Radius of curvature

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 $γ : ℝ \to ℝ n$ is a parametrized curve in $ℝ n$ then the radius of curvature at each point of the curve, $ρ : ℝ \to ℝ$ , is given by