Definition:Curvature/Cartesian Form
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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.
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:
\(\ds y'\) | \(=\) | \(\ds \dfrac {\d y} {\d x}\) | is the derivative of $y$ with respect to $x$ at $P$ | |||||||||||
\(\ds y' '\) | \(=\) | \(\ds \dfrac {\d^2 y} {\d x^2}\) | is the second derivative of $y$ with respect to $x$ 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