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My doubt is simple as that. When I have a smooth, regular curve (that is, its curvature is never zero), can I just assume that it is parameterized by arc-lenght, without any loss of generality? If not, is there any counter example?

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When you say "regular curve", don't you mean that its derivative is never $=0$? –  Olivier Bégassat Sep 28 '12 at 1:55
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I think you're going to have to say for which purposes you want to do this. Generally speaking, the properties of the image of a path are independent of the parametrization, whereas the properties of the path itself can depend on the parametrization. What sort of properties are you interested in? –  joriki Sep 28 '12 at 3:16
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A curve is regular if its derivative is never zero. When I asked it I had in mind the existence of the curvature and torsion functions, and the Frenet frame. When it is regular, can I just assume that it ir parameterized by arc lenght? Because bot curvature, torsion and the Frenet frame are well defines then! –  Marra Sep 28 '12 at 9:42
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up vote 2 down vote accepted

In a word, yes. If a curve is $C^k$ smooth with nonvanishing derivative, then its arclength parametrization is also $C^k$ smooth. Thus, from the theoretical point of view, nothing is lost in the change of parameter. And something is gained: the formulas for the TNB frame become simpler when the arclength parametrization is used.

However, from the practical point of view the arclength parametrization is usually messy: even for the simple cubic $y=x^3$ it involves an elliptic integral. Thus, for explicit computations one usually uses the simplest parametrization available, be it arclength or not.

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