Let $c(s), s>0$, be a curve of $\mathbb{R}^3$, ($s$ be the arc length) with curvature $k(s)>0,\forall s>0$. We consider a new curve with parametrization $\hat{c}(s)=c(s)-st(s),\ s>0$, which $t(s)$ is the unit tangent vector of $c(s)$. If the curvature $\hat{k}$ of $\hat{c}$ is positive and constant, what's the kind of curve ${c(s)}$? For example, $c$ is a cylindrical helix or a line (which is obviously not) or a circle or a plane curve?. Firstly, $s$ is not the arc length of $\hat{c}(s)$. So, $\hat{k}(s)=const=\dfrac{\|\hat{c}'\times \hat{c}''\|}{\|\hat{c}'\|^3}$. By evaluating $\hat{c}'$ and $\hat{c}''$ we have that $$const=\dfrac{\sqrt{k^2+\tau ^2}}{sk},\ s>0 $$ Now, how I suppose to find the torsion $\tau$ of $c$? Any help please, thanks!!
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