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I know solving the Frenet-Serret equations for a given curvature and torsion is possible, but only analytically possible under special cases like plane curves. Is there something I can exploit to calculate a unit-speed curve where, for instance, $\kappa(s)=\tau(s)=\sqrt{2}/(1+s^2)$?

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    $\begingroup$ See pages 32-33, 44-45 in Millman and Parker, Elements of Differential Geometry, Lancret's theorem. $\endgroup$
    – Will Jagy
    Mar 27, 2011 at 5:03

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See my answer on StackExchange: https://math.stackexchange.com/questions/28925/explicit-formula-for-space-curves/28931#28931

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