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Radius of curvature (mathematics)


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.

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

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):

and | z | denotes the absolute value of z.

If the curve is given parametrically by functions x(t) and y(t), then the radius of curvature is

Heuristically, this result can be interpreted as

If γ : ℝ → ℝn is a parametrized curve in n then the radius of curvature at each point of the curve, ρ : ℝ → ℝ, is given by


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