This seems to be a pretty silly or trivial doubt and I am probably having a brainfade, but I would like some clarity anyhow.
I basically have a space curve with curvature and torsion functions $\kappa, \tau$ satisfying a relation $$f(\kappa, \tau) = \lambda$$, $\lambda$ being a positive real constant. Now if on setting $\tilde{\kappa} = \mu \kappa$ and $\tilde{\tau} = \mu \tau$, where $\mu$ is a nonzero positive real, I had the above relation as $$f(\tilde{\kappa}, \tilde{\tau}) = \dfrac{\lambda}{\mu},$$ then does this mean I have a family of curves satisfying $f$ with varying constants, or have I just basically reparametrised the original curve?? It seems to me to be just a reparametrisaton of the original curve, but the fundamental theorem of space curves seems to suggest completely new curves.
HINT:
I believe the answer by achille hui implicitly contains a particular case given here $\kappa;\tau;$ for lines on a 1-sheet rotation hyperboloid. Please go through it and also his comments.