Definition:Curvature

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Definition

Let $C$ be a curve defined by a real function which is twice differentiable.

The curvature of a $C$ is the reciprocal of the radius of the osculating circle to $C$ and is often denoted $\kappa$ (Greek kappa).

Whewell Form

The curvature $\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as:

$\kappa = \dfrac {\d \psi} {\d s}$

where:

$\psi$ is the turning angle of $C$

$s$ is the arc length of $C$.

Cartesian Form

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 $P$.

Polar Form

Let $C$ be embedded in a polar plane.

The curvature $\kappa$ of $C$ at a point $P = \polar {r, \theta}$ is given by:

$\kappa = \dfrac {\paren {\map \arctan {\dfrac {r \theta'} {r'} } }' + \theta'} {\paren {r'^ + \paren {r \theta'}^2}^{1/2} }$

Parametric Form

Cartesian Coordinates

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:

$\kappa = \dfrac {x' y'' - y' x''} {\tuple {x'^2 + y'^2}^{3/2} }$

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 $P$.

Polar Coordinates

Let $C$ be embedded in a polar plane and defined by the parametric equations:

$\begin{cases} r = \map r t \\ \theta = \map \theta t \end{cases}$

The curvature $\kappa$ of $C$ at a point $P = \polar {r, \theta}$ is given by:

$\kappa = \dfrac {2 r'^2 \theta' + r r'' \theta' + r r' \theta'' + r^2 \theta'^3} {\paren {r'^2 + \paren {r \theta'}^2}^{1/2} }$

where:

$r' = \dfrac {\d r} {\d t}$ is the derivative of $r$ with respect to $t$ at $P$

$\theta' = \dfrac {\d \theta} {\d t}$ is the derivative of $\theta$ with respect to $t$ at $P$

$r''$ and $\theta''$ are the second derivatives of $r$ and $y$ with respect to $t$ at $P$.

Unit-Speed Parametric Form

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:

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

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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

• Equivalence of Definitions of Curvature

• Definition:Radius of Curvature

Sources

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