Proof of Epstein Gage Lemma: Aditya Tatu

[adityatatu]

Hi all,
The Epstein Gage Lemma states that a curve evolving under some given velocity vector V (V = VnN + VtT), where Vn is the normal velocity component and Vt is the tangential velocity component, N is the normal to the curve and T is the tangent to the curve, will give the same curves if evolved under only Vn, i.e. the normal velocity component. The Tangential component Vt affects only the parameterisation and not the shape of the curve.

Can somebody give me a simple enough proof of the above theorem?
thanks in advance..
Aditya Tatu
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[Reb]

The proof is, for the most part, a long and uninspired calculation. The basic idea is to use the fundamental theorem of curve geometry, which states that the curvature and the torsion of a space curve can characterize it - up to isometries. Extensively used are also the Serret-Frenet formulas.