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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 Spira mirabilis ： ウィキペディア英語版 Logarithmic spiral A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it ''Spira mirabilis'', "the marvelous spiral". ==Definition== In polar coordinates $(r,\; \backslash theta)$ the logarithmic curve can be written as :$r\; =\; ae^\backslash ,$ or :$\backslash theta\; =\; \backslash frac\; \backslash ln(r/a),$ with $e$ being the base of natural logarithms, and $a$ and $b$ being arbitrary positive real constants. In parametric form, the curve is :$x(t)\; =\; r(t)\; \backslash cos(t)\; =\; ae^\; \backslash cos(t)\backslash ,$ :$y(t)\; =\; r(t)\; \backslash sin(t)\; =\; ae^\; \backslash sin(t)\backslash ,$ with real numbers $a$ and $b$. The spiral has the property that the angle ''φ'' between the tangent and radial line at the point $(r,\; \backslash theta)$ is constant. This property can be expressed in differential geometric terms as :$\backslash arccos\; \backslash frac\text{'}(\backslash theta)\; \backslash rangle\}\text{'}(\backslash theta)\backslash |\}\; =\; \backslash arctan\; \backslash frac\; =\; \backslash phi.$ The derivative of $\backslash mathbf(\backslash theta)$ is proportional to the parameter $b$. In other words, it controls how "tightly" and in which direction the spiral spirals. In the extreme case that $b\; =\; 0$ ($\backslash textstyle\backslash phi\; =\; \backslash frac$) the spiral becomes a circle of radius $a$. Conversely, in the limit that $b$ approaches infinity (''φ'' → 0) the spiral tends toward a straight half-line. The complement of ''φ'' is called the ''pitch''. 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「Logarithmic 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.