Tangential angle

File:Intrinsic coordinates (Whewell equation).png

The tangential angle Template:Mvar for an arbitrary curve Template:Mvar in Template:Mvar.

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 Template:Mvar-axis. (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.)

Equations

Parametric

If a curve is given parametrically by (x(t), y(t)), then the tangential angle Template:Mvar at Template:Mvar is defined (up to a multiple of 2π) by

$\frac{(x^{\prime }(t),y^{\prime }(t))}{|x^{\prime }(t),y^{\prime }(t)|}=(\cos \phi ,\sin \phi ).$

Here, the prime symbol denotes the derivative with respect to Template:Mvar. Thus, the tangential angle specifies the direction of the velocity vector (x(t), y(t)), while the speed specifies its magnitude. The vector

$\frac{(x^{\prime }(t),y^{\prime }(t))}{|x^{\prime }(t),y^{\prime }(t)|}$

is called the unit tangent vector, so an equivalent definition is that the tangential angle at Template:Mvar is the angle Template:Mvar such that (cos φ, sin φ) is the unit tangent vector at Template:Mvar.

If the curve is parametrized by arc length Template:Mvar, so Template:Abs = 1, then the definition simplifies to

$(x^{\prime }(s),y^{\prime }(s))=(\cos \phi ,\sin \phi ).$

In this case, the curvature Template:Mvar is given by φ′(s), where Template:Mvar is taken to be positive if the curve bends to the left and negative if the curve bends to the right. Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point:

$\phi (s)=\underset{0}{\overset{s}{\int }}\kappa (s)ds+{\phi }_0$

$\phi (t)=\underset{0}{\overset{t}{\int }}\kappa (t)s^{\prime }(t)dt+{\phi }_0$

Explicit

If the curve is given by the graph of a function y = f(x), then we may take (x, f(x)) as the parametrization, and we may assume Template:Mvar is between −Template:Sfrac and Template:Sfrac. This produces the explicit expression

$\phi =\arctan f^{\prime }(x).$

Polar tangential angle

In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. If Template:Mvar denotes the polar tangential angle, then ψ = φ − θ, where Template:Mvar is as above and Template:Mvar is, as usual, the polar angle.

If the curve is defined in polar coordinates by r = f(θ), then the polar tangential angle Template:Mvar at Template:Mvar is defined (up to a multiple of 2π) by

$\frac{(f^{\prime }(\theta ),f(\theta ))}{|f^{\prime }(\theta ),f(\theta )|}=(\cos \psi ,\sin \psi )$.

If the curve is parametrized by arc length Template:Mvar as r = r(s), θ = θ(s), so Template:Abs = 1, then the definition becomes

$(r^{\prime }(s),r{\theta }^{\prime }(s))=(\cos \psi ,\sin \psi )$.

The logarithmic spiral can be defined a curve whose polar tangential angle is constant.

See also

• Differential geometry of curves
• Whewell equation
• Subtangent