Radius of curvature (mathematics)

Radius of curvature (mathematics)

In geometry, the radius of curvature, "R", of a curve at a point is a measure of the radius of the circular arc which best approximates the curve at that point. If this value taken to be positive when the curve turns anticlockwise and negative when the curve turns clockwise. It is given by

: R = frac{ds}{dvarphi} = 1/kappa,

where "s" is the arc length from a fixed point on the curve, "φ" is the tangential angle and scriptstylekappa is the curvature.

If the curve is given in Cartesian coordinates then the radius of curvature is

: R = frac { left(1 + left(dfrac{dy}{dx} ight)^2 ight)^{3/2{dfrac {d^2y}{dx^2.

If the curve is given parametrically with parameter "t" then the radius of curvature is

: R = frac{ds}{dvarphi} = frac {ig({dot{x}^2 + dot{y}^2}ig)^{3/2{dot {x}ddot{y} - dot{y}ddot{x,

where

:dot{x} = frac{dx}{dt} ext{ and }ddot{x} = frac{d^2x}{dt^2}.

See also

* Osculating circle

References

*cite book |title = Differential Geometry of Curves and Surfaces|first = Manfredo|last = do Carmo| id = ISBN 0-13-212589-7 | year = 1976


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