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Radius of Curvature

The radius of curvature is given by

R\equiv {1\over\kappa},
\end{displaymath} (1)

where $\kappa$ is the Curvature. At a given point on a curve, $R$ is the radius of the Osculating Circle. The symbol $\rho$ is sometimes used instead of $R$ to denote the radius of curvature.

Let $x$ and $y$ be given parametrically by

$\displaystyle x$ $\textstyle =$ $\displaystyle x(t)$ (2)
$\displaystyle y$ $\textstyle =$ $\displaystyle y(t),$ (3)

R={(x'^2+y'^2)^{3/2}\over x'y''-y'x''},
\end{displaymath} (4)

where $x'=dx/dt$ and $y'=dy/dt$. Similarly, if the curve is written in the form $y=f(x)$, then the radius of curvature is given by
R={\left[{1+\left({dy\over dx}\right)^2}\right]^{3/2}\over {d^2y\over dx^2}}.
\end{displaymath} (5)

See also Bend (Curvature), Curvature, Osculating Circle, Torsion (Differential Geometry)


Kreyszig, E. Differential Geometry. New York: Dover, p. 34, 1991.

© 1996-9 Eric W. Weisstein