Radius of Curvature

The radius of curvature is given by

$$R\equiv {1\over\kappa},$$ (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

$$x = x(t)$$ (2)

$$y = y(t),$$ (3)

then

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

$$R={\left[{1+\left({dy\over dx}\right)^2}\right]^{3/2}\over {d^2y\over dx^2}}.$$ (5)

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

References

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