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)

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

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

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

and this results in the normal

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

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

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

and by definition

$${𝐓}^{\prime }=k𝐍\therefore k=\frac{1}{r}$$

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
Canonical name	CurvatureOfACircle
Date of creation	2013-03-22 15:50:30
Last modified on	2013-03-22 15:50:30
Owner	cvalente (11260)
Last modified by	cvalente (11260)
Numerical id	9
Author	cvalente (11260)
Entry type	Example
Classification	msc 53A04
Related topic	circle
Related topic	curvature
Related topic	Connection
Related topic	CircleOfCurvature