Definition:Curvature

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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 curve
Seemingly, 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.
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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

  • Results about curvature can be found here.


Sources

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