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[parent] circle of curvature (Definition)

Let a given plane curve $ \gamma$ have a definite curvature $ \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}{\vert\kappa\vert}$ 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

$\displaystyle x = x(t),\;\; y ´= y(t),$
the coordinates of the center of curvature belonging to the point $ (x,\,y)$ of the curve are
$\displaystyle \xi = x-\frac{(x'^2+y'^2)y'}{x'y''-x''y'},\quad \eta = y+\frac{(x'^2+y'^2)x'}{x'y''-x''y'}.$

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$.



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"circle of curvature" is owned by rspuzio. [ full author list (3) | owner history (1) ]
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See Also: curvature of a circle, osculating curve

Other names:  osculating circle
Also defines:  radius of curvature, center of curvature, centre of curvature

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Cross-references: second-order, passing through, origin, parabola, curvature, coordinates, parametric form, center, side, neighbourhood, tangent, curve, radius, circle, point, plane curve
There are 10 references to this entry.

This is version 10 of circle of curvature, born on 2007-04-27, modified 2008-03-28.
Object id is 9277, canonical name is CircleOfCurvature.
Accessed 2673 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
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