Definition:Curvature/Parametric Form/Polar

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

Also see

 * Equivalence of Definitions of Curvature