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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 Twist (mathematics) ： ウィキペディア英語版 Twist (mathematics) In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve $X=X(s)$, where $s$ is the arc-length of $X$ and $U=U(s)$ a unit vector perpendicular at each point to $X$. Since the ribbon $(X,U)$ has edges $X$ and $X\text{'}=X+\backslash varepsilon\; U$ the twist (or ''total twist number'') $Tw$ measures the average winding of the curve $X\text{'}$ around and along the curve $X$. According to Love (1944) twist is defined by :$Tw\; =\; \backslash dfrac\; \backslash int\; \backslash left(\; \backslash dfrac\; \backslash times\; U\; \backslash right)\; \backslash cdot\; \backslash dfrac\; ds\; \backslash ;\; ,$ where $dX/ds$ is the unit tangent vector to $X$. The total twist number $Tw$ can be decomposed (Moffatt & Ricca 1992) into ''normalized total torsion'' $T$ and ''intrinsic twist'' $N$, that is :$Tw\; =\; \backslash dfrac\; \backslash int\; \backslash tau\; \backslash ;\; ds\; +\; \backslash dfrac\; =\; T+N\; \backslash ;\; ,$ where $\backslash tau=\backslash tau(s)$ is the torsion of the space curve $X$, and $\backslash left(\backslash Theta\; \backslash right)\_X$ denotes the total rotation angle of $U$ along $X$. The total twist number $Tw$ depends on the choice of the vector field $U$ (Banchoff & White 1975). When the ribbon is deformed so as to pass through an ''inflectional state'' (i.e. $X$ has a point of inflection) torsion becomes singular, but its singularity is integrable (Moffatt & Ricca 1992) and $Tw$ remains continuous. This behavior has many important consequences for energy considerations in many fields of science. Together with the writhe $Wr$ of $X$, twist is a geometric quantity that plays an important role in the application of the Călugăreanu–White–Fuller formula $Lk\; =\; Wr\; +\; Tw$ in topological fluid dynamics (for its close relation to kinetic and magnetic helicity of a vector field), physical knot theory, and structural complexity analysis. ==References== *Banchoff, T.F. & White, J.H. (1975) The behavior of the total twist and self-linking number of a closed space curve under inversions. ''Math. Scand.'' 36, 254–262. *Love, A.E.H. (1944) (''A Treatise on the Mathematical Theory of Elasticity'' ). Dover, 4th Ed., New York. *Moffatt, H.K. & Ricca, R.L. (1992) (Helicity and the Călugăreanu invariant ). ''Proc. R. Soc. A'' 439, 411–429. 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「Twist (mathematics)」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.