Frenet frame

Let I be an intervalMathworldPlanetmathPlanetmath and let γ:I3 be a parameterized space curve, assumed to be regularPlanetmathPlanetmathPlanetmath ( and free of points of inflection. We interpret γ(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) =γ(t)γ(t), the unit tangent;
N(t) =T(t)T(t), the unit normal;
B(t) =T(t)×N(t), the unit binormal.

A straightforward application of the chain ruleMathworldPlanetmath shows that these definitions are covariant with respect to reparameterizations. Hence, the above three vectors should be conceived as being attached to the point γ(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 γ(t) of the space curve. The osculating plane at the point γ(t) is the plane spanned by T(t) and N(t); the normal planeMathworldPlanetmath at γ(t) is the plane spanned by N(t) and B(t); the rectifying plane at γ(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