Definition:Curvature/Parametric Form

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Definition

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

Cartesian Coordinates

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

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

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 $P$.


Polar Coordinates

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

$\begin{cases} r = \map r t \\ \theta = \map \theta t \end{cases}$


The curvature $\kappa$ of $C$ at a point $P = \polar {r, \theta}$ is given by:

$\kappa = \dfrac {2 r'^2 \theta' + r r'' \theta' + r r' \theta'' + r^2 \theta'^3} {\paren {r'^2 + \paren {r \theta'}^2}^{1/2} }$

where:

$r' = \dfrac {\d r} {\d t}$ is the derivative of $r$ with respect to $t$ at $P$
$\theta' = \dfrac {\d \theta} {\d t}$ is the derivative of $\theta$ with respect to $t$ at $P$
$r''$ and $\theta''$ are the second derivatives of $r$ and $y$ with respect to $t$ at $P$.


Also see

  • Results about curvature can be found here.