PlanetMath: Serret-Frenet equations

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Serret-Frenet equations

(Theorem)

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, and free of points of inflection. Let , , 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.

$\displaystyle T'(s)$	$\displaystyle =$	$\displaystyle \kappa(s) N(s);$	(1)
$\displaystyle N'(s)$	$\displaystyle =$	$\displaystyle -\kappa(s) T(s) + \tau(s)B(s);$	(2)
$\displaystyle B'(s)$	$\displaystyle =$	$\displaystyle -\tau(s) N(s).$	(3)

Equation (1) follows directly from the definition of the normal and from the definition of the curvature, $\kappa(s)$ . Taking the derivative of the relation

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

gives

$\displaystyle N'(s)\cdot T(s) = - T'(s) \cdot N(s) = -\kappa(s).$

Taking the derivative of the relation

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

gives

$\displaystyle N'(s) \cdot N(s) = 0.$

By the definition of torsion, we have

$\displaystyle N'(s)\cdot B(s) = \tau(s).$

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

$\displaystyle B'(s) \cdot T(s) = -T'(s)\cdot B(s) = 0,$
$\displaystyle B'(s) \cdot N(s) = -N'(s)\cdot B(s) = -\tau(s),$
$\displaystyle B'(s)\cdot B(s) = 0.$

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

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

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

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

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

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"Serret-Frenet equations" is owned by rmilson. [ full author list (3) | owner history (1) ]

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Other names:

Frenet equations, Frenet-Serret equations, Frenet-Serret formulas, Serret-Frenet formulas, Frenet formulas

Attachments:: expressions for curvature and torsion (Theorem) by Simone

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Cross-references: structure, orthonormal, represent, mapping, special orthogonal group, matrix, derivative, equation, vectors, relations, moving trihedron, points of inflection, oriented space curve, arclength parameterization, interval
There are 4 references to this entry.

This is version 16 of Serret-Frenet equations, born on 2002-02-02, modified 2007-06-02.
Object id is 1636, canonical name is SerretFrenetFormulas.
Accessed 16867 times total.

Classification:

AMS MSC:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

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