PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (195)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
Frenet frame (Definition)

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 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:

$\displaystyle T(t)$ $\displaystyle = \displaystyle \frac{\gamma'(t)}{\Vert \gamma'(t) \Vert}\, ,$   the unit tangent;    
$\displaystyle N(t)$ $\displaystyle = \displaystyle \frac{T'(t)}{\Vert T'(t) \Vert} \, ,$   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)$.



Anyone with an account can edit this entry. Please help improve it!

"Frenet frame" is owned by rmilson. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: space curve

Other names:  moving trihedron, moving frame, repère mobile, Frenet trihedron
Also defines:  osculating plane, normal plane, rectifying plane, unit normal, unit tangent, binormal
Log in to rate this entry.
(view current ratings)

Cross-references: spanned by, passing through, planes, parameter, functions, point, vectors, reparameterizations, definitions, chain rule, application, orthonormal basis, trajectory, points of inflection, parameterized space curve, interval
There are 19 references to this entry.

This is version 12 of Frenet frame, born on 2002-02-02, modified 2006-12-09.
Object id is 1660, canonical name is TNBFrame.
Accessed 28108 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)