# 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 (http://planetmath.org/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 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 $B(t)$.

 Title Frenet frame Canonical name FrenetFrame Date of creation 2013-03-22 12:15:44 Last modified on 2013-03-22 12:15:44 Owner rmilson (146) Last modified by rmilson (146) Numerical id 16 Author rmilson (146) Entry type Definition Classification msc 53A04 Synonym moving trihedron Synonym moving frame Synonym repère mobile Synonym Frenet trihedron Related topic SpaceCurve Defines osculating plane Defines normal plane Defines rectifying plane Defines unit normal Defines unit tangent Defines binormal