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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 Hyperbolic law of cosines ： ウィキペディア英語版 Hyperbolic law of cosines In hyperbolic geometry, the law of cosines is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. Take a hyperbolic plane whose Gaussian curvature is $-\backslash frac1$. Then given a hyperbolic triangle ''ABC'' with angles ''α'', ''β'', ''γ'', and side lengths ''BC'' = ''a'', ''AC'' = ''b'', and ''AB'' = ''c'', the following two rules hold: :$\backslash cosh\backslash frac\; ak\; =\; \backslash cosh\backslash frac\; bk\; \backslash cosh\backslash frac\; ck\; -\; \backslash sinh\backslash frac\; bk\; \backslash sinh\backslash frac\; ck\; \backslash cos\backslash alpha,\; \backslash ,$ considering the sides, while :$\backslash cos\backslash alpha\; =\; -\backslash cos\backslash beta\; \backslash cos\backslash gamma\; +\; \backslash sin\backslash beta\; \backslash sin\backslash gamma\; \backslash cosh\backslash frac\; ak,\; \backslash ,$ for the angles. Christian Houzel (page 8) indicates that the hyperbolic law of cosines implies the angle of parallelism in the case of an ideal hyperbolic triangle: :When α = 0, that is when the vertex ''A'' is rejected to infinity and the sides ''BA'' and ''CA'' are ''parallel'', the first member equals 1; let us suppose in addition that γ = π/2 so that cos γ = 0 and sin γ = 1. The angle at ''B'' takes a value β given by 1 = sin β cosh(''a''/k); this angle was later called ''angle of parallelism'' and Lobachevsky noted it by ''F(a)'' or Π(''a''). ==Hyperbolic law of Haversines== In cases where ''a/k'' is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to rounding errors, for the exact same reason it does in the Spherical law of cosines. The hyperbolic version of the law of haversines can prove useful in this case: :$\backslash sinh^2\backslash frac\; =\; \backslash sinh^2\backslash frac\; +\; \backslash sinh\backslash frac\; bk\; \backslash sinh\backslash frac\; ck\; \backslash sin^2\backslash frac,\; \backslash ,$ 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「Hyperbolic law of cosines」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.