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?
Is there any loss in generality when I assume that a regular curve is arc-lenght parameterized?
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differential-geometry
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0When you say "regular curve", don't you mean that its derivative is never $=0$? – 2012-09-28
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1I 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? – 2012-09-28
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1A 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! – 2012-09-28
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0See also http://math.stackexchange.com/questions/683039/how-to-reparametrize-curves-in-terms-of-arc-length-when-arc-length-evaluation-ca/683230#683230 – 2016-07-02
1 Answers
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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.