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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 : 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'', : Explicitly, : The hyperbolic distance between two points ''u'' and ''v'' of ''S''+ is given by the formula : 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
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