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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 Lamé curve ： ウィキペディア英語版 Superellipse A superellipse, also known as a Lamé curve after Gabriel Lamé, is a geometric figure defined in the Cartesian coordinate system as the set of all points (''x'', ''y'') with : $\backslash left|\backslash frac\backslash right|^n\backslash !\; +\; \backslash left|\backslash frac\backslash right|^n\backslash !\; =\; 1,$ where ''n'', ''a'' and ''b'' are positive numbers. This formula defines a closed curve contained in the rectangle −''a'' ≤ ''x'' ≤ +''a'' and −''b'' ≤ ''y'' ≤ +''b''. The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve. If ''n'' < 2, the figure is also called a hypoellipse; if ''n'' > 2, a hyperellipse. When ''n'' ≥ 1 and ''a'' = ''b'', the superellipse is the boundary of a ball of R2 in the ''n''-norm. The extreme points of the superellipse are (±''a'', 0) and (0, ±''b''), and its four "corners" are (±''sa, ±sb''), where $s=2^$ (sometimes called the "superness"〔Donald Knuth: ''The METAFONTbook'', p. 126〕). ==Mathematical properties== When ''n'' is a nonzero rational number ''p''/''q'' (in lowest terms), then each quadrant of the superellipse is a plane algebraic curve. For positive ''n'' the order is ''pq''; for negative ''n'' the order is 2''pq''. In particular, when ''a'' = ''b'' = 1 and ''n'' is an even integer, then it is a Fermat curve of degree ''n''. In that case it is non-singular, but in general it will be singular. If the numerator is not even, then the curve is pasted together from portions of the same algebraic curve in different orientations. The curve is given by the parametric equations :$\backslash left.$ \begin x\left(\theta\right) &= \plusmn a\cos^} \theta \\ y\left(\theta\right) &= \plusmn b\sin^} \theta \end \right\} \qquad 0 \le \theta < \frac or :$$ \begin x\left(\theta\right) &= ^} \cdot a \sgn(\cos \theta) \\ y\left(\theta\right) &= ^} \cdot b \sgn(\sin \theta). \end where the sign function is :$\backslash sgn(t)\; =\; \backslash begin$ -1, & t < 0 \\ 0, & t = 0 \\ +1, & t > 0 . \end The area inside the superellipse can be expressed in terms of the gamma function, Γ(''x''), as :$\backslash mathrm\; =\; 4\; a\; b\; \backslash frac\backslash right)\backslash right)^2\}\backslash right)\}\; .$ The pedal curve is relatively straightforward to compute. Specifically, the pedal of : $\backslash left|\backslash frac\backslash right|^n\backslash !\; +\; \backslash left|\backslash frac\backslash right|^n\backslash !\; =\; 1,$ is given in polar coordinates by : $(a\; \backslash cos\; \backslash theta)^\}+(b\; \backslash sin\; \backslash theta)^\}=r^\}.$ 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「Superellipse」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.