Serret-Frenet equations

Let $I\subset ℝ$ be an interval, and let $\gamma :I\to {ℝ}^3$ 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

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