Definition:Curvature/Parametric Form/Cartesian
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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:
- $\begin{cases} x = \map x t \\ y = \map y t \end{cases}$
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 $P$.
Also see
- Results about curvature can be found here.
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid