Differential Geometry of Curves - Frenet Frame

See also: Frenet–Serret formulas

A Frenet frame is a moving reference frame of n orthonormal vectors ei(t) which are used to describe a curve locally at each point γ(t). It is the main tool in the differential geometric treatment of curves as it is far easier and more natural to describe local properties (e.g. curvature, torsion) in terms of a local reference system than using a global one like the Euclidean coordinates.

Given a Cn+1-curve γ in Rn which is regular of order n the Frenet frame for the curve is the set of orthonormal vectors

called Frenet vectors. They are constructed from the derivatives of γ(t) using the Gram–Schmidt orthogonalization algorithm with

\mathbf{e}_{j}(t) = \frac{\overline{\mathbf{e}_{j}}(t)}{\|\overline{\mathbf{e}_{j}}(t) \|} \mbox{, } \overline{\mathbf{e}_{j}}(t) = \mathbf{\gamma}^{(j)}(t) - \sum _{i=1}^{j-1} \langle \mathbf{\gamma}^{(j)}(t), \mathbf{e}_i(t) \rangle \, \mathbf{e}_i(t)

The real-valued functions χi(t) are called generalized curvatures and are defined as

The Frenet frame and the generalized curvatures are invariant under reparametrization and are therefore differential geometric properties of the curve.

Read more about this topic:  Differential Geometry Of Curves

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