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In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof.[1][2][3]

## Definition

In the case of a space curve, the radius of curvature is the length of the curvature vector.

In the case of a plane curve, then R is the absolute value of[3]

${\displaystyle R\equiv \left|{\frac {ds}{d\varphi }}\right|={\frac {1}{\kappa }},}$

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

If the curve is given in Cartesian coordinates as y(x), then the radius of curvature is (assuming the curve is differentiable up to order 2):