Curvature

#Math
Gives how much a smooth curve ($\displaystyle \vec{r}'(t)\neq 0$ and is continuous$) bends

$\displaystyle \kappa=\left\lVert \frac{\mathrm{d}\vec{T} }{ \mathrm{d}s}\right\rVert=\frac{\left\lVert \vec{T}'(t)\right\rVert}{\left\lVert \vec{r}'(t)\right\rVert}=\frac{\left\lVert \vec{r}'(t)\times \vec{r}''(t)\right\rVert}{\left\lVert \vec{r}'(t)\right\rVert^{3}}$

• $\displaystyle \vec{T}$ is the unit tangent of the arc
• $\displaystyle s$ is the arc length
• $\displaystyle \vec{r}(t)$ gives the position of the curve as a function of time traveled