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. 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 $\left( R = \frac{a^2}{b} \right)$.

See also

• Osculating circle