# curvature of a circle

Let $C_{r}$ be a circle of radius $r$ centered at the origin.

A canonical parameterization of the curve is (counterclockwise)

 $g(s)=r\left(\cos\left(\frac{s}{r}\right),\sin\left(\frac{s}{r}\right)\right)$

for $s\in(0,2\pi r)$ (actually this leaves out the point $(r,0)$ but this could be treated via another parameterization taking $s\in(-\pi r,\pi r)$)

Differentiating the parameterization we get

 $\mathbf{T}=g^{\prime}(s)=\left(-\sin\left(\frac{s}{r}\right),\cos\left(\frac{s% }{r}\right)\right)$

and this results in the normal

 $\mathbf{N}=J\cdot\mathbf{T}=-\left(\cos\left(\frac{s}{r}\right),\;\sin\left(% \frac{s}{r}\right)\right)=-\frac{g(s)}{r}$

Differentiating $g$ a second time we can calculate the curvature

 $\mathbf{T}^{\prime}=-\frac{1}{r}\left(\cos\left(\frac{s}{r}\right),\;\sin\left% (\frac{s}{r}\right)\right)=\frac{1}{r}\mathbf{N}$

and by definition

 $\mathbf{T}^{\prime}=k\mathbf{N}\;\;\therefore\;k=\frac{1}{r}$

and thus the curvature of a circle of radius $r$ is $\displaystyle{\frac{1}{r}}$ provided that the positive direction on the circle is anticlockwise; otherwise it is $\displaystyle{-\frac{1}{r}}$.

Title curvature of a circle CurvatureOfACircle 2013-03-22 15:50:30 2013-03-22 15:50:30 cvalente (11260) cvalente (11260) 9 cvalente (11260) Example msc 53A04 circle curvature Connection CircleOfCurvature