Curvature

The curvature, represented by $\kappa$ , of a smooth (that is, with no cusps or sharp corners) function $\vec r(t)$ is a measure of how fast the direction of the tangent vector is changing at a given point. It is equal to

$\kappa(t)=\frac{\left\|\vec{T}'(t)\right\|}{\left\|\vec{r}'(t)\right\|}= \frac{\bigl\|\vec{r}'(t)\times\vec{r}''(t)\bigr\|}{\left\|\vec{r}'(t)\right\|^3}$

where $T'(t)$ is the derivative of the unit tangent vector function, equal to

$\vec{T}(t)=\frac{\vec{r}'(t)}{\left\|\vec{r}'(t)\right\|}$

It is also equal to the reciprocal of the radius of curvature (represented with $\rho$), or the radius of the circle which has the same slope and concavity as the function. Curvature is often signed, especially in higher dimensions (see below), with a positive curvature representing the unit tangent vector rotating in the counterclockwise direction.

Other formulae

In the case of a planar parametric function, curvature is equal to

$\kappa(t)=\frac{x'y''-x''y'}{\sqrt{(x'^2+y'^2)^3}}$

If the curve can be rewritten explicitly as $y=f(x)$ , the curvature is equal to

$\kappa(x)=\frac{f''(x)}{\sqrt{\bigl(1+f'(x)^2\bigr)^3}}$

Curvature of a function in polar coordinates is equal to

$\kappa(\theta)=\frac{r(\theta)^2+2r'(\theta)^2-r(\theta)r''(\theta)}{\sqrt{\bigl(r(\theta)^2+r'(\theta)^2\bigr)^3}}$

Generalizations to higher dimensions

There are two common methods of generalizing curvature. The first method, known as the Gaussian curvature, is square of the geometric mean of the principle curvatures, or curvatures along each of the principle axes (the curvature in the direction of greatest and least curvature on the surface).

$\Kappa=\kappa_1\kappa_2$

If the curvature in any direction along a surface is zero, the Gaussian curvature will be zero as well (for example, a cylinder has zero Gaussian curvature). A negative Gaussian curvature represents a saddle point (such as the side of a torus facing the centre). The surface integral of the Gaussian curvature is known as the total curvature

The mean curvature is the arithmetic mean of the principle curvatures.

$H =\frac{\kappa_1+\kappa_2}{2}$

A surface with zero mean curvature is known as a minimal surface, and will have the minimum surface area possible for a given boundary. For example, solutions to Laplace's equation are minimal surfaces.