Frenet-Serret equations

The Frenet-Serret equations are a set of three equations for the derivatives of the Basis vectors of the Moving frame in three-dimensional Cartesian coordinates. Given the tangent, normal and binormal unit vectors T\mathbf{T}, N\mathbf{N} and B\mathbf{B}, we have

dTds=κN,dNds=−κT+τB,dBds=−τN\frac{d\mathbf{T}}{ds}=\kappa\mathbf{N}, \quad \frac{d\mathbf{N}}{ds}=-\kappa\mathbf{T}+\tau\mathbf{B}, \quad \frac{d\mathbf{B}}{ds}=-\tau\mathbf{N}

with κ\kappa as the curvature and τ\tau the torsion of the trajectory curve. They can be put into antisymmetric matrix form like

(T˙N˙B˙)=(0κ0−κ0τ0−τ0)(TNB)\begin{pmatrix}\dot{\mathbf{T}} \\ \dot{\mathbf{N}} \\ \dot{\mathbf{B}}\end{pmatrix}=\begin{pmatrix}0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0\end{pmatrix}\begin{pmatrix}\mathbf{T} \\ \mathbf{N} \\ \mathbf{B}\end{pmatrix}