# 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 $P$.

## Also see

## Sources

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