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. It is the inverse of the curvature.

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

$\frac{ds}{d\varphi} = \frac{1}{\kappa},$

where s is the arc length from a fixed point on the curve, φ is the tangential angle and $\scriptstyle\kappa$ 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):

$R =\left| \frac { \left(1 + y'^{\,2}\right)^{3/2}}{y''}\right|, \qquad\mbox{where}\quad y' = \frac{dy}{dx},\quad y'' = \frac{d^2y}{dx^2},$

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

$R = \;\left|\frac{ds}{d\varphi}\right| \;= \;\left|\frac {\big({\dot{x}^2 + \dot{y}^2}\big)^{3/2}}{\dot {x}\ddot{y} - \dot{y}\ddot{x}}\right|, \qquad\mbox{where}\quad \dot{x} = \frac{dx}{dt},\quad\ddot{x} = \frac{d^2x}{dt^2},\quad \dot{y} = \frac{dy}{dt},\quad\ddot{y} = \frac{d^2y}{dt^2}.$

Heuristically, this result can be interpreted as

$R = \frac{\left|\mathbf{v}\right|^3}{\left| \mathbf{v} \times \mathbf{ \dot v} \right|}, \qquad\mbox{where}\quad \left| \mathbf{v} \right| = \left| (\dot x, \dot y) \right| = R \frac{d\varphi}{dt}.$

## Examples

### Semicircles and circles

For a semi-circle of radius a in the upper half-plane

$y=\sqrt{a^2-x^2}, \quad y'=\frac{-x}{\sqrt{a^2-x^2}}, \quad y''=\frac{-a^2}{(a^2-x^2)^{3/2}},\quad R=|-a| =a.$
An ellipse (red) and its evolute (blue). The dots are the vertices of the ellipse, at the points of greatest and least curvature.

For a semi-circle of radius a in the lower half-plane

$y=-\sqrt{a^2-x^2}, \quad R=|a|=a.$

The circle of radius a has a radius of curvature equal to a.

### Ellipses

In an ellipse with major axis 2a and minor axis 2b, the vertices on the major axis have the smallest radius of curvature of any points $\left( R = \frac{b^2}{a} \right)$, and the vertices on the minor axis have the largest radius of curvature of any points $\left( R = \frac{a^2}{b} \right)$.