Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/180

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

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

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

101. Curvature at a point. Consider any curve. As in section,

	$\Delta \tau$ = total curvature of the arc PP′,
and	${\frac {\Delta \tau }{\Delta s}}$ = average curvature of the arc PP′.

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 $\left(={\tfrac {\Delta \tau }{\Delta s}}\right)$ as P′ approaches P along the curve is defined as the curvature at P, that is,

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

(39)	∴ $K={\frac {d\tau }{ds}}$ = curvature.

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.

102. Formulas for curvature. It is evident that if, in the last section, instead of measuring the angles which the tangents made with OX, we had denoted by $\tau$ and $\tau +\Delta \tau$ the angles made by the tangents with any arbitrarily fixed line, the different steps would in no wise have been changed, and consequently the results are entirely in:dependent of the system of coordinates used. However, since the equations of the curves we shall consider are all given in either rectangular or polar coördinates, it is necessary to deduce formulas for K in terms of both. We have

	$\ \tan \tau$	$={\frac {dy}{dx}}$,	§ 32
or	$\ \tau$	$\arctan {\frac {dy}{dx}}$.