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In physics and in the mathematics of plane curves, Cotes's spiral (also written Cotes' spiral and Cotes spiral) is a spiral that is typically written in one of three forms : : : where ''r'' and ''θ'' are the radius and azimuthal angle in a polar coordinate system, respectively, and ''A'', ''k'' and ''ε'' are arbitrary real number constants. These spirals are named after Roger Cotes. The first form corresponds to an epispiral, and the second to one of Poinsot's spirals; the third form corresponds to a ''hyperbolic spiral'', also known as a ''reciprocal spiral'', which is sometimes not counted as a Cotes's spiral.〔 〕 The significance of Cotes's spirals for physics is in the field of classical mechanics. These spirals are the solutions for the motion of a particle moving under an inverse-cube central force, e.g., : where ''μ'' is any real number constant. A central force is one that depends only on the distance ''r'' between the moving particle and a point fixed in space, the center. In this case, the constant ''k'' of the spiral can be determined from μ and the areal velocity of the particle ''h'' by the formula : when ''μ'' < ''h'' 2 (cosine form of the spiral) and : when ''μ'' > ''h'' 2 (hyperbolic cosine form of the spiral). When ''μ'' = ''h'' 2 exactly, the particle follows the third form of the spiral : ==See also== * Archimedean spiral * Hyperbolic spiral * Newton's theorem of revolving orbits * Bertrand's theorem 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「Cotes's spiral」の詳細全文を読む スポンサード リンク
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