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Radius of Curvature

The radius of curvature is given by

R=1/(|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 (e.g., Lawrence 1972, p. 4).

Let x and y be given parametrically by

x=x(t)
(2)
y=y(t),
(3)

then

R=((x^('2)+y^('2))^(3/2))/(|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=([1+((dy)/(dx))^2]^(3/2))/(|(d^2y)/(dx^2)|).
(5)

In polar coordinates r=r(theta), the radius of curvature is given by

R=((r^2+r_theta^2)^(3/2))/(|r^2+2r_theta^2-rr_(thetatheta)|),
(6)

where r_theta=dr/dtheta and r_(thetatheta)=d^2r/dtheta^2(Gray 1997, p. 89).

See also

Bend, Curvature, Osculating Circle, Radius of Gyration, Radius of Torsion, Torsion

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, 1997.Kreyszig, E. Differential Geometry. New York: Dover, p. 34, 1991.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972.

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Radius of Curvature

Cite this as:

Weisstein, Eric W. "Radius of Curvature." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RadiusofCurvature.html

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