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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 sinusoidal spiral ： ウィキペディア英語版 sinusoidal spiral In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates :$r^n\; =\; a^n\; \backslash cos(n\; \backslash theta)\backslash ,$ where ''a'' is a nonzero constant and ''n'' is a rational number other than 0. With a rotation about the origin, this can also be written :$r^n\; =\; a^n\; \backslash sin(n\; \backslash theta).\backslash ,$ The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including: * Equilateral hyperbola (''n'' = −2) * Line (''n'' = −1) * Parabola (''n'' = −1/2) * Tschirnhausen cubic (''n'' = −1/3) * Cayley's sextet (''n'' = 1/3) * Cardioid (''n'' = 1/2) * Circle (''n'' = 1) * Lemniscate of Bernoulli (''n'' = 2) The curves were first studied by Colin Maclaurin. ==Equations== Differentiating :$r^n\; =\; a^n\; \backslash cos(n\; \backslash theta)\backslash ,$ and eliminating ''a'' produces a differential equation for ''r'' and θ: :$\backslash frac\backslash cos\; n\backslash theta\; +\; r\backslash sin\; n\backslash theta\; =0$. Then :$\backslash left(\backslash frac,\backslash \; r\backslash frac\backslash right)\backslash cos\; n\backslash theta\; \backslash frac$ = \left(-r\sin n\theta ,\ r \cos n\theta \right) = r\left(-\sin n\theta ,\ \cos n\theta \right) which implies that the polar tangential angle is :$\backslash psi\; =\; n\backslash theta\; \backslash pm\; \backslash pi/2$ and so the tangential angle is :$\backslash varphi\; =\; (n+1)\backslash theta\; \backslash pm\; \backslash pi/2$. (The sign here is positive if ''r'' and cos ''n''θ have the same sign and negative otherwise.) The unit tangent vector, :$\backslash left(\backslash frac,\backslash \; r\backslash frac\backslash right)$, has length one, so comparing the magnitude of the vectors on each side of the above equation gives :$\backslash frac\; =\; r\; \backslash cos^\; n\backslash theta\; =\; a\; \backslash cos^\}\; n\backslash theta$. In particular, the length of a single loop when $n>0$ is: :$a\backslash int\_\}^\}\; \backslash cos^\}\; n\backslash theta\backslash \; d\backslash theta$ The curvature is given by :$\backslash frac\; =\; (n+1)\backslash frac\; =\; \backslash frac\; \backslash cos^\}\; n\backslash theta$. 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「sinusoidal spiral」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.