How to proof curvature and torsion are independent

Question

As we know that curvature describes the change of curve in tangent and normal plane, while torsion describes the change the curve in binomial and normal plane. Assume we have a trajectory with length of $T$, then we can compute its curvature and torsion at time $t$ as $\kappa(t)$ and $\tau(t)$, how can we prove that $\kappa(t)$ and $\tau(t)$ are independent? Or since they are scale, it is not necessary to prove?

Thank you

Ben

Answer 1

It's easy to prove that curvature and torsion are independent by looking at the curves $$ \gamma_{a,b} = C(a \cos t, a \sin t, bt) $$ where $C = \frac{1}{\sqrt{a^2 + b^2}}$. The curvature and torsion for this curve are both constant, and the (constant) torsion depends only on $b$, and hence can be varied independently of the curvature.

If you mean the curvature and torsion at a particular point $t$ are "independent," I think I need to know more clearly what your notion of "independence" is.

Answer 2

The fundamental theorem of curve theory says that if $\kappa(s) >0$ and $\tau(s)$ are given differentiable functions then there is a regular curve with $\kappa(s)$ and $\tau(s)$ as curvature and torsion.