# 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.

${T}^{\prime}(s)$ | $=$ | $\kappa (s)N(s);$ | (1) | ||

${N}^{\prime}(s)$ | $=$ | $-\kappa (s)T(s)+\tau (s)B(s);$ | (2) | ||

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

$$T(s)\cdot B(s)=0,$$ | ||

$$N(s)\cdot B(s)=0,$$ | ||

$$B(s)\cdot B(s)=1,$$ |

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

$${B}^{\prime}(s)\cdot T(s)=-{T}^{\prime}(s)\cdot B(s)=0,$$ | ||

$${B}^{\prime}(s)\cdot N(s)=-{N}^{\prime}(s)\cdot B(s)=-\tau (s),$$ | ||

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

This proves equation (3).

It is also convenient to describe the Serret-Frenet equations by using
matrix notation. Let $F:I\to \mathrm{SO}(3)$ (see - special orthogonal
group^{}), the mapping defined by

$$F(s)=(T(s),N(s),B(s)),s\in I$$ |

represent the Frenet frame as a $3\times 3$ orthonormal matrix. Equations (1) (2) (3) can be succinctly given as

$$F{(s)}^{-1}{F}^{\prime}(s)=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill \kappa (s)\hfill & \hfill 0\hfill \\ \hfill -\kappa (s)\hfill & \hfill 0\hfill & \hfill \tau (s)\hfill \\ \hfill 0\hfill & \hfill -\tau (s)\hfill & \hfill 0\hfill \end{array}\right)$$ |

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 |