\; =\; x\_0y\_0\; -\; x\_1\; y\_1\; -\; \backslash ldots\; -\; x\_n\; y\_n\; .$ The hyperbolic distance between two points ''u'' and ''v'' of ''S''+ is given by the formula :$d(\backslash mathbf,\; \backslash mathbf)\; =\; \backslash operatorname(B(\backslash mathbf,\; \backslash mathbf))\; ,$ where is the inverse function of hyperbolic cosine. 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「Hyperboloid model」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.