Curvature at a point

Consider any curve. As in the last section, $\Delta \tau$ = total curvature of the arc PP$'$, and $\frac{\Delta \tau}{\Delta s}$ = average curvature of the arc PP$'$.

Figure 12.2: Geometry of the curvature at a point.

More important, however, than the notion of the average curvature of an arc is that of curvature at a point. This is obtained as follows. Imagine P to approach P along the curve; then the limiting value of the average curvature $\left( = \frac{\Delta \tau}{\Delta s} \right)$ as P$'$ approaches P along the curve is defined as the curvature at P, that is,

Curvature at a point = $\lim_{\Delta s \to 0} \left( \frac{\Delta \tau}{\Delta s} \right) = \frac{d\tau}{ds}$.

Thefore,

$$K = \frac{d\tau}{ds} = \ {\rm curvature}.$$ (12.2)

Since the angle $\Delta \tau$ is measured in radians and the length of arc $\Delta s$ in units of length, it follows that the unit of curvature at a point is one radian per unit of length.