Frenet-Serret ApparatusDataA space curve given by the position vector r(t).ObjectiveFind the functions for arc length, s(t), and curvature, κ(t). Find the unit vector functions T(t), N(t) and B(t).MethodDifferentiate rwith respect to t, drdt, which is the velocity vector. Its length is the speed, so the arc length satisfies dsdt=∣drdt∣and st=∫t0t∣drd∣d, using τ as the variable of integration. This is a strictly increasing function of t and so there is an inverse function t=t(s). Reparameterize by arc length and we have the result from the definitions.Ts=drds, sNs=dTdsfrom which we get s=∣dTds∣and Ns=1dTds, then B=T×Ncompleting the work.DifficultyThe integral ∫t0t∣drd∣dcan only be expressed in a useful form for a few very special cases.FixDifferentiate with respect to t rather than s and use the chain rule. Doing this generates a number of formulas which we solve opportunistically for the required objects.
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