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'Frenet frame'
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| Title of object: |
Frenet frame |
| Canonical Name: |
TNBFrame |
| Type: |
Definition |
| Created on: |
2002-02-02 17:02:19 |
| Modified on: |
2006-03-24 08:54:32 |
| Classification: |
msc:53A04 |
| Defines: |
osculating plane, normal plane, rectifying plane, unit normal, unit tangent, binormal |
| Synonyms: |
Frenet frame=moving trihedron
Frenet frame=moving frame
Frenet frame=rep\`ere mobile
Frenet frame=Frenet trihedron |
Revision comment (for changes between this and next version):
| Changes for correction #11039 ('"R"'). |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}} |
Content:
Let $I\subset R$ be an interval and let $\gamma:I\to\reals^3$ be a
parameterized space curve, assumed to be
\PMlinkname{regular}{SpaceCurve} and free of points of inflection. We
interpret $\gamma(t)$ as the trajectory of a particle moving through
3-dimensional space. The moving trihedron (also known as the Frenet
frame, the Frenet trihedron, the rep\`ere mobile, and the moving
frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined
and named as follows:
\begin{align*}
T(t) &= \displaystyle \frac{\gamma'(t)}{\Vert \gamma'(t) \Vert}\, ,
&&
\text{the unit tangent;}\\
N(t) &= \displaystyle \frac{T'(t)}{\Vert T'(t) \Vert} \, ,&&
\text{the unit normal;}\\ \\
B(t) &= T(t)\times N(t) \, ,&& \text{the unit binormal.}\\
\end{align*}
A straightforward application of the chain rule shows that these
definitions are covariant with respect to reparameterizations. Hence,
the above three vectors should be conceived as being attached to the
point $\gamma(t)$ of the oriented space curve, rather than being
functions of the parameter $t$.
Corresponding to the above vectors are 3 planes, passing through each
point $\gamma(t)$ of the space curve. The \emph{osculating plane} at
the point $\gamma(t)$ is the plane spanned by $T(t)$ and $N(t)$; the
\emph{normal plane} at $\gamma(t)$ is the plane spanned by $N(t)$ and
$B(t)$; the rectifying plane at $\gamma(t)$ is the plane spanned by
$T(t)$ and $B(t)$. |
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