# Serret-Frenet equations

Let $I\subset\mathbb{R}$ be an interval, and let $\gamma:I\to\mathbb{R}^{3}$ be an arclength parameterization of an oriented space curve, assumed to be regular (http://planetmath.org/Curve), 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 (http://planetmath.org/CurvatureOfACurve) and torsion functions (http://planetmath.org/Torsion). The following differential relations, called the Serret-Frenet equations, hold between these three vectors.

 $\displaystyle T^{\prime}(s)$ $\displaystyle=$ $\displaystyle\kappa(s)N(s);$ (1) $\displaystyle N^{\prime}(s)$ $\displaystyle=$ $\displaystyle-\kappa(s)T(s)+\tau(s)B(s);$ (2) $\displaystyle B^{\prime}(s)$ $\displaystyle=$ $\displaystyle-\tau(s)N(s).$ (3)

Equation (1) follows directly from the definition of the normal (http://planetmath.org/MovingFrame) $N(s)$ and from the definition of the curvature (http://planetmath.org/CurvatureAndTorsion), $\kappa(s)$. Taking the derivative of the relation

 $N(s)\cdot T(s)=0,$

gives

 $N^{\prime}(s)\cdot T(s)=-T^{\prime}(s)\cdot N(s)=-\kappa(s).$

Taking the derivative of the relation

 $N(s)\cdot N(s)=1,$

gives

 $N^{\prime}(s)\cdot N(s)=0.$

By the definition of torsion (http://planetmath.org/CurvatureAndTorsion), we have

 $N^{\prime}(s)\cdot B(s)=\tau(s).$

This proves equation (2). Finally, taking derivatives of the relations

 $\displaystyle T(s)\cdot B(s)=0,$ $\displaystyle N(s)\cdot B(s)=0,$ $\displaystyle B(s)\cdot B(s)=1,$

and making use of (1) and (2) gives

 $\displaystyle B^{\prime}(s)\cdot T(s)=-T^{\prime}(s)\cdot B(s)=0,$ $\displaystyle B^{\prime}(s)\cdot N(s)=-N^{\prime}(s)\cdot B(s)=-\tau(s),$ $\displaystyle B^{\prime}(s)\cdot B(s)=0.$

This proves equation (3).

It is also convenient to describe the Serret-Frenet equations by using matrix notation. Let $F:I\to\operatorname{SO}(3)$ (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 (1) (2) (3) can be succinctly given as

 $F(s)^{-1}F^{\prime}(s)=\begin{pmatrix}0&\kappa(s)&0\\ -\kappa(s)&0&\tau(s)\\ 0&-\tau(s)&0\end{pmatrix}$

In this formulation, the above relation is also known as the structure equations of an oriented space curve.

 Title Serret-Frenet equations Canonical name SerretFrenetEquations Date of creation 2013-03-22 12:15:13 Last modified on 2013-03-22 12:15:13 Owner rmilson (146) Last modified by rmilson (146) Numerical id 20 Author rmilson (146) Entry type Theorem Classification msc 53A04 Synonym Frenet equations Synonym Frenet-Serret equations Synonym Frenet-Serret formulas Synonym Serret-Frenet formulas Synonym Frenet formulas Related topic SpaceCurve Related topic Torsion Related topic CurvatureOfACurve