Generic properties of helices and Bertrand curves (Q1865874)

The aim of this paper is the study of generic properties of cylindrical helices and Bertrand curves as applications of singularity theory for plane curves and spherical curves. For a plane curve \(\vec \gamma(t)\) the authors consider the curve \[ \vec {\widetilde\gamma} =\vec \gamma (t)+ (\cot\theta \int^1_{t_0} \bigl\|\vec \gamma(\tau) \bigr\|d\tau)\vec \alpha+ \vec c \] where \(\theta\) is a constant and \(\vec\alpha\), \(\vec c\) constant vectors with \(\dot{\vec\gamma} (t).\vec\alpha =0\) and \(\|\vec\alpha \|=1\). Theorem: ``\(\vec\gamma\) is a cylincrical helix. Moreover all cylindrical helices can be constructed by the above method''. Let \(\vec\gamma: I \to S^2\) be a unit speed spherical curve, \[ \vec{\widetilde \gamma}=\alpha \int^\sigma_{\sigma_0} \vec\gamma (\tau)d\tau+ \alpha\cot\theta \int^\sigma_{ \sigma_0} \vec s(\tau) d\tau+\vec c \] where \(\alpha,0\) are constants \(\vec c\) is a constant vector and \(\vec s=\vec \gamma\times\vec t\), \(\vec t=\dot {\vec \gamma}\). Theorem: ``\(\vec{\widetilde\gamma}\) is a Bertrand curve and all Bertrand curves can be constructed by the above method''.