| Version current |
Version 11 |
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Let $I\subset \reals$ be an interval and let $\gamma:I\to\reals^3$ be a
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Let $I\subset R$ be an interval and let $\gamma:I\to\reals^3$ be a
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| parameterized space curve, assumed to be |
parameterized space curve, assumed to be |
| \PMlinkname{regular}{SpaceCurve} and free of points of inflection. We |
\PMlinkname{regular}{SpaceCurve} and free of points of inflection. We |
| interpret $\gamma(t)$ as the trajectory of a particle moving through |
interpret $\gamma(t)$ as the trajectory of a particle moving through |
| 3-dimensional space. The moving trihedron (also known as the Frenet |
3-dimensional space. The moving trihedron (also known as the Frenet |
| frame, the Frenet trihedron, the rep\`ere mobile, and the moving |
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 |
frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined |
| and named as follows: |
and named as follows: |
| \begin{align*} |
\begin{align*} |
| T(t) &= \displaystyle \frac{\gamma'(t)}{\Vert \gamma'(t) \Vert}\, , |
T(t) &= \displaystyle \frac{\gamma'(t)}{\Vert \gamma'(t) \Vert}\, , |
| && |
&& |
| \text{the unit tangent;}\\ |
\text{the unit tangent;}\\ |
| N(t) &= \displaystyle \frac{T'(t)}{\Vert T'(t) \Vert} \, ,&& |
N(t) &= \displaystyle \frac{T'(t)}{\Vert T'(t) \Vert} \, ,&& |
| \text{the unit normal;}\\ \\ |
\text{the unit normal;}\\ \\ |
| B(t) &= T(t)\times N(t) \, ,&& \text{the unit binormal.}\\ |
B(t) &= T(t)\times N(t) \, ,&& \text{the unit binormal.}\\ |
| \end{align*} |
\end{align*} |
| A straightforward application of the chain rule shows that these |
A straightforward application of the chain rule shows that these |
| definitions are covariant with respect to reparameterizations. Hence, |
definitions are covariant with respect to reparameterizations. Hence, |
| the above three vectors should be conceived as being attached to the |
the above three vectors should be conceived as being attached to the |
| point $\gamma(t)$ of the oriented space curve, rather than being |
point $\gamma(t)$ of the oriented space curve, rather than being |
| functions of the parameter $t$. |
functions of the parameter $t$. |
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| Corresponding to the above vectors are 3 planes, passing through each |
Corresponding to the above vectors are 3 planes, passing through each |
| point $\gamma(t)$ of the space curve. The \emph{osculating plane} at |
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 |
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 |
\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 |
$B(t)$; the rectifying plane at $\gamma(t)$ is the plane spanned by |
| $T(t)$ and $B(t)$. |
$T(t)$ and $B(t)$. |
