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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 Tangential angle ： ウィキペディア英語版 Tangential angle In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the ''x''-axis.〔"Natural Equation" at MathWorld〕 (Note, some authors define the angle as the deviation from the direction of the curve at some fixed starting point. This is equivalent to the definition given here by the addition of a constant to the angle or by rotating the curve.〔For example W. Whewell "Of the Intrinsic Equation of a Curve, and its Application" ''Cambridge Philosophical Transactions'' Vol. VIII (1849) pp. 659-671. (Google Books ) uses φ to mean the angle between the tangent and tangent at the origin. This is the paper introducing the Whewell equation, an application of the tangential angle.〕) == Equations == If a curve is given parametrically by $(x(t),\backslash \; y(t))$, then the tangential angle $\backslash varphi$ at $t$ is defined (up to a multiple of $2\backslash pi$) by〔MathWorld "Tangential Angle"〕 : $\backslash frac\; =\; (\backslash cos\; \backslash varphi,\backslash \; \backslash sin\; \backslash varphi).$ Here, the prime symbol denotes derivative. Thus, the tangential angle specifies the direction of the velocity vector $(x\text{'}(t),\backslash \; y\text{'}(t))$, while the speed specifies its magnitude. The vector $\backslash frac$ is called the unit tangent vector, so an equivalent definition is that the tangential angle at $t$ is the angle $\backslash varphi$ such that $(\backslash cos\; \backslash varphi,\backslash \; \backslash sin\; \backslash varphi)$ is the unit tangent vector at $t$. If the curve is parameterized by arc length $s$, so $|x\text{'}(s),\backslash \; y\text{'}(s)|\; =\; 1$, then the definition simplifies to $(x\text{'}(s),\backslash \; y\text{'}(s))\; =\; (\backslash cos\; \backslash varphi,\backslash \; \backslash sin\; \backslash varphi)$. In this case, the curvature $\backslash kappa$ is given by $\backslash varphi\text{'}(s)$, where $\backslash kappa$ is taken to be positive if the curve bends to the left and negative if the curve bends to the right.〔MathWorld "Natural Equation" differentiating equation 1〕 If the curve is given by $y\; =\; f(x)$, then we may take $(x,\backslash \; f(x))$ as the parameterization, and we may assume $\backslash varphi$ is between $-\backslash pi/2$ and $\backslash pi/2$. This produces the explicit expression $\backslash varphi\; =\; \backslash arctan\; f\text{'}(x)$. 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「Tangential angle」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.