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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 Universal parabolic constant ： ウィキペディア英語版 Universal parabolic constant The universal parabolic constant is a mathematical constant. It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter (twice the focal length). It is denoted ''P''.〔, a Wolfram Web resource.〕〔 American Mathematical Monthly, 120 (2013), 929-935.〕 In the diagram, the latus rectum is pictured in blue, the parabolic segment that it forms in red and the focal parameter in green. (The focus of the parabola is the point ''F'' and the directrix is the line ''L''.) The value of ''P'' is〔See Parabola#Length of an arc of a parabola. Use p=2f, the length of the semilatus rectum, so h=f and q=f.sqrt(2). Calculate 2s in terms of f, then divide by 2f, which is the focal parameter.〕 : $P\; =\; \backslash ln(1\; +\; \backslash sqrt2)\; +\; \backslash sqrt2\; =\; 2.29558714939\backslash dots$ . The circle and parabola are unique among conic sections in that they have a universal constant. The analogous ratios for ellipses and hyperbolas depend on their eccentricities. This means that all circles are similar and all parabolas are similar, whereas ellipses and hyperbolas are not. ==Derivation== Take $y\; =\; \backslash frac$ as the equation of the parabola. The focal parameter is $p=2f$ and the semilatus rectum is $\backslash ell=2f$. :$$ \begin P & := \frac\int_^\ell \sqrt\right)^2}\, dx \\ & = \frac\int_^\sqrt}\, dx \\ & = \int_^\sqrt\, dt \quad (x=2ft) \\ & = \operatorname(1)+\sqrt\\ & = \ln(1+\sqrt)+\sqrt. \end 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「Universal parabolic constant」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.