Serret-Frenet equations

Let $I\subset ℝ$ be an interval, and let $\gamma :I\to {ℝ}^3$ bean arclength parameterization of an oriented space curve, assumed tobe 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 followingdifferential relations, called the Serret-Frenet equations, holdbetween 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 thenormal (http://planetmath.org/MovingFrame) $N(s)$ and from the definition of thecurvature (http://planetmath.org/CurvatureAndTorsion), $\kappa (s)$. Taking the derivative ofthe 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 usingmatrix notation. Let $F:I\to \mathrm{SO}(3)$ (see - special orthogonalgroup), the mapping defined by

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

represent the Frenet frame as a $3\times 3$ orthonormalmatrix. Equations (1) (2) (3) can besuccinctly given as

In this formulation, the above relation is also known as the structureequations 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