TITLE: The knowledge of $n=n(s)$ can be used to determine the curvature $k(s)$ and the torsion $\tau (s)$
QUESTION [7 upvotes]: Question: 

Show that the knowledge of the vector function $n=n(s)$ of a curve $\alpha:I\rightarrow \mathbb{R^3}$ with nonzero torsion everywhere, determines the curvature $k(s)$ and the torsion $\tau (s)$ of $\alpha$. 

Notes: $n$ is the normal versor to $\alpha$. 
Attempt: I tried using Frenet-Serret formulas, and then using the vector product between $n$ and $n'$, but it seems like I can't get to any result.

REPLY [3 votes]: The Question as posed here is not solvable (as is indicated by the comments in the answer above). One needs the knowledge of the function $\frac{\kappa}{\tau}$ at one point $t_0$.
Lets give a counterexample: 
Consider the Helix
$$ c(s) := (a \cdot \cos(s) , a \cdot \sin(s) , b \cdot s) \text{ for } s \in \mathbb{R}$$
with $a^2 +b^2 = 1$, $a,b>0$. Then $c(s)$ ist parametrized by arclength.
The normal vector is given by
$$ n(s) = (- \cos(s) , -\sin(s), 0) \text{ for all } s\in \mathbb{R}.  $$
One has in general $\kappa=a, \tau= -b$ and $\frac{\kappa}{\tau}=-a/b$.
The choices $a_1=b_1= 1/\sqrt{2}$ and $a_2= 1/2 ,b_2= \sqrt{3}/2$ give two different curves parametrized by arclength. Both curves have the same normal vector for all times and non-vanishing torsion. And both curves have different curvature and torsion.