curvature of a circleCurvatureOfACircle2006-04-10 17:21:532008-03-19 22:45:12ExampleSerret-Frenet equations in $\mathbb{R}^2$% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands hereLet $C_r$ 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
$$ \mathbf{T} = g'(s) = \left( -\sin \left( \frac{s}{r} \right), \cos \left( \frac{s}{r}\right) \right)$$
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
$$\mathbf{T}' = -\frac{1}{r}\left(\cos\left(\frac{s}{r}\right),\;\sin\left(\frac{s}{r}\right)\right) = \frac{1}{r} \mathbf{N}$$
and by definition
$$ \mathbf{T}' = k\mathbf{N}\;\; \therefore\; k = \frac{1}{r} $$
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 $\displaystyle{-\frac{1}{r}}$.