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What is the differential equation of a 3D space curve whose varying curvature and torsion are given as functions of arc length? (up to rotation and translation of Euclidean motions?)

EDIT1:

Motivation in asking the question is also about uniqueness of associated curve embeddings like e.g., on the following two $ R^2$ surfaces containing lines with zero curvature and non-zero torsion... which I hoped would be associated with some constant of integration.

1) Straight line rulings of a One sheet hyperboloid ruled surface, skewed to the central axis.

2) Central axis $ u = 0 $ of helicoid

$$ ( u \cos t, u \sin t , c t) $$

I am teased by the generalized Frenet-Serret formulation that includes geodesic torsion, geodesic curvature and normal curvature.

One reference to generalized Frenet-Serret is among last pages of DJStruik's Differential Geometry Lectures Book 2nd Edition.

Even at risk of attracting some adverse reaction for vagueness, I ask : Why is there no concept of a scalar torsion component of normal torsion? This remark may however be ignored for this discussion.

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All spaces curves parametrized by arc length satisfy a system of structural differential equations, called Frenet-Serret formulas.

These equations show that the curvature and torsion determine the curve up to a rigid motion.

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  • $\begingroup$ Not quite: you can get nonuniqueness at points where the curvature and torsion are $0$. $\endgroup$ – Robert Israel Feb 16 '15 at 22:42
  • $\begingroup$ For example, with $\kappa = 3 s^2$ and $\tau = 0$, you could have $T = [\cos(s^3), \sin(s^3), 0]$ or $T = [\cos(s^3), \text{sgn}(s) \sin(s^3), 0]$, producing non-congruent curves. $\endgroup$ – Robert Israel Feb 16 '15 at 22:56

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