Curvature

Gives how much a smooth curve (${\vec{r}}^{\prime }(t)\ne 0$ and is continuous$) bends

$\kappa =\Vert \frac{\mathrm{d}\vec{T}}{\mathrm{d}s}\Vert =\frac{\Vert {\vec{T}}^{\prime }(t)\Vert }{\Vert {\vec{r}}^{\prime }(t)\Vert }=\frac{\Vert {\vec{r}}^{\prime }(t)\times {\vec{r}}^{\prime \prime }(t)\Vert }{{\Vert {\vec{r}}^{\prime }(t)\Vert }^3}$

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