\begin{definition}[Definition:Curvature/Parametric Form/Polar]
Let $C$ be a curve defined by a real function which is twice differentiable.
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 $r$ {{WRT|Differentiation}} $t$ at $P$
:$\theta' = \dfrac {\d \theta} {\d t}$ is the derivative of $\theta$ {{WRT|Differentiation}} $t$ at $P$
:$r''$ and $\theta''$ are the second derivatives of $r$ and $y$ {{WRT|Differentiation}} $t$ at $P$.
\end{definition}