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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),\ y(t)), then the tangential angle \varphi at t is defined (up to a multiple of 2\pi) by〔MathWorld "Tangential Angle"〕
: \frac = (\cos \varphi,\ \sin \varphi).
Here, the prime symbol denotes derivative. Thus, the tangential angle specifies the direction of the velocity vector (x'(t),\ y'(t)), while the speed specifies its magnitude. The vector \frac is called the unit tangent vector, so an equivalent definition is that the tangential angle at t is the angle \varphi such that (\cos \varphi,\ \sin \varphi) is the unit tangent vector at t.
If the curve is parameterized by arc length s, so |x'(s),\ y'(s)| = 1, then the definition simplifies to (x'(s),\ y'(s)) = (\cos \varphi,\ \sin \varphi). In this case, the curvature \kappa is given by \varphi'(s), where \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,\ f(x)) as the parameterization, and we may assume \varphi is between -\pi/2 and \pi/2. This produces the explicit expression \varphi = \arctan f'(x).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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