curvature of a circle

Let Cr be a circle of radius r centered at the origin.

A canonical parameterization of the curve is (counterclockwise)


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

Differentiating the parameterization we get


and this results in the normal


Differentiating g a second time we can calculate the curvaturePlanetmathPlanetmath


and by definition


and thus the curvature of a circle of radius r is 1r provided that the positive direction on the circle is anticlockwise; otherwise it is -1r.

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