# 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:

- $y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ with respect to $x$ at $P$
- $y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ with respect to $x$ at $P$.

## Also see

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid