# Definition:Curvature

## 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 $P$.

### Parametric Form

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

$\begin{cases} x = \map x t \\ y = \map y t \end{cases}$

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 $P$.

### 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:

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

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 $P$.