# Definition:Curvature

## Definition

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

The **curvature of $C$** is the rate of change of turning angle with respect to the arc length of $C$.

The **curvature of $C$** is often denoted **$\kappa$ (Greek kappa)**.

### Reciprocal of Radius of Osculating Circle

The **curvature** of $C$ is defined as:

- the reciprocal of the radius of the osculating circle to $C$.

### 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:

\(\ds y'\) | \(=\) | \(\ds \dfrac {\d y} {\d x}\) | 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 $P$. |

### Polar Form

Let $C$ be embedded in a polar plane.

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The **curvature** $\kappa$ of $C$ at a point:

- $P = \polar {r, \theta}$

is given by:

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

### 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:

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

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:

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

Let $C$ have the unit-speed parametrization:

- $x'^2 + y'^2 = 1$

The **curvature** $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by:

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

This page needs the help of a knowledgeable authority.In particular: Clearly this is unsigned curvature; the sign is somehow determined by the orientation of the plane and the curveSeemingly, just different styles. Signed curvature is a special for plane curves, and should be called 'signed' curvature. 'Sign' is only useful if the orientation of the parameterization is of interest. If you are knowledgeable in this area, then you can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Help}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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

- Results about
**curvature**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**curvature** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**curvature**