Frenet frame
Let be an interval and let be aparameterized space curve, assumed to beregular
(http://planetmath.org/SpaceCurve) and free of points of inflection. Weinterpret as the trajectory of a particle moving through3-dimensional space. The moving trihedron (also known as the Frenetframe, the Frenet trihedron, the repère mobile, and the movingframe) is an orthonormal basis of 3-vectors definedand named as follows:
the unit tangent; | ||||
the unit normal; | ||||
the unit binormal. |
A straightforward application of the chain rule shows that thesedefinitions are covariant with respect to reparameterizations. Hence,the above three vectors should be conceived as being attached to thepoint of the oriented space curve, rather than beingfunctions of the parameter .
Corresponding to the above vectors are 3 planes, passing through eachpoint of the space curve. The osculating plane atthe point is the plane spanned by and ; thenormal plane at is the plane spanned by and; the rectifying plane at is the plane spanned by and .
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 |