Curvature

Curvature is a property of curves — think of it as the roundness of a curve at any given point. Informally also, we can consider it the reciprocal of the radius of the largest circle that can fit into that point. Linking with the unit tangential vector:

$$\kappa (t)=\frac{\Vert {\vec{T}}^{\prime }(t)\Vert }{\Vert {\vec{r}}^{\prime }(t)\Vert }$$

If $\vec{r}(s)$ is in arc length parametrisation, the curvature is:

$$\kappa (t)=\Vert \frac{d\vec{T}}{ds}\Vert$$