One of my homework problems asks us to show that the curvature and the torsion of a regular parameterised curve with non-zero torsion everywhere are uniquely determined, when we already have the knowledge of its normal vector function. This in term is equivalent to determine a curve from a given normal vector function $n(s)$, by use of the fundamental theorem of the local theory of curves.
Attempt:
From the Frenet formulas we conclude that
$-\langle n',b\rangle n=b'$.
This is a first-order differential equation, which hence must possess a unique solution. Could one conclude from here that the normal function can uniquely determine the binormal vector, hence the torsion, hence the curvature of a curve?
Inform me of any error that occurs, thanks in advance.
\langleand\ranglefor the angle brackets $\langle\rangle$. $<$ and $>$ are for inequalities. (@martini and I made identical edits to your posts fixing this, 3 seconds apart.) – Harald Hanche-Olsen Nov 7 '12 at 12:28