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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 Lemniscate of Bernoulli ： ウィキペディア英語版 Lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points ''F''1 and ''F''2, known as foci, at distance 2''a'' from each other as the locus of points ''P'' so that ''PF''1·''PF''2 = ''a''2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from ''lemniscus'', which is Latin for "pendant ribbon". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4. This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed ''focal points'' is a constant. A Cassini oval, by contrast, is the locus of points for which the ''product'' of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form of Watt's linkage, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a crossed square.〔.〕 == Equations == * Its Cartesian equation is (up to translation and rotation): :$(x^2\; +\; y^2)^2\; =\; 2a^2\; (x^2\; -\; y^2)\backslash ,$ * In polar coordinates: :$r^2\; =\; 2a^2\; \backslash cos\; 2\backslash theta\backslash ,$ * As a parametric equation: :$x\; =\; \backslash frac;\; \backslash qquad\; y\; =\; \backslash frac$ In two-center bipolar coordinates: :$rr\text{'}\; =\; a^2\backslash ,$ In rational polar coordinates: :$Q\; =\; 2s-1\backslash ,$ 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「Lemniscate of Bernoulli」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.