Tangential Angle

For a Plane Curve, the tangential angle $\phi$ is defined by

$$\rho\,d\phi=ds,$$ (1)

where $s$ is the Arc Length and $\rho$ is the Radius of Curvature. The tangential angle is therefore given by

$$\phi=\int_0^t s'(t)\kappa(t)\,dt,$$ (2)

where $\kappa(t)$ is the Curvature. For a plane curve ${\bf r}(t)$, the tangential angle $\phi(t)$ can also be defined by

$${{\bf r}'(t)\over \vert{\bf r}'(t)\vert}=\left[{\matrix{\cos[\phi(t)]\cr \sin[\phi(t)]\cr}}\right].$$ (3)

Gray (1993) calls $\phi$ the Turning Angle instead of the tangential angle.

See also Arc Length, Curvature, Plane Curve, Radius of Curvature, Torsion (Differential Geometry)

References

Gray, A. ``The Turning Angle.'' §1.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 13-14, 1993.