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# Frenet frame

Let $I\subset\mathbb{R}$ be an interval and let $\gamma:I\to\mathbb{R}^{3}$ be a parameterized space curve, assumed to be regular 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 repère mobile, and the moving frame) is an orthonormal basis of 3-vectors $T(t),N(t),B(t),$ defined and named as follows:

$\displaystyle T(t)$ | $\displaystyle=\displaystyle\frac{\gamma^{{\prime}}(t)}{\|\gamma^{{\prime}}(t)% \|}\,,$ | the unit tangent; | |||

$\displaystyle N(t)$ | $\displaystyle=\displaystyle\frac{T^{{\prime}}(t)}{\|T^{{\prime}}(t)\|}\,,$ | the unit normal; | |||

$\displaystyle B(t)$ | $\displaystyle=T(t)\times N(t)\,,$ | the unit binormal. |

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 *osculating plane* at
the point $\gamma(t)$ is the plane spanned by $T(t)$ and $N(t)$; the
*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 $B(t)$.

## Mathematics Subject Classification

53A04*no label found*

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