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.