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Frenet-Serret_Apparatus - Frenet-Serret Apparatus Data A...

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Frenet-Serret Apparatus Data A space curve given by the position vector r (t). Objective Find the functions for arc length, s(t), and curvature, κ(t). Find the unit vector functions T (t), N (t) and B (t). Method Differentiate r with respect to t, d r dt , which is the velocity vector. Its length is the speed, so the arc length satisfies ds dt = d r dt and s t = t 0 t d r d 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. T s = d r ds ,  s N s = d T ds from which we get  s = d T ds and N s = 1 d T ds , then B = T × N completing the work. Difficulty The integral t 0 t d r d d can only be expressed in a useful form for a few very special cases. Fix Differentiate 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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