Definition:Curvature/Unit-Speed Parametric Form

Definition
Let $C$ be a curve defined by a real function which is twice differentiable. Let $C$ be embedded in a cartesian plane and defined by the parametric equations:
 * $\begin{cases} x = \map x t \\ y = \map y t \end{cases}$

Suppose the curve has the unit-speed parametrization:


 * $x'^2 + y'^2 = 1$

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


 * $\kappa = \sqrt{x^2 + y^2}$

where:
 * $x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ $t$ at $P$
 * $y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ $t$ at $P$
 * $x$ and $y$ are the second derivatives of $x$ and $y$ $t$ at $P$.

Also see

 * Equivalence of Definitions of Curvature