Frenet frame

Let $I\subset ℝ$ be an interval and let $\gamma :I\to {ℝ}^3$ 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:

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

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

$B(t)$ $=T(t)\times N(t),$ 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