Torsion of Curves

Definition

Let $\mathbf{T}(s)$ be a Unit Tangent Vector of a smooth curve, $\mathbf{N}$ be the Principal Unit Normal Vector, and $\mathbf{B}$ be the Binormal Vector $\tau =-\mathbf{N}\cdot \frac{d\mathbf{B}}{ds}\iff -\tau \mathbf{N}=\frac{d\mathbf{B}}{ds}=\frac{\det [{\mathbf{r}}^{\prime },{\mathbf{r}}^{\prime \prime },{\mathbf{r}}^{\prime \prime \prime }]}{|{\mathbf{r}}^{\prime }\times {\mathbf{r}}^{\prime \prime }{|}^2}$