PlanetMath: Serret-Frenet equations in $\mathbb{R}^2$

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Serret-Frenet equations in $\mathbb{R}^2$

(Definition)

Given a plane curve, we may associate to each point on the curve an orthonormal basis consisting of the unit normal tangent vector and the unit normal. In general, different points will have different bases associated to them, so we may ask how the basis depends upon the choice of point. The Serret-Frenet equations answer this question by relating the rte of change of the basis vectors to the curvature of the curve.

Suppose is an open interval and $c\colon I \to \mathbbmss{R}^2$ is a twice continuously differentiable curve such that $\Vert c'\Vert = 1$ . Let us then define the tangent vector and normal vector as

$\displaystyle \mathbf{T}$	$\displaystyle =$	$\displaystyle c',$
$\displaystyle \mathbf{N}$	$\displaystyle =$	$\displaystyle J \cdot \mathbf{T},$

where $J=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$ is the rotational matrix that rotates the plane

degrees counterclockwise.

Curvature

Differentiating $\langle c',c'\rangle=1$ yields $\langle \mathbf{T}',\mathbf{T}\rangle=0$ , so $\mathbf{T}'$ is in the orthogonal complement of $\mathbf{T}$ , which is

-dimensional. Since $J\cdot \mathbf{T}$ is also in the orthogonal complement, it follows that there exists a function $\kappa\colon I \to \mathbbmss{R}$ such that

$\displaystyle \mathbf{T}'=\kappa J \cdot \mathbf{T}.$

Furthermore, $\kappa$ is uniquely determined by this equation. We define this unique $\kappa$ to be the curvature of

. Explicitly,

$\displaystyle \kappa = \langle \mathbf{T}',J \cdot \mathbf{T}\rangle.$

Serret-Frenet equations

By the definition of curvature

$\displaystyle \mathbf{T}'$

$\displaystyle =$

$\displaystyle \kappa J\cdot \mathbf{T} = \kappa \mathbf{N}$

and so

$\displaystyle \mathbf{N}'$

$\displaystyle =$

$\displaystyle J\cdot \mathbf{T}' = \kappa J \mathbf{N} = -\kappa \mathbf{T}$

since $J^2=-\operatorname{I}$ . These are the Serret-Frenet equations

$\displaystyle \begin{pmatrix}\mathbf{T}\\ \mathbf{N}\end{pmatrix}' = \begin{pma... ...\ -\kappa & 0 \end{pmatrix}\begin{pmatrix}\mathbf{T}\\ \mathbf{N}\end{pmatrix}.$

"Serret-Frenet equations in $\mathbb{R}^2$ " is owned by rspuzio. [ full author list (3) | owner history (2) ]

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See Also: Serret-Frenet equations

Attachments:: curvature of a circle (Example) by cvalente

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Cross-references: equation, function, orthogonal complement, degrees, plane, rotates, rotational matrix, normal, tangent, continuously differentiable, open interval, curvature, vectors, Serret-Frenet equations, basis, bases, tangent vector, unit normal, orthonormal basis, curve, point, associate, plane curve
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This is version 5 of Serret-Frenet equations in $\mathbb{R}^2$ , born on 2005-05-18, modified 2007-12-15.
Object id is 7072, canonical name is SerretFrenetEquationsInMathbbR2.
Accessed 1289 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

None.[ View all 2 ]

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