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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

Ben
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2 Answers2

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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.

John Hughes
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  • Hi John, thank you for your reply. My independent is for $\kappa$ and $\tau$, not for a particular time $t$. Your first half answered my question. Just wondering if we can have a more general proof, instead of an equation based one. – Ben Jul 15 '14 at 23:49
  • Rene's answer points to the more general proof, but if you want to know things are independent, an example showing it seems like the simplest possible demonstration. (At least to me!) – John Hughes Jul 15 '14 at 23:56
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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.

Rene Schipperus
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