Definition:Radius of Curvature

Definition

The radius of curvature of a curve $C$ at a point $P$ is defined as the reciprocal of the absolute value of its curvature:

$\rho = \dfrac 1 {\size k}$

Cartesian Coordinates

The radius of curvature of $C$ at a point $P$ can be expressed in cartesian coordinates as:

$\rho = \size {\dfrac {\paren {1 + y'^2}^{3/2} } {y} }$

Parametric Cartesian Form

Let $C$ be embedded in a cartesian plane and defined by the parametric equations:

$\begin{cases} x = \map x t \\ y = \map y t \end{cases}$

The radius of curvature $\rho$ of $C$ at a point $P = \tuple {x, y}$ is given by:

$\rho = \dfrac {\paren {x'^2 + y'^2}^{3/2} } {\size {x' y - y' x} }$

where:

$x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ with respect to $t$ at $P$

$y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ with respect to $t$ at $P$

$x$ and $y$ are the second derivatives of $x$ and $y$ with respect to $t$ at $P$.

Also see

• Results about radius of curvature can be found here.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): curvature
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): curvature
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): radius of curvature