Curvature of a circle

Consider a circle of radius $R$.

Figure 12.1: The curvature of a circle.

Let

$\tau$ = angle that the tangent at P makes with the $x$-axis,

and

$\tau + \Delta \tau$ = angle made by the tangent at a neighboring point P$'$.

Then we say $\Delta \tau$ = total curvature of arc PP$'$. If the point P with its tangent be supposed to move along the curve to P$'$, the total curvature ( $= \Delta \tau$) would measure the total change in direction, or rotation, of the tangent; or, what is the same thing, the total change in direction of the arc itself. Denoting by $s$ the length of the arc of the curve measured from some fixed point (as A) to P, and by $\Delta s$ the length of the arc P P$'$, then the ratio $\frac{\Delta \tau}{\Delta s}$ measures the average change in direction per unit length of arc^12.1. Since, from Figure 12.1, $\Delta s = R \cdot \Delta \tau$, or $\frac{\Delta \tau}{\Delta s} = \frac{1}{R}$, it is evident that this ratio is constant everywhere on the circle. This ratio is, by definition, the curvature of the circle, and we have

$$K = \frac{1}{R}.$$ (12.1)

The curvature of a circle equals the reciprocal of its radius.