Curvature

The curvature of a Space Curve

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

where $\vec{T}(s)$ is the Unit Tangent Vector and $s$ is Arc Length

Osculating Circle

The circle whose curvature is the same as a Space Curve at a particular point, and whose radius $R$ is reciprocal to the curvature at that point. That is to say:

$$\kappa =\frac{1}{R}$$