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:

\(\ds y'\)

\(=\)

\(\ds \dfrac {\d y} {\d x}\)

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

• Equivalence of Definitions of Curvature

• 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