circle of curvature

Let a given plane curve $\gamma$ have a definite curvature (http://planetmath.org/CurvaturePlaneCurve) $\kappa$ in a point $P$ of $\gamma$.  The circle of curvature of $\gamma$ in the point $P$ is the circle which has the radius $\frac{1}{|\kappa |}$ and which has with the curve the common tangent in $P$ and which in a neighbourhood of $P$ is on the same side as the curve.

The radius $\varrho$ of the circle of curvature is the radius of curvature of $\gamma$ in $P$.  The center of the circle of curvature is the center of curvature of $\gamma$ in $P$.

When the curve $\gamma$ is given in the parametric form

$$x=x(t),yÃ\mathrm{‚}Â\mathrm{´}=y(t),$$

the coordinates of the center of curvature belonging to the point  $(x,y)$  of the curve are

$$\xi =x-\frac{(x^{\prime 2}+y^{\prime 2})y^{\prime }}{x^{\prime }{y}^{\prime \prime }-x^{\prime \prime }{y}^{\prime }},\eta =y+\frac{(x^{\prime 2}+y^{\prime 2})x^{\prime }}{x^{\prime }{y}^{\prime \prime }-x^{\prime \prime }{y}^{\prime }}.$$

Example.  Since the curvature of the parabola  $y=x^2$  in the origin is  $-2$,  the corresponding radius of curvature is $\frac{1}{2}$ and the center of curvature  $(0,\frac{1}{2})$.

Furthermore, it is possible to define the circle of curvature without first knowing about curvature of the curve. (In fact, using this definition, one could reverse the procedure and define curvature as the radius of the circle of curvature.) We may define the circle of curvature a point $P$ of $\gamma$ as the unique circle passing through $P$ which makes a second-order contact with $\gamma$ at $P$.

Title	circle of curvature
Canonical name	CircleOfCurvature
Date of creation	2013-03-22 16:59:46
Last modified on	2013-03-22 16:59:46
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	13
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 53A04
Synonym	osculating circle
Related topic	CurvatureOfaCircle
Related topic	OsculatingCurve
Defines	radius of curvature
Defines	center of curvature
Defines	centre of curvature