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?)


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.


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.

  • $\begingroup$ Not quite: you can get nonuniqueness at points where the curvature and torsion are $0$. $\endgroup$ 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. $\endgroup$ Feb 16 '15 at 22:56

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