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6
of
'moving trihedron'
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| Title of object: |
moving trihedron |
| Canonical Name: |
TNBFrame |
| Type: |
Definition |
| Created on: |
2002-02-02 17:02:19-05 |
| Modified on: |
2003-01-26 18:37:45.567401-05 |
| Classification: |
msc:53A04 |
| Defines: |
osculating plane, normal plane, rectifying plane, unit normal, unit tangent, binormal |
| Synonyms: |
moving trihedron=moving trihedron
moving trihedron=moving frame
moving trihedron=Frenet frame
moving trihedron=rep\`ere mobile
moving trihedron=Frenet trihedron |
Preamble:
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\natnums}{\mathbb{N}}
\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\lb}{\left[}
\newcommand{\rb}{\right]}
\newcommand{\supth}{^{\text{th}}}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{theorem}[proposition]{Theorem}
\newcommand{\bg}{\boldsymbol{\gamma}}
\newcommand{\dbg}{\bg'}
\newcommand{\ddbg}{\bg''}
\newcommand{\dddbg}{\bg'''}
\newcommand{\der}[1]{#1{}'}
\newcommand{\bT}{\mathbf{T}}
\newcommand{\bN}{\mathbf{N}}
\newcommand{\bB}{\mathbf{B}} |
Content:
Let $\bg:I\to\reals^3$ be a parameterized space curve, assumed to be
regular and free of points of inflection. The moving trihedron,
also known as the Frenet frame\footnote{Other names for this include
the Frenet trihedron, the rep\`ere mobile, and the moving frame.} is
an orthonormal basis of vectors $(\bT(t),\bN(t),\bB(t))$ defined and
named as follows:
\begin{align*}
\bT(t) &= \displaystyle \frac{\dbg(t)}{\Vert \dbg(t) \Vert}\, , &&
\text{the unit tangent;}\\
\bN(t) &= \displaystyle \frac{\der{\bT}(t)}{\Vert \der{\bT}(t) \Vert} \, ,&&
\text{the unit normal;}\\ \\
\bB(t) &= \bT(t)\times \bN(t) \, ,&& \text{the unit binormal.}\\
\end{align*}
A straightforward application of the chain rule shows that these
definitions are invariant with respect to reparameterizations. Hence,
the above three vectors should be conceived as being attached to the
point $\bg(t)$ 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 of the space curve. The \emph{osculating plane} is the
plane spanned by $\bT$ and $\bN$; the \emph{normal plane} is the plane
spanned by $\bN$ and $\bB$; the rectifying plane is the plane spanned
by $\bT$ and $\bB$. |
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