Definition:Curvature/Cartesian Form

Definition
Let $C$ be a curve defined by a real function which is twice differentiable. Let $C$ be embedded in a cartesian plane.

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


 * $\kappa = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$

where:
 * $y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ $x$ at $P$
 * $y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ $x$ at $P$.

Also see

 * Equivalence of Definitions of Curvature