Finding a space curve given some conditions on curvature and torsion

Question

I have a space curve where the curvature is $\kappa$ and torsion $\tau = \kappa'$. An example of this would be a curve with curvature $\kappa = 1 - \cos s$ , $\tau = \sin s$. Is it possible to find the possible parametric equation of the curve? Either generally or at least for the particular example mentioned?

Answer 1

Solving the Frenet-Serret equations with given functions $\kappa$ and $\tau$ will always give you a solution, namely a curve with the said curvature and torsion functions. The condition for this to be well-defined is that the curvature should be positive. In the case you mentioned one will therefore obtain a solution so long as $s$ is less than $\pi/2$. A standard reference is the book by Millman and Parker:

Millman, Richard S.; Parker, George D. Elements of differential geometry. Prentice-Hall Inc., Englewood Cliffs, N. J., 1977.