Definition:Radius of Curvature/Parametric Form/Cartesian

Definition

Let $C$ be a curve defined by a real function which is twice differentiable.

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

• 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