Let $I\subset\reals$ be an interval, and let $\gamma:I\to\reals^3$ be an arclength parameterization of an oriented space curve, assumed to be regular, and free of points of inflection. Let $T(s)$ $N(s)$ $B(s)$ denote the corresponding moving trihedron, and $\kappa(s), \tau(s)$ the corresponding curvature and torsion functions. The following differential relations, called the Serret-Frenet equations, hold between these three vectors. \begin{eqnarray} \label{eq:dT} T'(s) &=& \kappa(s) N(s);\\ \label{eq:dN} N'(s) &=& -\kappa(s) T(s) + \tau(s)B(s); \\ \label{eq:dB} B'(s) &=& -\tau(s) N(s). \end{eqnarray} Equation () follows directly from the definition of the normal $N(s)$ and from the definition of the curvature, $\kappa(s)$ Taking the derivative of the relation $$N(s)\cdot T(s) = 0,$$ gives $$N'(s)\cdot T(s) = - T'(s) \cdot N(s) = -\kappa(s).$$ Taking the derivative of the relation $$N(s)\cdot N(s) = 1,$$ gives $$N'(s) \cdot N(s) = 0.$$ By the definition of torsion, we have $$N'(s)\cdot B(s) = \tau(s).$$ This proves equation (). Finally, taking derivatives of the relations

It is also convenient to describe the Serret-Frenet equations by using matrix notation. Let $F:I \to \SO(3)$ (see - special orthogonal group), the mapping defined by $$F(s) = (T(s),N(s),B(s)),\quad s\in I$$ represent the Frenet frame as a $3\times 3$ orthonormal matrix. Equations () () () can be succinctly given as $$F(s)^{-1} F'(s) = \begin{pmatrix} 0 & \kappa(s) & 0 \\ -\kappa(s) & 0 & \tau(s) \\ 0 & -\tau(s) & 0 \end{pmatrix} $$ In this formulation, the above relation is also known as the structure equations of an oriented space curve.