This is a problem carried over from theoretical mechanics. The moving axode of a rigid body traversing a given curve $\gamma$ is essentially a "conoid" given as : $${x}(s,t) = \dfrac{\kappa}{\kappa^2 +\tau^2} N_0(s) + t (\tau T_0 +\kappa B_0)$$ where $\kappa, \tau, T,N,B$ are all quantities associated with the Frenet frame of the curve $\gamma$.
If the $x,y,z$ are coordinates along $T_0,N_0,B_0$ respectively,I get \begin{align} x &= t\tau \\ y &= \dfrac{\kappa}{\kappa^2 +\tau^2} \\ z & = t\kappa, \end{align}
where $T_0, N_0, B_0$ are the directions of the vectors at an initial point when the moving frame and the coordinate frame coincide. I am interested in the particular case when $y = \dfrac{\kappa}{\kappa^2 +\tau^2}$ is a constant. I think the surface would then be a cone. Am I right? or is my intuition wrong?
This is a link to the paper I took this problem from: Selig