Tangential angle

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The tangential angle \varphi for an arbitrary curve P

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.[1] (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.[2])

Equations[edit]

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[3]

\frac{(x'(t),\ y'(t))}{|x'(t),\ y'(t)|} = (\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{(x'(t),\ y'(t))}{|x'(t),\ y'(t)|} 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.[4]

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).

Polar tangential angle[edit]

In polar coordinates, define the polar tangential angle as the angle between the tangent line to the curve at the given point and ray from the origin to the point.[5] If \psi denotes the polar tangential angle, then \psi = \varphi - \theta, where \varphi is as above and \theta is, as usual, the polar angle.

If the curve is defined in polar coordinates by r = f(\theta), then polar tangential angle \psi at \theta is defined (up to a multiple of 2\pi) by

\frac{(f'(\theta),\ f(\theta))}{|f'(\theta),\ f(\theta)|} = (\cos \psi,\ \sin \psi).

If the curve is parameterized by arc length s as r = r(s),\ \theta = \theta(s), so |r'(s),\ r\theta'(s)| = 1, then the definition becomes (r'(s),\ r\theta'(s)) = (\cos \psi,\ \sin \psi).

The logarithmic spiral can be defined a curve whose polar tangential angle is constant.[5][6]

See also[edit]

References[edit]

  1. ^ "Natural Equation" at MathWorld
  2. ^ 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.
  3. ^ MathWorld "Tangential Angle"
  4. ^ MathWorld "Natural Equation" differentiating equation 1
  5. ^ a b "Logarithmic Spiral" at Planet Math
  6. ^ Williamson for section unless otherwise noted.