Frenet frame
Let be an interval and let be a
parameterized space curve, assumed to be
regular
(http://planetmath.org/SpaceCurve) and free of points of inflection. We
interpret 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 defined
and named as follows:
the unit tangent; | ||||
the unit normal; | ||||
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 of the oriented space curve, rather than being
functions of the parameter .
Corresponding to the above vectors are 3 planes, passing through each
point of the space curve. The osculating plane at
the point is the plane spanned by and ; the
normal 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 |